Nonlinear Equations: Methods, Models and Applications (Progress in Nonlinear Differential Equations and Their Applications)

dvy = 0

(a,) 6- 1",

-A ----7)

aa`

((P_1)sn2 / JaAI a1 bai,A + a2ba2,A

The continuity of uy with respect to y implies the existence of > 0 such that

1
x2 E B,1(al)

if

ua2

Thus if d(a2i a2) < 77, we have p

J(E 1Pa:.A)

(6)

i=1

((P_1)sn-2+

<1+

n_2

\

al6ai.A

aM

aL

1

al da1,a + a2ba2.A a1 aal.A

- IOMflB1(p) al aai.a + 6a2.a

bn2

2

At this point we state the following claim whose proof is postponed until the end of this section.

Claim: There exists co small enough, and a positive constant i7' such that ^ -' i 6-2-1\

J

A! 2ba2.A + & 2.A

b°1.a2

C eo JM Wa

dvgo

1P-2.a + Pal.\cPa

dvgo.

From another part, by assumption, we know that

Lr

I

O

,.a

P

2

M

a1.a+ 1Pn2.adv9.

A Riemann Mapping Type Theorem in Higher Dimensions, Part I

11

Thus we derive from the above claim and (6), that: P

J( aiVa;.A) 5 (1 + O(

1

2

))

PSn-2 -

i=1

2 C

J tIM cO a Pa,,Ad+r

P

p.'

(7)

clearly implies (i), which is therefore proven if d(a1i a2) < n. Now we rule out the case where d(al,a2) > 77 as follows:

If d(al,a2) > ii then Va2,A + Va2

JeM

= 0(\n-2

Wai,A)

Therefore 1

Oa

Paj.a !5 P2.0( \n_2)

1

- 0( \n_2 )

cpa,,a < el therefore the proof of (i) is Taking A very large we have Ei¢j cp reduced to the case d(al, a2) < j7. The proof of (i) is thereby established. The proofs of (ii) and (iii) will be completed together since they rest on the same expansion of the functional J. Using Lemma 6.2 we derive the following inequality P

J(E ai Va:.A)

(8)

i-1 P 1

EPi=2 ai

ai f

#j aiaj LAI as P 7 n-1 + ["Lij4 n-2 ai

JOM Va_A

r aj JtA4'Pa;.a Va) a

Then (8) implies using Lemma 2.2 and Lemma 2.4 P

J(E ai W0;.a) i-1

Lapa; .a + E

[ Pi=1 a2Sn-2+ +(1+B) at P

i=2 ai

z

S

n-2+[EP P

L i#j ai s7 f am 'Pa ;

7 n-1 ai=a a7 faA1 n-2

(9)

P{=

1 aiz Ac A_

Q Va; ,a - ["Li=1 P ai

where C and C' are positive constants independent of (a 1, ... , a,), A and p. Let us assume that

i#j

8A1

-A n---

M.O. Ahmedou

12

If e1 is chosen small enough, then for .A large enough, we have P

J(E ai 0a..A)

(10)

i-1

EP

a?Sn-2+ +(1+9)E#jaia f 'papa + EP a2 c 1° s - S+n_y + P 7 n-1 a a7 ( P 'Pa - Ei Cli 2 a' #j n-2 i -" s=1 am _

1

Making an expansion we have P

J(F' ai c2...\)

(11)

i-1

(EP a, Sn-2 + +(1 + B) E''j aiaj JaM'Po P 'Paj.a + EP a? ° 3) 1

n-1

1

aj J8M a;.a Paj,a -E' i=1 ai

'Y n-1 P s n-2 + Eig6j n-2 ai

P Ei=2 ai

Finally we obtain:

((''

P a?n-1 S 1+ n- 2 n-I

P

ai Pa,.a) i-1

C`P 1

-

i

C

aiaj E a?,Sn-2 ' i#j

18A1

Wai.aiPaj.J1

Cn

aaj

n - 1

a

n - 2 i#j Ea?Sn-2 i '

8h1

Wa;.a Paj.a +

(12)

\n-2

Let us observe that (iii) follows from (12), so it remains only to prove (ii).

Proof of (ii). Let us now assume that there exists 00, 0 < 00 < 1 and 1 > 00 for any (i, j), then a`ai P=

Lr-l O r

<

ai

r=1 ary"

and

,,,p2

aj

f"'0

P

Now we choose 0 < 60 < 1 and B > 0 such that (1 + 8) Let

b = Sn_2 Cry(n

n-2

-

B 0

- 00-P >

- -y90

< 0.

n- n-11+0 1

n-2 6o )

We then derive P

J(E ai 0 ,.a)

(13)

i-1

(Fp a?) EP 1 ayE-411

P i#j

C

aa

b

'

'a2Sn-2

aM

a.,A(Paj,a + ),n-2

A Riemann Mapping Type Theorem in Higher Dimensions, Part I

13

Then using (iii) of Lemma 2.4, we derive (ii) from (13). The proof of Lemma 4.1 is thereby complete. Proof of the claim. Let EO > 0 be given and let such that ba,,a(x) > Co (26a,,a + Saz.A) (x)}.

E° = {x E 13M We have:

J

bax.a

> E°

M 2ba,,a + baz,a

Jo ba,nA bax.

\ dv,0

(14) t

>

Co

- ( 1-E02E°

bax.,\ d t-,0

\J8M

ba

1.9m

ba1

.

bat a

Let us observe that bax.a =

eM

'AI

bnz,A bat.a

and a

a

ba a bn

L

2

bat.Abaz.A + bat,abax.a

--2- )

L'-

-

Then (14) becomes bax.a

J m 26a,,,\ + 6-2,1\

>

EO

2

EO

(1 - ( 1 - EO

)

J

M

ba, abax.a + bat,abaz.a

Thus for co small enough, we have: bax,a nz > CEO bQ, baz.a + aM 19At 2ba,.a + baz.a

>- C'EO J

Mt1B,(at )

> C"60 (".11EO

(15)

z

b,,. ibnx.A + bnt.ab° i

at.a'Paz.a + J M'Pa,

ax.a +

2

(16)

0

Hence our claim is proved. Lemma 4.1 implies the following proposition:

Proposition 4.1. There exists an integer p0 and a positive real number A0 > 0 such

that for any (a1 i ... , ap) satisfying ai > 0, EP=j ai = 1, for any (a1, ... , a,) E 8M, for any A > A0, we have

i=1

M.O. Ahmedou

14

The proof of Proposition 4.1 follows from (i), (ii) and (iii) of Lemma 4.1. We first choose 0 < e 1 < 11 and AO so that: (FP aZ )Proof.

C1

\

+ O( -n 2) + C, I < p

(17)

Ei=1 a= Considering (a 1, . . . , ap ) , ( a1 ,... , ap) and A > A0, we study various cases:

1st case: There exists (io,jo) such that

I

Qla

< Bo, then taking, A -> Apo =

sup(A(po, E), Ao) where A(po, E1) is given by (i) of Lemma 4.1, we derive: P

P

p"-2 S if

J(E ai oa;.A)

dog > E

Wa,.a

i#j JM If on the contrary F-#3 faM Oa;.A cpa _,,\ da9 <_ ei we apply (iii). Since we have chosen E1 such that: i=1

Ep

1

2

(1 + O(

1 2) +

CJ<pa2 E11

(18)

E, lai we derive that J(Ep 1 ai

p 1-2 S and the proof of Proposition 4.1 is

established in this case. 2nd case: Let us assume that > eo for any (i, j), then either (i) or (ii) of Lemma 4.1 holds. If (i) holds then Proposition 4.1 holds, so let us assume that (ii) holds and then choose po such that: (po + 1)c2 > 1 then P

J(E ai c a,.a) <- po

S.

i=1

0

The proof of Proposition 4.1 is thereby complete.

5. Proof of Theorem 1.2 For the proof of Theorem 1.2, we introduce the following notations: For any p > 1 and A > 0, let P

P

Bp =Bp(OM)={Ectik,ai>0,Eai=1,aiEOM } i=1

i=1

and Bo = Bo(OM) = 0. Set also fp(A) to denote the map from Bp(OM) to E+ defined by

f'(A)(P E aibai) _ i=1

Lp

1'Pa,,A,

IIEi=1 Ta,.A

Clearly we have Bp_1 C Bp and Wp_1 C Wp. Moreover fp(A) enjoys the following properties:

II

A Riemann Mapping Type Theorem in Higher Dimensions, Part I

15

Proposition 5.1. The function fp(A) has the following properties:

(i) For any integer p > 1, there exists a real number Ap > 0 such that

fp(A) : Bp(8M) -' W, for any A > Ap. (ii)

There exists an integer po > 1, such that for any integer p > po, and for any A > AN, the map of pairs fp(A) : (Bp, Bp_ 1) - (Wp, Wp_ 1) satisfies (fp). (.A) - 0 where

(fp(A)).: H.(BP,BP-1) - H.(Wp,WP-1) and H. is the *th homology group with Z2 coefficients. Proof.

(i) is a direct consequence of the inequalities (i), (ii) and (iii) of Lemma

4.1, indeed: P

P

J(fp(A)(Y' ai 6-i)) = J(> ai oa-a. ) i=1

i=1

(ii) follows from Proposition 4.1.

11

For the sequel we need the following notations: Let Ap_ 1 = { (a], ... , ap),

ai > 0, EP =j ai = 1} and Fp = { (a, , ... , a,) E (8M)P such that 3 i 0 j with ai = aj}. Let o-p be the symmetric group of order p, which acts on Fp, and let TP be a op equivariant tubular neighborhood of Fp, in (8M)P. (The existence of such a neighborhood is derived in the book of G. Bredon [81.) From another part, considering the topological pair (Bp,Bp_1) we observe that (Bp\Bp_1) can be described as ((8M)P)` x, (Op \80p_1) where (8M)P)' = {(al,...,ap) E (OM)P such that ai 54 aj,di j}. We notice that ((OM)")` x,

(Ap \ 8ip_1) is a noncompact manifold of dimension (n - 1)p + p - 1. Let for

0<0<1,,Mp=VpxopOp_1,where Vp=MP\TpandOp_1={(a1i...,ap)E such that Q E [1 - 8,1 +0),Vi,Vj}. Mp is a manifold which can be seen as a subset of Bp, and the topological pair (Bp, M`) retracts by deformation onto Op-1

(Bp, Bp_ 1), we thus have

H. (Bp, Bp-

H. (Bp,.A4P').

Thus by excision we have

H. (Bp, Bp- 1) = H. (m, am P) Since any manifold is orientable modulo its boundary with 7L2 coefficients, we have a nonzero orientation class in H(n-1)p+p-1(Bp, Bp_1) which we denote by wp.

In contrast with Proposition 5.1, we have the following proposition:

Proposition 5.2. Under the assumption that (P) has no solution, we have, for every p E N*

(f,(A)).(w,) $ 0.

M.O. Ahmedou

16

Proof. An abstract topological argument displayed in [7], pp. 260-265, see also [6], which extends virtually to our framework, shows that:

If (f, (A).0- 0 then (ff(A)). # O for every p > 2. Since Js+E, for e > 0 small enough satisfies Js+E C V(1, 6), where 6 -+ 0 if e - 0, one can define using Lemma 3.1 a continuous map s : Js+E -+ OM which associates to u = acpQ a + v E Js+E - a E OM. Here (a, a, A) are the unique solution of the minimization: min{Ilu - a0a,AII, a > 0, A > 0, a E 8M}. So if r : Wl - Js+E denotes the retraction by deformation of WI onto Js+E, the existence of such a retraction by deformation follows from the assumption that

(P) has no solution from one part and from Proposition 3.1 from another part. Let us observe that s o r o f1(A) = idOM hence (f1(A)).(w1) $ 0, where w1 is the orientation class of 8M. Therefore the proof of Proposition 5.2 is reduced to the abstract topological argument of Bahri-Coron [7].

Proof of Theorem 1.2 completed. Proposition 5.2 is in contradiction with Proposition 5.1. Therefore (P) has a solution and Theorem 1.2 is thereby established.

6. Appendix Lemma 6.1. There holds 12-

Proof.

,

f

6a

MfBo(aj)

6a,'.,dvgo

=

o(

6a

6az,advgo)

J 8M

For e > 0 a fixed number, let AE = {x E Bp(a1) n 8M; 6a,,a >

1

£A-s

}.

Then 1

6as,,a ;

6a, advgo

aMnBP(a,) C

JAe

<£

6al.a 6as,adv9o +

s

6a,.2 6a, advgo +

JA

<£ J

6Q

aA. z

Since z + n2n

1-s s

1\1 2

J8MOBp(a,)\A, 1£^JOMflBp(ai)\Ac \£

02Adv90+ yO( AZ+ 1^ ss £^ Z

> n - 2 and since C 8M

then our Lemma follows.

ban,-'A 602,Advgo > An-2

6as,ad'aso

6a2.,\dv9o

A Riemann Mapping Type Theorem in Higher Dimensions, Part 1

17

Lemma 6.2. [5] Let q > 2 be given. There exists ry >I such that for any (a,,. .., a,), ai > 0, we have

(p)('

+ i=1

>aad. i0i

Lemma 6.3. [16] [Maximum Principle] Under the assumption that R9 > 0, h9 > 0 and R. > 0, or hg > 0, let u E C2(M) f1 C' (8M) satisfying

L9u > 0 on M and B9u < 0

on 8M.

Then u<0 on M. Acknowledgements The author is indebted to Prof. Abbas Bahri for teaching him his theory of critical

point at infinity and he is grateful to Prof. Antonio Ambrosetti for his interest in this work and his constant support.

References [1] Ambrosetti, A., Li, Y.Y., Malchiodi, A., Yamabe and scalar curvature problems under boundary conditions, Math. Ann. 322 (2002), 667-699. [2] Aubin T., Equations differentielles non lineaires et Probleme de Yamabe concernant la courbure scalaire, J. Math. Pures et Appl. 55 (1976), 269-296. [3] Aubin T., Some Nonlinear Problems in Differential Geometry, Springer, (1998). [4] Bahri, A., Critical points at infinity in some variational problems, Pitman Research Notes in Mathematics Series no 182, Longman, London (1988). [5] Bahri A., Proof of the Yamabe conjecture, without the positive mass theorem, for locally conformally flat manifolds. In Einstein metric and Yang-Mills connections (Mabuchi T., Mukai S., eds.), Marcel Dekker, New York, (1993). [6] Bahri A., Brezis H., Elliptic Differential Equations involving the Sobolev critical exponent on Manifolds, PNLDE 20, Birkhauser. [7] Bahri A., Coron J.M., On a nonlinear elliptic equation involving critical Sobolev exponent: The effect of the topology of the domain, Comm. P.A.M. 31 (1988) 253294.

[8] Bredon, G., Introduction to compact transformations Groups, Academic Press, New York (1972).

[9] Chang, S.A., Yang, P., A perturbation result in prescribing scalar curvature on on S", Duke Math. J. 64 (1991), 27-69. [10] Chen, Y., On a nonlinear elliptic equation involving critical Sobolev exponent, Nonlinear Anal., T.M.A., 33 (1998), 41-49. [11] P. Cherrier, Problemes de Neumann nonlineaires sur les varietes Riemanniennes, J. Funct. Anal. 57 (1984), 154-207. [12] Z. Djadli, A. Malchiodi, M. Ould Ahmedou, Prescribing scalar and mean curvature on the three-dimensional half-sphere, J. Geom. Anal. 13 (2003), 233-267.

18

M.O. Ahmedou

[13] Escobar J., Sharp constant in a Sobolev trace Inequality, Indiana Univ. Math. J., 37 (1988). 687-698. [14] J. Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. Math. 136 (1992), 1-50. [15] J. ESCOBAR, The Yamabe problem on manifolds with boundary, J. Diff. Geom., 35 (1992), no. 1, 21-84. [16] J. Escobar, Conformal metrics with prescribed mean curvature on the boundary, Cal. Var. and PDE's 4 (1996), 559-592. [17] V. Felli, M. Ould Ahmedou, Compactness result in conformal deformations on manifolds with boundaries, preprint (2001). [18] Hamza, H., Sur les transformations conformes des varietes Riemanniennes a bord, J. Funct. Anal. 92 (1990), 403-447. [191 Z.C.Han, Yan Yan Li, The Yamabe problem on manifolds with boundaries: existence and compactness results, Duke Math. J. 99 (1999), 489-542. [20] Z.C. Han, Yan Yan Li. The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature, Comm. Anal. Geom., to appear. (211 Lee J.M., Parker T.H., The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), 37-91. [2.2] P.L. Li, J. Q. Liu, Nirenberg's problem on the 2-dimensional hemisphere, Int..1. Math. 4 (1993), 927-939.

1231 Li Y.Y., The Nirenberg problem in a domain with boundary, Top. Meth. Nonlin. Anal. 6 (1995), 309-329. [241 Li Y.Y., Zhu, M., Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), 383-417. (25] Ma, L., Conformal metrics with prescribed mean curvature on the boundary of the

unit ball, to appear. (26] Morse M., Van Schaak G., The critical point theory under general boundary conditions, Annals of Math. 35 (1934), 545-571. [27) Schoen R., Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479-495. [28] Schoen R., Yau S.T., Conformally flat manifolds, Kleinian groups, and scalar curvature, Invent. Math. 92 (1988), 47-71. (291 Schoen R., Yau S.T., On the Proof of Positive Mass Conjecture in General Relativity, Comm. Math. Phys. 65 (1979), 45-76. [30] Spivak M., A comprehensive introduction to differential geometry, 2nd edition, Houston, Publish or Perish (1970).

Mohameden Ould Ahmedou Rheinische Friedrich-Wilhelms-Universitat Bonn Mathematisches Institut Beringstrasse 4 D-53115 Bonn, Germany E-mail address: [email protected]

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 19-31 © 2003 Birkhauser Verlag Basel/Switzerland

Computable Information Content and a Simple Application to the Study of DNA Vieri Benci and Giulia Alenconi

1. Introduction We present some new ideas about complexity and entropy of a dynamical system. These ideas have been developed by an interdisciplinary group in Pisa ([5], [7], [16]. [17], [3]) and the theoretical work is supported by application to several complex

systems, as to turbulent regimes or DNA sequences. Here we present the main theoretical concepts and some simple applications to the study of DNA sequences. In the first part, we will give the definition of complexity of a single orbit and

entropy of a dynamical system and discuss their relation with the KolmogorovSinai entropy (KS-entropy) which is the most important invariant of a dynamical system. Finally, we will present an application to the analysis of the entropy of genomes on an evolutive basis. The advantages of adopting this point of view are the following: we give a new computable definition of information and complexity which allows to give a computable characterization of the KS-entropy; these definitions make sense even for a single orbit and can be measured by suitable data compression algorithms; hence they can be used in simulations and in the analysis of experimental data; the asymptotic behavior of these quantities allows to compute not only the Kolmogorov-Sinai entropy but also other quantities which give a measure of the chaotic behavior of a dynamical system even in the case of null KSentropy.

2. Information, complexity and entropy Information, complexity and entropy are words which in the mathematical and physical literature are used with different meanings and sometimes even as synonyms (e.g., the Algorithmic Information Content of a string is called Kolmogorov complexity; the Shannon entropy sometimes is confused with Shannon information, etc.). In our approach the basic notion is the notion of information. Given a finite string s (namely a finite sequence of symbols taken in a given alphabet),

20

V. Benci and G. Menconi

the intuitive meaning of quantity of information I(s) contained in s is the following one:

I(s) is the length of the smallest binary message from which you can reconstruct s. In his pioneering work, Shannon defined the quantity of information as a statistical notion using the tools of probability theory ([21]). Thus in Shannon framework, the quantity of information which is contained in a string depends on its context. For example the string `pane' contains a certain information when it is considered as a string coming from the English language. The same string `pane' contains much less Shannon information when it is considered as a string coming from the Italian language because it is more frequent in the Italian language (in Italian it means "bread" and, of course, it is very frequent). Roughly speaking, the Shannon information of a string is the absolute value of the logarithm of its probability. However there are measures of information which depend intrinsically on the string and not on its probability within a given context. We will adopt this point of view. An example of these measures of information is the Algorithmic Information Content (AIC). In order to define it, it is necessary to define the partial recursive functions. We limit ourselves to give an intuitive idea which is very close to the formal definition. We can consider a partial recursive function as a computer C

which takes a program p (namely a binary string) as an input, performs some computations and gives a string s = C(p), written in the given alphabet, as an output. The AIC of a string s is defined as the shortest binary program p which gives s as its output, namely 4,1ic(s, C) = min{Ip1 : C(p) = s}, where IpI means the length of the string p. A computing machine is called universal (namely it is a computer in the usual sense of the word) if it can simulate any other

machine (for a precise definition see any book on recursion). In particular if U and U' are universal then JI;trr(s, U) - I,41r(s, U') I < const., where the constant depends only on U and U'. This implies that, if U is universal, the complexity of a with respect to U depends only on a up to a fixed constant and then its asymptotic behavior does not depend on the choice of U. The shortest program p which outputs the string s is a sort of optimal encoding of s. The information that is necessary to reconstruct the string is contained in the program. Unfortunately this coding procedure cannot be performed by any algorithm. This is a very deep statement and, in some sense, it is equivalent to the 'hiring halting problem or to the Godel incompleteness theorem. Then the Algorithmic Information Content is not computable by any algorithm. Another measure of the information content of a finite string can also be defined by a lossless data compression algorithm Z which satisfies some suitable properties which we will not specify here. Details are discussed in [7]. We can define the information content of the string s as the length of the compressed string Z(s),

Application to the Study of DNA

21

namely

Iz (s) = IZ(s)I The advantage of using a compression algorithm lies in the fact that, in this way, the information content Iz (s) turns out to be a computable function. For this reason we will call it Computable Information Content (CIC). In any case, given any string s, we assume to have defined the quantity I(s) via AIC or via

CIC. 2.1. Infinite strings If w is an infinite string, in general, its information is infinite; however it is possible to define another notion: the complexity. The complexity K(w) of an infinite string w is the average information I contained in a single digit of w, namely limsupl(w")

K(w) =

T1-00

n

(1)

where w" is the string obtained taking the first n elements of w. If we equip the set of all infinite strings 12 with a probability measure p, the couple (S2,p) can be viewed as an information source, provided that p is invariant under the natural shift map o, which acts on a string w = (w,)iE as follows: o(w) = , where wi = wi_1 Vi E N. The entropy h of (S2.µ) can be defined as the expectation value of the complexity: (2) hµ = K(w) dµ .

J

If I (w) = JAW (w) or I (w) = Iz (w), under suitable mild assumptions on Z and p, hµ turns out to be equal to the Shannon entropy. Notice that, in this approach, the probabilistic aspect does not appear in the definition of information or complexity, but only in the definition of entropy.

3. Dynamical systems and chaos Chaos, unpredictability and instability of the behavior of dynamical systems are strongly related to the notion of information. The Kolmogorov-Sinai entropy can be interpreted as an average measure of information that is necessary to describe a step of the evolution of a dynamical system. The traditional definition of the Kolmogorov-Sinai entropy is given by the methods of probabilistic information theory. It is the translation of the Shannon entropy into the world of dynamical systems. We have seen that the information content of a string can be defined either with probabilistic methods or using the AIC or the CIC. Similarly, also the KSentropy of a dynamical system can be defined in different ways. The probabilistic method is the usual one, the AIC method has been introduced by Brudno [101; the CIC method has been introduced in [161 and [51. So, in principle, it is possible to define the entropy of a single orbit of a dynamical system (which we will call, as sometimes it has already been done in the literature, complexity of the orbit).

V. Benci and G. Menconi

22

There are different ways to do this (see [10], [15], [18], [8], [17]). In this paper, we consider a method which can be implemented in numerical simulations. Now we will describe it briefly. Using the usual procedure of symbolic dynamics, given a partition a of the phase space of the dynamical system (X,,u,T), it is possible to associate a string (x) to the orbit having x as initial condition. If a = (A1,. .. , Al), then 4 , (x) _ (so, s i ,

.... sk,...) if and only if T k x E A,,,,

dk.

If we perform an experiment, the orbit of a dynamical system can be described

only with a given degree of accuracy related to the partition of the phase space X. A more accurate measurement device corresponds to a finer partition of X. The symbolic orbit 46 (x) is a mathematical idealization of the results of these measurements. We can define the complexity K(x, a) of the orbit with initial condition x with respect to the partition a in the following way

K(x,a) =

maxlimsupl(x'a,n)

n-.m

n

where

1(x, a, n) := I(4 (x)"); (3) here 4% (x)" represents the first n digit of the string,, (x). Letting a to vary among all the computable partitions, we set

K(x) = sup K(x, a) . 0

The number K(x) can he considered as the average amount of information necessary to "describe" the orbit in the unit time when you use a sufficiently accurate measurement device.

3.1. Computable partitions Now it is necessary to spend few words on the notion of "computable partition". This notion is based on the idea of computable structure which relates the abstract notion of metric space with computer simulations. The formal definitions are given in [14]. We limit ourselves to describe its motivation. Many models of the real world

use the notion of real numbers or more in general the notion of complete metric spaces. Even if you consider a very simple complete metric space, as, for example, the interval [0, 1] it contains a continuum of elements. This fact implies that most of these elements (numbers) cannot be described by any finite alphabet. Nevertheless, in general, the mathematics of complete metric spaces is simpler than the "discrete mathematics" in making models and the relative theorems. On the other hand the discrete mathematics allows to make computer simulations. A first connection between the two worlds is given by the theory of approximation. But this connection becomes more delicate when we need to simulate more sophisticated objects of continuum mathematics. For example an open cover or a measurable partition of [0, 1] is very hard to be simulated by computer: nevertheless, these

Application to the Study of DNA

23

notions are crucial in the definition of many quantities as, e.g., the KS-entropy of a dynamical system or the Brudno complexity of an orbit. For this reason, we have introduced the notion of "computable structure" which is a new way to relate the world of continuous models with the world of computer simulations. The intuitive idea of computable partition, is the following one: a partition is computable if it can be recognized by a computer. For example, an interval [a, b] might belong to a computable partition if both a and

b are computable real numbers 1. In particular, a computable partition contains only a denumerable number of elements. In the above construction, the complexity of each orbit K(x) is defined independently of the choice of an invariant measure. In the compact case, if µ is an invariant measure on X then fX K(x) dµ equals the KS-entropy. Then, in an ergodic dynamical system, for almost all points x E X, and for suitable choice of a, I(x, a, n) - h,n. Notice that this result holds for a large class of Information

functions I as for example the AIC and the CIC. Thus we have obtained an alternative way to understand the meaning of the KS-entropy. The above construction makes sense also for a non-stationary system. Its average over the space X is a generalization of the KS-entropy to the non-stationary case. Moreover, the asymptotic behavior of I (x, a, n) gives an invariant of the dynamics which is finer than the KS-entropy and is particularly relevant when the KS-entropy is null.

3.2. Initial data sensitivity and entropy It is well known that the Kohnogorov-Sinai entropy is related to the instability of the orbits. The exact relations between the KS-entropy and the instability of the system is given by the Ruelle-Pesin theorem. We will recall this theorem in the one-dimensional case. Suppose that the average rate of separation of nearby starting orbits is exponential, namely Ax(n) ^- Ax(0)2A" for 0x(0) K 1,

where Ox(n) denotes the distance of these two points at time n. The number A is called Lyapunov exponent; if A > 0 the system is unstable and \ can be considered as a measure of its instability (or sensibility with respect to the initial conditions). The Ruelle-Pesin theorem implies that, under some regularity assumptions, A equals the KS-entropy.

There are chaotic dynamical systems whose entropy is null: usually they are called weakly chaotic. Weakly chaotic dynamics arises in the study of selforganizing systems, anomalous diffusion, long range interactions and many others. In such dynamical systems the amount of information necessary to describe n steps of an orbit is less than linear in n, then the KS-entropy is not sensitive enough to distinguish the various kinds of weakly chaotic dynamics. Nevertheless, IA computable number is a real number whose binary (or decimal) expansion can be obtained at any given accuracy by an algorithm

24

V. Bend and G. Menconi

using the ideas we presented here, the relation between initial data sensitivity and information content of the orbits can be generalized to these cases. To give an example of such a generalization, let us consider a dynamical system ([0, 1], T) where the transition map T is constructive2, and the function I (x, a, n) is defined using the AIC in a slightly different way than before (use open coverings instead of partitions, see [17]). If the speed of separation of nearby

starting orbits goes like Ax(n)

0x(0) f (x, n), then for almost all the points

x E [0, 11 we have

I (x, a, n)

log(f (x, n)).

In particular, if we have power law sensitivity ( Ax(n) ^_information content of the orbit is

I(x,a,n) - plog(n) .

.x(0)nP), the (4)

If we have a stretched exponential sensitivity (Ox(n) ^_- Ax(0)2a"°, p < 1) the information content of the orbits will increase with the power law: I(x, a, n) - nP. An example of stretched exponential is provided by the Manneville map. The Manneville map is a discrete time dynamical system which was introduced by [25] as an extremely simplified model of intermittent turbulence in fluid dynamics. The Manneville map is defined on the unit interval to itself by T (x) = x + x` (mod. 1).

When z > 2 the Manneville map is weakly chaotic and non-stationary. It can be proved [18], [8], [17] that for almost each x (with respect to the Lebesgue measure) IA1c(x,a,n)-n`-1

`

.

(5)

4. Analysis of experimental data By the previous consideration, the analysis of I (x, a, n) gives useful information on the underlying dynamics. Since I (x, a, n) can be defined through the CIC, it turns out that it can be used to analyze experimental data using a compression algorithm which satisfies the property required by the theory and which is fast enough to analyze long strings of data. We have implemented a particular compression algorithm we called CASToRe. We have used CASToRe on the Manneville map and we have checked that

the experimental results agree with the theoretical one, namely with equation (5). Then we have used it to analyze the behavior of I(x, a, n) for the logistic map at the chaos threshold (see [7] and [9]). The numerical results in physics literature suggest that in the case of logistic map at the chaos threshold the speed of separation of nearby starting orbits is a power law. Equation (4) implies that the Information Content of such an orbit increases proportionally to the logarithm

of time. The Information function as it has been calculated by our numerical experiments increases below any power law, thus confirming the theory. In the following, we will always refer to the algorithm CASToRe, so that we

will omit to indicate Z in the CIC and K formulas. 2 A constructive map is a map that can be defined using a finite amount of information, see [17].

Application to the Study of DNA

25

We now give a short description of the internal running of the algorithm. CASToRe is a compression algorithm which is a modification of the LZ78 algorithm ([23]). Its theoretical advantages with respect to LZ78 are shown in [5], [9] and the Appendix of [7]: it is a sensitive measure of the Information content of sequences. That is why it is called CASToRe: Compression Algorithm, Sensitive To Regularity. The algorithm CASToRe is based on an adaptive dictionary. At the beginning of encoding procedure, the dictionary is empty. In order to explain the principle of encoding, let us consider a point within the encoding process, when the dictionary already contains some words. We start analyzing the input stream, looking for the longest word W, which is already contained in the dictionary, matching the stream. Then, we look for the longest word Y (already contained in the dictionary and not necessarily different form the word W) where the joint word WY matches the stream. The output file contains an ordered sequence of pairs (iw, iy) such that icy and iy are the dictionary index numbers corresponding to the words W and Y, respectively. The pair (iw,iy) is referred to the new encoded word WY and has its own index number. The following example shows how the algorithm CASToRe encodes the input stream $ =' AGCAGAGCCAG' . The first column is the dictionary index number of the codeword which is showed in the same line, second column. For an easier reading, we add a third column which shows each encoded word in the original stream s, but it is not contained in the output file: 1

2 3 4 5 6

(0, A) (0, G) (0, C) (1,2) (4,3) (3,4)

[A]

[G]

[C] [AG]

[AGC] [CAG]

4.1. The window segmentation A modified version of the algorithm CASToRe can be useful in studying correlated and repetitive sequences within the genomes. We performed a window segmentation option which partitions the stream into segments of a fixed size and proceeds to encode them separately, as the following scheme shows: 1. choose a window range L; 2. construct the set { W,} of all the frames of size L within the sequence s under examination; 3. calculate the CIC(W;) for each frame W;; 4. calculate the complexity at range L, K(L) = (K(W;)), which is defined as the mean computable complexity over all the frames W.

V. Benci and G. Menconi

26

At this point, there are two possible results which can be extracted from the window segmentation performed on the sequence s:

(i) we can study the mean complexity K(L) as a function of the range L and obtain a sort of spectrum of the sequence; (ii) we can study how the complexity K(W) of the windows W varies along the sequence, at a fixed window range L, this way identifying the regions at lowest complexity.

4.2. Complexity of complete genomes We have analyzed the complexity of 13 complete genomes of some Archaea, Bacteria and Eukaryotes, together with chromosomes H and N of Arabidopsis thaliana. The following tables show the results for what concerns the complexity values (first table) and redundancy values (second table). We remark that the maximal complexity is achieved when the occurrences

of the symbols are independent from each other and its maximal value equals loge #A, where A is the alphabet and #A is the number of symbols in the alphabet. For quaternary sequences, like the genomes, this means maximal complexity is 2 bits per symbol. In the case of the genomes, we notice that the values of the complexity K are significantly different from 2, so that the genomes are not equivalent to random sequences.

This is also supported by the analysis of the redundancies of 0th, first and third order. We recall that the kth order redundancy Rk is defined as Rk

= 1- f7K,

where K is the complexity obtained by the use of the algorithm CASToRe and Hk is the usual kth order Shannon entropy. By definition, HO = loge #A = 2. The more the redundancy is different from the value 0, the more correlated is the sequence under examination. It is also possible to clearly recognize that some genomes have very low complexity (smaller than 1.84 bits per symbol), which means that their internal structure presents mid and long range correlations. Another reason to justify the low complexity is the presence of repetitive sequences in the genome: the algorithm CASToRe is forced by its own nature to find out previous occurrences of words, so that the Information content decreases. In Section 4.3, we will investigate where the low complexity regions are located. Finally, we underline that the algorithm CASToRe has not specialistic features for the analysis of genomes, so this result is impressive and highlights the differences with previous application of compression methods on genome analysis ([4], [19], [27], [30]), which were focused not on complete genomes.

4.3. Long range correlated repetitions In order to identify the amplitude of the repetitive sequences within a genome and their role in the Information content behavior, we have studied the value of the complexity K as a function of the window range L, performing the window segmentation (see Section 4.1).

Application to the Study of DNA

27

Genome Methanococcus Jannaschii

Complexity K

Archeoglobus fulgidus

1.90939

Methanobacterium thermoautrophicum

1.90707

Pyrococcus abyssi

1.90068

Aquifex aeolicus

1.88282

Escherichia coli

1.89303

Bacillus subtilis Haemophylus influenzae Mycoplasma genitalium Rickettsia prowazekii Thermotoga maritima

1.87038 1.86589 1.84845 1.82346

Arabidopsis thaliana (chr. II and N)

1.89176

Saccharomyces cerevisiae

1.88927 1.77681

1.79411

1.89342

Caenorhabditis elegans

Genome

Red. Ro

Red. Rl

Red. R3

Methanococcus Jannaschii

0.105315

0.051585

0.667289

Archeoglobus fulgidus

0.045137

0.038684

0.663734

Methanobacterium thermoautrophicum Pyrococcus abyssi

0.047918

0.041293

0.662792

0.051693

0.041831

0.665146

Aquifex aeolicus

0.055837 0.053906 0.073215 0.066079 0.075091

0.044202 0.047592 0.061247 0.040465 0.055844

0.664216 0.667213 0.671370 0.664568 0.669820

Escherichia coli Bacillus subtilis Haemophylus influenzae Mycoplasma genitalium Rickettsia prowazekii Thermotoga maritima

Arabidopsis thaliana (chr. II and N) Saccharomyces cerevisiae

0.087918 -0.016497

0.62499

0.053790

0.045976

0.664012

0.054287 0.023967

0.670512

0.023721

0.663491

0.048641

This spectrum shows the extent of the correlated repetitions within the genome. As an example, consider Figure 1 on the left. The graph shows the function K(L) versus the window range L for the genome of the yeast Saccharomyces cerevisiae. As the window range increases, the mean complexity decreases. It follows that the extent of correlated repetitive sequences increases, so that we can conclude that the yeast genome presents correlations at a range of at least 2 x 106 base pairs! This analysis has been repeated for all the 21 genomes and this way it has been possible to detect how long the correlations can be in those genomes.

28

V. Benci and G. Menconi

e

FIGURE 1. Left: the complexity K(L) vs. the window range L for the complete genome of Saccharomyces cerevisiae. Right: the complexity K(L) vs. the window range L for the different functional regions of the genome of Saccharornyces cerevisiae. The solid line is the complexity of the coding regions, the dashed line is the complexity of intergenic regions, the dotted line is the complexity of intronic regions. Figure 1 on the right shows that complexity behaves in clearly different ways in the three different functional regions of the genome of Saccharomyces cerevisiae: exons (coding), intergenic and intronic regions (both non-coding). The coding re-

gions are the least correlated and the most random (the complexity decrease is early interrupted), while both non-coding regions present longer correlated repetitions, which are more evident in the intronic regions. Finally, we remark that, even in the cases where the complexity K of the complete genome is not significantly small, the analysis of the window complexity along the genome, at some fixed window range, reveals interesting characteristics of the genome and allows to localize the regions which present lower complexity. Figure 2 on the left shows the behavior of the complexity K(W) as a function of the windows W with fixed window range L = 10000 base pairs, for the complete genome of Methanobacterium autotrophicum. The complexity of the whole genome is 1.90707 bits per symbol, but the window analysis highlights two regions where the complexity intensely decreases. The enhancement pictured in Figure 2 on the right is referred to the first low complex region (from 960000 by to 1040000 bp). The dotted lines at level 1.6 and at level 1.2 respectively represent the non-coding regions and the coding regions. It is impressive that the low complexity region matches a non-coding region with the greatest extent. This result can be found also in the other genomes. On conclusion, this technique can be useful in identifying different functional intervening regions and can be easily applied to complete genomes with no need of specialistic tools.

Application to the Study of DNA

29

FIGURE 2. Left: Window analysis of the complete genome of Methanobacterium autotrophicum. The window range is fixed at L = 10000 bp. Right: the enhancement of the first region where the complexity rapidly decreases, compared with its functional meaning (see text).

References [1] Allegrini P., Barbi M., Grigolini P., West B.J.,

"Dynamical model for DNA sequences", Phys. Rev. E, 52, 5281-5297 (1995). [2] Allegrini P., Grigolini P., West B.J., "A dynamical approach to DNA sequences", Phys. Lett. A, 211, 217-222 (1996). [3] Allegrini P., Benci V., Grigolini P., Hamilton P., Ignaccolo M., Menconi G., Palatella L., Raffaelli G., Scafetta N., Virgilio M., Yang J., Compression and Diffusion: A Joint Approach to Detect Complexity, work in preparation, submitted to Discrete and Continuous Dynamical Systems - B (2001), electronic preprint: cond-mat/0202123. [4] Allison L., Stern L., Edgoose T., Dix T.I., "Sequence complexity for biological sequence analysis", Comput. Chem., 24: 43-55, 2000. [5] Argenti F., Benci V., Cerrai P., Cordelli A., Calatolo S., Menconi G., "Information and dynamical systems: a concrete measurement on sporadic dynamics", to appear in Chaos, Solitons and Fractals (2001). [6] Batterman R., White H., "Chaos and algorithmic complexity", Found. Phys. 26, 307-336 (1996). [7] Benci V., Bonanno C., Galatolo S., Menconi G., Ponchio F., "Information, complexity and entropy: a new approach to theory and measurement methods", http://arXiv.org/abs/math.DS/0107067 (2001). [8] Bonanno C., "The Manneville map: topological, metric and algorithmic entropy", work in preparation (2001).

[9] Bonanno C., Menconi G., "Computational information for the logistic map at the chaos threshold", arXiv E-print no. nhin.CD/0102034 (2001), to appear on Discrete and Continuous Dynamical Systems. [10] Brudno A.A., "Entropy and the complexity of the trajectories of a dynamical system", Trans. Moscow Math. Soc. 2, 127-151 (1983).

V. Benci and G. Menconi

30

[11] Buiatti M., Acquisti C., Mersi G., Bogani P., Buiatti M., "The biological meaning of DNA correlation", Mathematics and Biosciences in interaction, Birkhauser ed., in press (2000). [12] Buiatti M., Grigolini P., Palatella L., "Nonextensive approach to the entropy of symbolic sequences", Physics A 268, 214 (1999).

[13] Chaitin G.J., Information, randomness and incompleteness. Papers on algorithmic information theory., World Scientific, Singapore (1987). [14] Galatolo S., "Pointwise information entropy for metric spaces", Nonlinearity 12, 1289-1298 (1999). (15] Galatolo S., "Orbit complexity by computable structures", Nonlinearity 13, 1531-1546 (2000).

[16] Galatolo S., "Orbit complexity and data compression", Discrete and Continuous Dynamical Systems 7, 477-486 (2001). (17] Galatolo S., "Orbit complexity, initial data sensitivity and weakly chaotic dynamical systems", arXiv E-print no. math.DS/0102187 (2001). [18] Gaspard P., Wang X.J., "Sporadicity: between periodic and chaotic dynamical behavior", Proc. Natl. Acad. Sci. USA 85, 4591-4595 (1988). [19] Gusev V.D., Nemytikova L.A., Chuzhanova N.A., "On the complexity measures of genetic sequences", Bioinformatics, 15: 994-999, 1999. (20] Katok A., Hasselblatt B., Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press (1995). [21] Khinchin A. L, Mathematical foundations of Information Theory, Dover Publications, New York.

[22] Lempel A., Ziv A., "A universal algorithm for sequential data compression", IEEE Trans. Information Theory IT 23, 337-343 (1977). [23] Lempel A., Ziv J., "Compression of individual sequences via variable-rate coding", IEEE Transactions on Information Theory IT 24, 530-536 (1978). [24] Li W., "The study of DNA correlation structures of DNA sequences: a critical review", Computers chem. 21, 257-271 (1997). [25] Manneville P., "Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems", J. Physique 41, 1235-1243 (1980). [26] Kosaraju S. Rao, Manzini G., "Compression of low entropy strings with Lempel-Ziv algorithms", SIAM J. Comput. 29, 893-911 (2000). [27] Milosavljevic A., Jurka J., "Discovering simple DNA sequences by the algorithmic significance method", CA BIOS, 9: 407-411, 1993. [28] Pesin Y.B., "Characteristic Lyapunov exponents and smooth ergodic theory", Russ. Math. Surv. 32, 55-112 (1977). [29] Politi, A.; Badii, R., "Dynamical `strangeness' at the edge of chaos". J. Phys. A 30 (1997), no. 18, L627-L633. [30] Rivals E., Delgrange 0., Delahaye J: P., Dauchet M., Delorme M: O., Henaut A., Ollivier E., "Detection of significant patterns by copression algorithms: the case of approximate tandem repeats in DNA sequences", CABIOS, 13: 131-136, 1997. [31] Tsallis C., "Possible generalization of Boltzmann-Gibbs statistics", J. Stat. Phys. 52, 479 (1988).

Application to the Study of DNA

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[32] Tsallis C., Plastino A.R., Zheng W.M., "Power-law sensitivity to initial conditions - new entropic representation", Chaos Solitons Fractals 8, 885-891 (1997).

Vieri Benci, Giulia Menconi Dipartimento di Matematica Applicada "U. Dini" University degli Studi di Pisa Via Bonanno 25 b 1-56126 Pisa, Italy

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 33-51 © 2003 Birkhauser Verlag Basel/Switzerland

Bounded Positive Critical Points of Some Multiple Integrals of the Calculus of Variations Lucio Boccardo and Benedetta Pellacci

1. Introduction Let SI be a bounded domain in IiN, (N > 2) and a : St x JR smooth real function. We recall that simple functionals as

R be a bounded

J(v) = 2 Ja(x,v)IVvl2, n

are differentiable only along directions of 0.2(12) n L°°(1l). The directional de rivative is given by

(J'(u),v) = f a(x, u)VuVv + n

Jn

ae(x,u)IVuI2v,

where the lower order term a.(x,u)IVuI2, in general, belongs to L'(Il) and not to W-1,2(12). Existence results of critical points of such functionals have been proved in [2], [3], [4], [5], [7], [8], [9], [10], [11]. In [7], [9], [10] some multiplicity result are proved by using a "weak" notion of derivative for continuous functions in complete

metric spaces. In [2], [3], [4], some existence results are proved for bounded and unbounded coefficients a(x, s). Here we will present some existence results for this problem, proved in [2], [5] and [11]. We will present new proofs, which hinge on a different (simpler) order of the steps in the proof of the Palais-Smale sequences compactness. We consider the following class of integral functionals

I(v) = 2 f a(x,v)IVvI2 n

-PJ

(v+)p,

n

but we want to point out that our method works for general functionals as T (V) =

J?/.i(x,vVv) - Ji,12(xv),

This work contains the unpublished part of the lecture held by the first author at Bergamo* 2001 Conference.

' Siede Peschiera, beilo e forte arnese da fronteggiar Bresciani e Bergamaschi (Inf. XX).

L. Boccardo and B. Pellacci

34

and we assume that

2
(1)

{ 8a x, s

are functions, which are measurable with respect to x and continuous with respect to s. The paper is organized as follows. In Section 2 we will present a new proof of two existence results proved in [2] in the case of bounded coefficients (studied in Subsection 2.1) and in [5] in the case of unbounded coefficients (studied in Subsection 2.3). Finally, in Section 3 we will consider non-increasing coefficients and we will obtain a new proof of an existence result proved in [11].

2. Non-decreasing coefficients 2.1. Bounded coefficients We assume that the following conditions are satisfied for almost every x in 11 and for every s in 1R+ (for a,,3, ry, 6 E IR+) 0 < a < a(x,s)

(2)

a,(x,s)s > 0,

(3)

Ia.(x, s) I

(4)

'Y,

(p - 2)a(x, s) - a.(x,s)s > b.

(5)

a(x,s) - a(x,0)

(6)

For s < 0, we define

so that a, (x, s) - 0 for every s < 0. Let us recall the definition of a critical point. Definition 2.1. A function u is a critical point of I if

u E W01.2(1)nL°°(1):

f a(x,u)VuVcp+J V ,p E

W,1,2

s1

J(u4)p n

(0) n L°°(I)

In this section we will present a new proof of an existence result proved in [2]. In order to study the existence of critical points, we need a version of the Ambrosetti and Rabinowitz Theorem [1] for functionals not differentiable in all directions. The proof can be found in [2].

Theorem 2.2. Assume (1), (2), (3), (4), (5). Then the functional I has at least one nontrivial (positive) critical point.

Multiple Integrals of the Calculus of Variations

35

Proof. The proof (see [2]) is done by dividing it into two steps. In the first one, only the geometric hypotheses are used to deduce the existence of a sequence in W01,2 (0) n L°°(S2) satisfying the following condition (9). The second (longer) step is the proof of the compactness of this sequence {un} in W01'2(S1). In this way,

condition (9) can be considered as a compactness condition on our functional, which substitutes in the non-differentiable case the role done by the well-known Palais-Smale condition in the regular classical (semilinear) case. As in the semilinear case (i.e., a(x, s) = a(x)), assumptions (1) and (2) imply

that uo = 0 is a strict local minimum of I: there exist p, R E R+ such that I(v) > p > 0, for IIvIIw01.2(0) = R > 0.

(7)

Remark that I (Tcpj) < 0, for some T E R+, where 'Pl is the first eigenfunction of the Dirichlet problem for the Laplacian in S2. Moreover T can be chosen large enough, such that 2R.

(8)

From now on, we follow the proof of Theorem 3.3 of [2]. We apply Theorem 2.1 of [2]. Consider X = W01'2(0) and Y = W01'2(SZ) n L°°(f ), endowed with the norm

=II'Ilw,.z(n)+II'1100.Let

r = {1 : [0,1] - (Y'11 ' 11,,)

:

7 continuous and y(0) =0,-y(l) = T
Observe that every 'y E r is continuous from [0,1] to W01'2(0), so that, by (7) and (8), for every ry E r there exists t E [0, 1] such that I17(OIlwo.3(n) = R.

Thus, I(Tcpl) < 0 implies that o =- inf max I(y(t)) > p > max{I(0),I(T,pl)} = 0. 7Er tE(o,11

Let {-yn} c r be a sequence of paths for which

a< max(I(7n(t))TIIw11I00 tE(0,1)

Note that

L ha + II

II

is a norm in WW'P(1) n L°°(S2) which is equivalent

' I I oo + I I ' II By applying Theorem 2.1 of [2] we deduce the existence of a path ryn E r and a function un = ryn(tn) E y ([0,1]) satisfying o7nct)-,n(n)uL-(0) + 11%(t) -'rn(t)11 1.4 )

to II

n,

O - n < I(un) < o + 2n' 1(

<

I' (un),z)1 - V n ( 1

IIvIIL M

(9)

St +

IlvII

.z Wj(n))

Vv E

W01, 2

o'

n L°°(1),

L. Boccardo and B. Pellacci

36

and for n E IN large enough,

II7n(tn)IIL-(n) < 2M,.

= PFn(tn)IIL-(Q) <- III-Yn(tn) With this setting, we shall use the results of the following section concerning the sequence { u,, 1. Proposition 2.5 implies that {un } possesses a subsequence IIunIIL-(12)

{un, } which (thanks to Theorem 2.10) is strongly convergent in W01'2(fl) to some

u E Wj'2(Sl) n L°°(f ), I(u) = a > 0 and u is a nontrivial critical point of I. Remark 2.3. If p < 2, assumption (2) implies that I is coercive, so that there exists a (nontrivial) bounded minimum. If assumption (3) is not satisfied, some existence results of critical points can be found in [4] and [11]. 2.2. Compactness of the sequence {un} This section is the core of the paper, since our aim is the new proof of the compactness of Palais-Smale sequences.

Recall that, on the functional I, we assume (1), (2), (3), (4), (5). Let un satisfy (9). We write again our framework as: un E Wa'2(i1) n L°°(1),

IIunhIL-(n) < 2Mn,

II(un)I < R, (10)

r IIyII I(I'(un), v>I <_

6n

l

1

M (n' +

J , Vv E

n

W,1.2

(n) n L°°(s2),

(11)

where R is a positive constant, {Mn} C &+ - {0} is any sequence and {en} C I+ is a sequence converging to zero.

In the first step the boundedness of the sequence {un} is proved (un,, - u). is not bounded in L°°(f ), the boundedness of u is In the second step, even if proved. In the third step, it is proved that the sequence {un } converges strongly to zero in W01'2(1 ). Thus, u > 0. Then, roughly speaking, only the contribution of the sequence {un} on the subsets {x : 0 < un(x) < u(x)} is not negligible, that is where {un} is positive and bounded in L°°(1 ). Proposition 2.4. The sequence {un} is bounded in

Proof. Take -pun as test function in (11). From (10) and (11) we derive

Ja(x,un)IVunI2

-p

(u,+,)p
S2

J all un)IQu,,I2 S]

p`

J/

2p J unas(x,un)IVunI2 + p / (un)p St

(2 + IIunli«;o,z( )]'

Sl

Multiple Integrals of the Calculus of Variations

37

We sum the two previous inequalities and we get

f [(p - 2)a(x, un) - unae(x, un)llounl2 < 2e,(2 + Ilunllwa.z(ul)) + 2pR. SZ

Then assumptions (1) and (5) imply that there exists a positive constant L, such

that Ilunlly{o.2 < L.

(12)

0 As a consequence of the previous proposition, we get that there exist a func-

tion u E

W,1,2

(Q) and a subsequence of {un} (still denoted by {un}) such that

un converges to u weakly in W01'2(S2), strongly in L2(Sl) and almost everywhere.

Moreover, we take v = un in (11) and we use conditions (2) and (12) in order to deduce that for some co E 1R//+,

J

unae(x,un)IVu1I2 < Co.

91

Now we shall prove that the function u is bounded. For every k > 0, we define

Tk(s) = max(-k, min(k, s)) ,

Gk(v) = v - Tk(v).

(13)

Proposition 2.5. The function u belongs to L°°(S2).

Proof. Taking v = Gk(un) as test function in (11), we deduce that

J a(x,un)IVGk(un)12 +2 S3

J

a.(x,un)Gk(un)IVGk(U.)I2

J/S2

J (un )p-1 Gk(un) + (2 + L)en . 52

From (2) and (3), the Sobolev inequality we obtain that s

as

(j) IGk(un)12.

:5

J(u)P'Gk(un)+En t2

< where en

2p_2

f IGk(un)Ip +

2p_2

kp-1 f IGk(un)I +£n,

0 and S is the Sobolev constant. Now the Holder inequality implies

that f IGk(un)Ip < sl

(JGk(un)I 2sl

meas(An.k)1-

(JIGkI2)

L. Boccardo and B. Pellacci

38

where An,k = {x E SZ : Iu, (x) I > k}. Thanks to estimate (12) we have that z

r

IGk(U.)12

J

< C1

Sl

and that there exists k' > 0 such that, for k > k* = k*(L), CI

2P-2

meal (An.k)1-` < 2 S.

Moreover (thanks again to estimate (12) and the Holder and Young inequalities) 2

1

J

C2

(Jck(u)2)

kP-1

SI

+£n

SI

<2

(fic.ni2

so

2V

+C3 k2(P-1)[meas(An.k)]1+

+£n

z C4 I

(JIGk(uflI2.)

< k(P- 1)[meas(An.k)]+p

,

S2

where en =

C4

r

J

e;,. Since 11U111,2'

(0)

< c(L), we obtain

T z / IGk(Un)12


,+,£n

S2


+£n.

--

Remark that , since p < N + 2. Thus, a classical result due <2+ F. to Stampacchia ([13]) implies that c(IIuIIjt l 2(u)) = M.

IIUIIL-(SJ)

(14)

0 Proposition 2.6. The sequence {u,-, } converges strongly to zero in W01'2(1). Moreover

f a.,(x,un)u

Ja

IVu11I2 --+0.

Proof. Taking v _ -u; as test function in (11), we deduce that

Ja(x,un)IVu;12 - 2 J a8(x,u)un Iounl2 < (2+L)£n. SI

SI

Then the assumptions on a(x, a) and a, (x, s) imply the conclusion.

13

Multiple Integrals of the Calculus of Variations

39

Remark 2.7. As a consequence of the previous proposition we get that u > 0.

Lemma 2.8. The sequence {[un - u}+} converges strongly to zero in W12()).

Proof. We take v = Gk(u,+,) as test function in (11). In virtue of the Sobolev embedding theorem, we deduce that

CJ IVGk(u')I2 <-

f(u)P_1Gk(u)+e'n N+9

J ICk(un

C

(.u+)(P-1)

)I2

+E n'

n

1

(xEi2:u,,(x)>k)

S1

N+2

2N

1

4

(Jlvck(ut)I2)

< cl

S2

f

J {xES2:un(x)>k}

(un)(P-1) 2N

+ En-

Now, by the Young inequality, we obtain

(.u+)(P-1)

2 fIvck(u)I2
n

+ en.

{xEt2:un (x)>k}

S2

< 2', we deduce that, for

Since NO is bounded in W01'2(f) and as (p - 1) any fixed e > 0, there exists k1 > 0 such that, N+2

2C2

r J {xES2:un(x)>k}

(un)(P-1)7F+3

< e,

d k> kl and V n E IV.

So that

a

JIVCk(u)I2 <e+En,

Vk>k1,`dnEW.

(15)

12

Since IVu12 E L1(Il), there exists k2 > 0 such that, for every k > k2, IVu12 < E.

a

(16)

1

{zES2:k
Now we study the behavior of the positive part of u - u. The positivity of u yields un(x) > k in the subset {x E Sl : k < un(x) - u(x)}.

L. Boccardo and B. Pellacci

40

Therefore, we have

afIVGkEufl - uJI2 S2

I

=a

I V [un - u112 < a

2a

IVu, I2+2a

J

I V [un - u]12

J

f

IVu12.

Fix k() = max{kl, k2}. From by (15) and (16) we deduce that

a

J

I VGk [un - u]+12 < 2(E + En) + 2E, `dk > ka, do E

N.

(17)

s2

The use of Tk[un - u]+ as test function in (11) leads to

- u]+ +

2J

a.(x, un) Tk[un - u]+ 1oun12

J( St

S2

(ui )p-' Tk[un - u]+ + (2 + 4L)E,, st

From (2) and (3) we obtain that

.

If IVTk lfun - u]+12 - / (un )p-1 Tk[Un - u]+ SZ

(18)

< 2 + 4L)e + fa(x , un ) VuVTk[un - u]+. s:

We point out that the last integral is different from zero only on the subset {x E 11 : u(x) C un(x) < u(x) + k}; and here the sequence {un} is bounded, because u E L°O(S2). Thus, there exists

n, E N such that Ja(xun)V1LVTk[un-u]4 < E,

`dn > n,.

se

Then (17), (18) and (19) imply that a

IV[un - u]+ I2 < En + 5E,

dk > kV, Vn > n,

(En -' 0),

that is 11[U. - u]+IIWI.2(n)

0.

Now we study the behavior of the negative part of un - u.

(19)

Multiple Integrals of the Calculus of Variations

41

Lemma 2.9. The sequence {[u, -u]-} converges strongly to zero in W0.2(11).

Proof. First, we will show that Define spa(s) = se

A=

,

16a2'

Recall that llullc-(S)) < M. Note that w,\(TAI [un - u] ) =

0

in {x u(x)

WA(TAfju,+, - u]-)

in {x : 0 < un(x) < u(x)},

cpA(u)

in {x : U" (X) < 0}.

Since u belongs to L°"(1l), the sequence {un} is bounded in the set {x : 0 < un(x) < u(x)}. Let zn = Tftf[u,,, - u] We use cOA(TM[u, - u]-) as test function in (11), from (1) and (3) we get

J

a(x,un)VunVZW'\(zn) +

J

2

as(x,un)IVunl2wA(zn) > -clen.

But 2

a.(x,un)lVn l2wa(zn) = J {x:un(x)>0}

f

J

(J

2

IV(un-u)l2tpa(zn)+2

J

{x:O
T

a (x, un )DunV uccA(Zn )

J

and the last two integrals converge to zero. From now on we will denote with = W;,, for i = 1, 2, ... , quantities converging to zero.

It results

a(x,un)VunVznA (zn) =

a(x,un)lV(u - un)12V, (zn)

S)

J {x:O
J

a(x, un)VuV(u - un)cpA(zn)

a(x,un)IVznl2WA(zn)

-

a(x, un)VunVuwa(u) a(x, un)VuV(u - u'++)V,\(zn)

Sl

J

a(x,un)VunVWa(u) -

J

(x:un
42

L. Boccardo and B. Pellacci

Since u E L' (Q), un

0 strongly in Wa'2(S2) and u,,

u in W0 '2(SZ), we have

J

J

a(x, un)VunVuti, (u) - 0,

{z:u,,<0}

a(x,un)IVuI2V (u) - 0.

J {x:u
Jj

a(x, un)Vu0(u - un)So (zn)

+ 0.

Then the choice of A implies that a 2 and a 2

fIVTsiFun _U)-12 < CIE. +W,la +W +wn +W,a, +Wn. 0

Finally the inequality

J

f

IV[u;, -u] I2 =

r

IDuJ2+

J

IVTs4un -u] I2

J

{x:u,,(x)<0}

St

IVuI2+J IOT,,,[u+n -u]-I2 S2

gives the result lim

n-oo

f lV[u; - uJ-I2 = 0.

(20)

Moreover, we have

J0 <2

IV[un - u] I1 S

J

J

IVuI2+2

lV(u - un)I2 + r IQ[un - uJ-I2 Q

f

IVunI2+ rIv[un -uJ-I2 S]

We take into account (20) and that un converges to the positive function u E W,I'2(fl) strongly in L2(St) and almost everywhere joint with the fact that un converges strongly to 0 in W01'2(f ). Thus 11[U. - u]

Iltit,

2(0)

0.

D

Lemma 2.8 and Lemma 2.9 give the following theorem.

Theorem 2.10. The sequence (u,) converges strongly to u in W( 2(SZ). Moreover ?a,,(x,un)IQunl2 converges to Za,(x,u)IDul2 and {a(x,un)IVunl2} converges to a(x,u)IDul2 in L1(f) (which implies that I(un) converges to I(u)).

Multiple Integrals of the Calculus of Variations

43

2.3. Unbounded coefficients Here, we will study the existence of critical points of the integral functionals with unbounded (from above) coefficients a(x, s) whose model example is I2(V)

2

J(l + Ivlm)IovI2 n

1(,+)p,

p

m > 0.

n

In this section we will present a new proof of an existence result proved in [3), (5]. Notice that when we consider unbounded coefficients the functional is well defined only on a subset of W01'2(1l). More precisely, let us consider a function a(x, s) : 1 x p3+ - 1R+ measurable with respect to x E fl, derivable with respect to s E IR+ and such that hypotheses (3) and (5) are satisfied. Moreover, we will suppose that there exist positive constants a, 5 satisfying the following hypotheses a < a(x, s) < /3(s),

(21)

Ia8(x,s)I <_ y(s),

(22)

lira (/3(s) - sp-2) < 0, 8-+00

(23)

where 3,,y are continuous, increasing (possibly unbounded) functions of a real variable.

Remark 2.11. If we suppose (22) and that a(x, 0) < /3o we deduce that (21) is satisfied with /3(s) =,3o + F(s), where r(s) = fog y(t)dt. We consider an exponent p that satisfies (1) and we define the functional I : W01'2(1)

IR U {+001 by

I(v) =

2J

a(x,v)IVvI2

a

p

f(v,

if I(v) < +oo,

n otherwise,

+oo, We will prove the following theorem.

Theorem 2.12. Assume (1), (3), (5), (21), (22), (23). Then the functional I has at least one nontrivial (positive) critical point.

Proof. We follow the outline of the proof of Theorem 2.2 and we notice that, thanks to conditions (1), (21) and (23), the functional I satisfies the geometrical hypotheses of "Mountain Pass" type, i.e., uO = 0 is a strict local minimum in the topology of W01'2(S2) and I(Tcpl) < 0 for T sufficiently large. As before we set

Y = W01.2 (SZ) n L°°(fl). Condition (21) and (22) imply that for every u, v E Y there exists (J'(u),v), moreover for every v E Y the map v (J'(u),v) is continuous for every fixed u E Y and for every u E Y the map u (J'(u), v) is continuous for every fixed * E Y. This regularity properties of the functional I are enough to apply Theorem X = WW '2(1),

2.1 in [2]. Then, we get that there exists u,, E

W01,2 (n) n

and (11). Propositions 2.13 and 2.14 imply that

L°°(52) that satisfies (10) possesses a subsequence

L. Boccardo and B. Pellacci

44

{u,,,,}, which (thanks to Theorem 2.17) is strongly convergent in W01'2(11) to some u E 14o'2(S1) n L- (n). Lemma 2.19 implies that u is a critical point of I and from

Lemma 2.20 we deduce that I(u) = a > 0 so that u is nontrivial. Proposition 2.13. The sequence {un} is bounded in W01'2(51). Moreover, it results

Junas(x,uy)IVunI2 < Cl,

(24)

for some cl E 1R+.

Proof. The proof is the same as that of Proposition 2.4. Thus, there exist a positive function u E W01'2(1l) and a subsequence of {un} (still denoted by {un }) such that u, converges to u weakly in W01'2(S1) and strongly in L2(Sa).

Now we shall prove that the function u is bounded. Lemma 2.14. The function u belongs to L°°(SZ).

Proof. The proof is the same as that of Proposition 2.5. Indeed, note that for every n u E L°°(S1). So that we can take v = Gk(un) as test function in (11) in order to obtain (14). Proposition 2.15. The sequence {u,-,. } converges strongly to zero in W01'2()) (thus

u > 0). Proof. The proof is the same as that of Proposition 2.6. Remark 2.16. Notice that, even if un -* 0, we cannot deduce that

Ja(x,u,i)IVui2 _ as in Proposition 2.6. because the coefficients a(x, s) and a,, (x, s) are not uniformly bounded in L°'(S1). Theorem 2.17. The sequence {un} converges strongly to u in WO'2(S1). Proof. In order to prove that un converges to u in WW `2(S1) we can follow the same

procedure as in Subsection 2.1. Indeed, notice that all the test.functions taken in Lemma 2.9 and in Lemma 2.8 are continuous functions of un and u. Moreover, they are different from zero only in subsets of 11 where un is uniformly, with respect

to n, bounded in L°°(fl). This permits us to pass to the limit and conclude that

u - u in W0' (S1). Now we want to show that u is a critical point of the functional I. Lemma 2.18. The sequence Zas(x, un)IVunI2 converges to 2as(x, u)IVuI2 in L' (fl).

Multiple Integrals of the Calculus of Variations

45

We want to point out that, since a.(x, un) is not bounded in L°°(fl), the strong convergence in Ll(S2) of a,(x,un)IVunI2 to a,(x,u)IVuI2 is not a consequence of the strong convergence in L1((1) of IDunI2. Now we will prove that a e (x, u,,) I Dun I2 is uniformly equiintegrable. For any measurable subset E of fl and for any m 1R we have (thanks to (4) and (24)) f Eae(x,un)IVun12 = E

J

+

as(x,un)Ivu12

J

(xEE:O
aa(x,un)IVunI2 < y(m) J IVunl2+,nt J ua3(x, u)IVuI2 0

E

< 7(m)

JE

IDunI2 + ci m,

which proves the uniform equiintegrability of a8 (x, u,, ) I Vun 12 as a consequence of the uniform equiintegrability of IDunI2, i.e., lim

ae (x, un) IVun I2 = 0,

(EJ-.O

uniformly with respect to n,

E

Therefore, we have a. (x, un) I Vun l2 - a, (x, u) I VuI2 strongly in L1(11).

0

Lemma 2.19. The function u is a solution of the Dirichlet problem up-1(PI

Ja(xu)VuVP + f a,(x,u)IVuI2W=

(25)

V p E W0,2(S2) n L°°(Sl).

Proof. Notice that, since a(x,un) is not bounded in L°°(SZ), up to now we are not able to prove that a(x, un)IVun12 converges to a(x, u)IVuI2 in L1(1l). Now we take

v = Tk[un - 0] in (11) (k E R+, 0 E

W01,2

(n) n L°°(S2)) and pass to the limit. Thanks to Theorem 2.17, Lemma 2.18 and since u E L°°(0), we obtain

Ja(x,uu[u -

]+

J

<

n n `d¢ E W01'2(S2) n L- (0).

Then, it is enough to take order to obtain (25).

= u + V and

fu'Eu-0]

sl

u-

(cp E W0,2(S2) n L- (Q)) in

0

Finally, let us prove that u # 0. This will be a consequence of the following Lemma.

Lemma 2.20. The sequence {a(x, un)IVun12} converges to a(x, u)IVu12 in L1(S2).

46

L. Boccardo and B. Pellacci

Proof. Take v = u in (11) and use the Fatou Lemma. We have

Ja(x,u)IVuI2 +2 Jas(x,u)ulVuI2 S2

S2


Jas(xun)un 0

11

/

J

sz

n


< lim

(2 + L)e =

Jsi

Jn

up =

= u)

(thanks to (25), with

a(x,u) IVU12 +2

_

f a.(x u)uIVul2 0

Q

So we proved that {a(x,un)IVu,,12} converges to a(x,u)IVu12 in L'(1) and that converges to uas(x,u)IVu12 in L' (n). 0

3. Non-increasing coefficients In this section we will consider non-increasing coefficients a(x, s). We will present one of the existence results proved in [11]. Our proof will be different from [11] and we will follow the same outline as in the previous section. We will assume that a(x, s) satisfies (2), (4). Moreover, we will suppose the following hypotheses on as(x, s) (for I3o and Ro E iR+) Vs E IR+,

as(x, s)s < 0,

fo r every s > Rc,

las(x,s)Is < 0o ,

ip(s) zf'(s)

(26) (27)

susplas(x,s)I, (28)

E L'(f2).

Remark 3.1. Notice that condition (2) implies that

supJ

+las(x,s)I


n o while condition (28) is slightly stranger and it is always satisfied in the model case

a(x,s) = a(s). We will prove the following theorem.

Theorem 3.2. Assume (1), (2), (4), (26), (27) and (28). Then the functional I has at least one nontrivial (positive) critical point.

Multiple Integrals of the Calculus of Variations

47

Proof. We follow the outline of the proof of Theorem 2.2 and we notice that, thanks to conditions (1) and (2), the functional I satisfies the geometrical hypotheses of "Mountain Pass" type, i.e., u0 = 0 is a strict local minimum in the topology of W01'2(5i) and I(TWa) < 0 for T sufficiently large. Moreover, note that, as before, I satisfies all the assumptions requested by Theorem 2.1 in [2]. Then we get that there exists un E Wo'2(S2) n L°°(Q) that satisfies (10) and (11). Propositions 3.3 and 3.5 imply that {un} possesses a subsequence {unk}, which (thanks to Theorem 3.11) is strongly convergent in W01'2(0) to some u E W01 12 (n) n L- (0), I(u) = o > 0 and u is a nontrivial critical point of I. Proposition 3.3. The sequence {un} is bounded in W01'2(Sl).

Proof. As in Proposition 2.4 we take --lun as test function in (11). From (10) and

(11) we getJa(x,un)IVunl2_ p

2

f (/

(un )p < R

n

S2

pJ a(x,u,,)IVu, 2

J..pJ unas(x,un)IV nI2+pJ (fin )p

S2

<

En

(29)

Sa

S2

[2 + IINnII

p Notice that (6) and (26) imply

-2 J unas(x,un)IVunI210, sa

so that, when we sum the two inequalities in (29) we obtain

(2 -1)1

a(x,un)IVunI2 <2En(2+IIun1I,,.2(())+2pR.

sa

Then assumption (26) implies that there exists a positive constant L, such that IIunhI

Wo.2(n)

< L.

(30)

Thus, there exist a function u E and a subsequence of {un} (still denoted by {un}) such that un converges to u weakly in W1,2(n) and strongly in L2(SZ).

Remark 3.4. Notice that in the previous proposition we did not use hypothesis (5) because we took advantage of condition (26). Lemma 3.5. The function u belongs to L°O(1l).

Proof. Note that (27) implies that r/i(s)s <,%,

for every s E R.

(31)

48

L. Boccardo and B. Pellacci

We follow the argument of Lemma 3.5 in [[11] and we define %D (s) = 2-

J0

A

Vi(t) dt.

(32)

We notice that condition (28) implies that %P(s) is a bounded function. Let us consider v = e'O(u")Gk(un), where Gk(un) is defined in (13). Conditions (4) and (31) imply that v E L-(S2) n W1'2(Sl), then we can take v as test function in (11) and we obtain e*(u.,)a(x,Un)IVU.IZGk(un)'p(un)

f e°(u")a(x, u,,)VunVGk(un)+2a f St

52

J/

+2 J

f (un)p-tGk(un)+CSEn. R

St

Condition (26) and the definition of Vi(s) imply that

fenGk(ufl)IVuflI2

> 0.

Therefore, we obtain

f

Co(on)--'Gk (un)+C1En. 0

S!

From now on it is possible to follow the same argument as in Proposition 2.5 in order to obtain that u belongs to L°0(11). Remark 3.6. The use of the function Vi combined with exponential functions has been introduced in [12] and in [6] in order to study problems of this kind without assuming any sign condition on the quadratic gradient term. Proposition 3.7. The sequence {un } converges strongly to zero in W01'2(1l). Moreover

fa(xuu) IVun I2 -i 0,

2

f a,(x,un)un IVunI2 -+0. it

St

Proof. The same as for Proposition 2.6. Remark 3.8. As a consequence of the previous proposition we get that u > 0. Lemma 3.9. The sequence {[u,, - u]+} converges strongly to zero in W0"2(fl).

Proof. We take v = e*(u")Gk(u,+,) as test function in (11). As in Proposition 3.5 we get

a f IOGk(un)I2< f(un')P-'Gi.(un+)+Enr n

it

where {e;,} E Df+ is a sequence converging to zero. From now on it is possible to follow the same argument as in Lemma 2.8 to obtain (17). Then, in order to show

Multiple Integrals of the Calculus of Variations

49

that (un - u) + -> 0, it is left to prove that Tk(un - u)+ -+ 0, as n tends to infinity. Let us consider v = oa[Tk(un - u)+], where k > k0 (ki) is fixed in (17)) is fixed and z

V'\(s) := sea8 ,

A

'2 Y

16a2

Note that in {x : un(x) < u(x)1,

10 ,PA(Tk]un - u] +) =

WA(u. - u)

in {x:u(x) < un(x) < u(x) + k},

W,\ (k)

in {x un(x) > u(x) + k}.

Therefore, v E W01'2(Sl) n L- (Q) and we can take v as test function in (11). We k-(un - u) + and we obtain set zn = Tfa(xun)VunVznIzn]

+ ft

J

J/

2

as(x,un)IDunI2Wa[zn1

Wn.k,

0

wherenlim wn k = 0 for fixed k and for i = 1,2,.... Since un - u and from cc condition (2) we get

=

J a(x,u.n)VunVzn,Q'\]zn]

J

a(x,un)V(un - u)VznV'\]zn)

11

+J a(x,un)VuVTk(un - u)+A ]zn] f1r

=

J f1

a(x,u,,)IVTk(un - u)+I2A[Zn] +Wn k.

Thus, condition (4) yields J/

Ja(x, un)IVTk(un -

2J

IVunI2Wa]Zn] + Wn.k + Wn.k

(33)

n

f1

From (17) we deduce

2

Jn

+)

IVunl2Wa]Zn] = 2

r

J

IV(Un - u)I240A[Zn] - 2

L1

r IDup2f/Ia]zn] J S2

VunVuccA[zn]

+2 I IVTk(un - u)+I2wAlSZn], SZ

The last inequality together with (33) imply

JQ

IVTk(un 2`Wa]Zn]}

Wn.k +Wn.k +Wn,k +'Gaak)1(E+E"n).

L. Boccardo and B. Pellacci

50

From the choice of A we deduce

(e+e"n).

(34)

S2

Finally, (17) and (34) yields the conclusion. Lemma 3.10. The sequence {[un -u]-} is strongly convergent to zero in W0''2(Q).

Proof. Let us take as test function v = -(un - u)-. From conditions (1) and (26) we get

- J U,(2, un)QunO(6n - u) G En, 52

where {e;, } E IRis a sequence c onverging to zzero. Note that

- Ja(xun)VunV(un - u) _- J a(2, un)07Ln 0(un + f a(x, u,,)Vun V(u,, -u)-

tL)-

=I, un)IV(un - U)-12

Q

- I a(x, un)VuV(uy+, - u)2 + Ja(x.u)VuV(u n - U)n

11

Proposition 3.7 joint with the fact that u,+, - u implies that the last two integrals tend to zero. Then, from condition (2) we have

f IV(un - u)-I2 <

(35)

Q

Finally from (35) we have

f IV(un S2

- u) I =

f

f

IV(u,+, - u) I2 +

{s : u >0}

<en +2 f IVuI2+2 11

IV(un - u)

I2

(X: u <0} IVuI2

f

0

Proposition 3.7 yields the conclusion. Lemma 3.9 and Lemma 3.10 give the following theorem.

Theorem 3.11. The sequence {un} tends strongly to u in

In addition,

'a,(x,un)IVunl2 tends to za,(x,u)IDul2 in L'()) and

con-

verges to a(x,u)IVuI2 in L'(f2) (so that implies that I(un) converges to I(u) ).

Multiple Integrals of the Calculus of Variations

51

References [1] A. Ambrosetti, P.H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Flint. Anal. 14 (1973), 349--381. [2] D. Arcoya, L. Boccardo: Critical points for multiple integrals of the Calculus of Variations. Arch. Rational Mech. Anal. 134 (1996), 249-274. [3] D. Arcoya, L. Boccardo: Some remarks on critical point theory for nondifferentiable functionals. NoDEA 6 (1999), 79-100. [4] D. Arcoya, L. Boccardo, L. Orsina: Existence of critical points for some noncoercive functionals. Ann. Inst. H. Poincare Anal. Non Lineaire, 18 (2001), 437-457. [5] L. Boccardo: The Bensoussan & Co. technique for the study of some critical points problems. In Optimal Control and PDE, J.L. Menaldi et al. ed. Book in honour of Pressor Alain Bensoussan's 60`h birthday. IOS Press (2000). [6] L. Boccardo, S. Segura, C. Trombetti: Existence of bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term. J. Math. Pures et Appl., 80 (2001), 919-940. [7] A. Canino, M. Degiovanni: Nonsmooth critical point theory and quasilinear elliptic equations. Topological methods in differential equations and inclusions (Montreal, PQ, 1994), 1-50, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 472, Kluwer Acad. Publ., Dordrecht, 1995. [8] J.N. Corvellec, M. Degiovanni: Nontrivial solutions of quasilinear equations via nonsmooth Morse theory. J. Differential Equations 136 (1997), no. 2, 268-293. [9] J.N. Corvellec, M. Degiovanni, M. Marzocchi: Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlinear Anal. 1 (1993), no. 1, 151-171. [10] B. Pellacci: Critical points for non-differentiable functionals. Boll. Un. Mat. Ital. B (7) 11 (1997), 733-749. [11] B. Pellacci: Critical points for some integral functionals. To appear on Top. Meth. in Nonlinear Anal.

[12] S. Segura Existence and uniqueness for L' data of some elliptic equations with natural growth. To appear on Adv. in Diff. Eqn. [13] G. Stampacchia: Le Probli me de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus. Ann. Inst. Fourier Grenoble, 15 (1965), 189-258.

Lucio Boccardo, Benedetta Pellacci Dipartimento di Matematica University di Roma I Piazza A. Moro 2 1-00185 Roma, Italia

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 53-60 © 2003 Birkhauser Verlag Basel/Switzerland

Hilbert Type Numbers for Polynomial ODE's Marta Calanchi and Bernhard Ruf

Introduction Consider the polynomial ordinary differential equation of first order dt

+a,(t)u"

+a"-I(t)u"-' +...+a1(t)u

= f(t) , 0 < t < 1

(1)

where a1, ... , a", and f : [0,1] IR are continuous functions. We will say that a solution u(t) of (1) is a closed solution if it is defined in the interval [0, 1] and u(0) = u(1). If the coefficients f, a1i ... , a are 1-periodic, then a closed solution is a 1periodic solution. In this special case, equation (1) can be considered on the cylin-

der and the closed solutions correspond to periodic orbits on the cylinder. An isolated periodic solution will be called a limit cycle. According to A.L. Neto [8] the following problem was proposed by C. Pugh (see also N.G. Lloyd [7]): "Does there exist a number N = N(n) which depends only on the degree n such that (1) has at most N limit cycles?" For equations of degree n = 2 and n = 3 the problem was considered by S. Smale and A.L. Neto (see [81): - for the Riccati equation (n = 2), the number of limit cycles is at most 2 - for Abel's equation (n = 3) the number of limit cycles is at most 3, provided that a3(t) does not change sign.

On the other hand, it was shown in [8] that in general the answer to Pugh's problem is negative: in fact, for n > 3 equation (1) may have an arbitrary number of limit cycles if no restrictions on the coefficients are made. Neto gave examples of Abel equations which have, for any given integer k, at least k closed solutions, with coefficients ao, ... , a3 which are polynomials in t or in cos(irt) and sin(irt). It then becomes a natural problem to consider Pugh's problem under suitable restrictions on the coefficients of (1). In [4] Yu. Ilyashenko obtained the following result: If (Il) ao,... , an-1 are 1-periodic continuous functions and (12) maxtelo,1] lai(t)I 5 c j = 0,...,n - 1

M. Calanchi and B. Ruf

54

then the generalized Abel equation du

dt = un + an-1(t)un-1 + ... + al (t)u + ao(t)

(2)

has at most A(n, c) limit cycles, with A(n, c.) < 8 exp { (3c + 2) exp

[(2c+3J}.

The proof of this result is based on the study of the Poincare map associated to (2).

The number A(n, c) is called a Hilbert type number for equation (2). This is motivated by the fact that the problem of finding bounds for the number of limit cycles is related to the 16t" Hilbert Problem; indeed, it can be viewed as a particular case of it (see [8]). In the second part of Hilbert's 16th problem we read "This is the question as to the maximum number and position of Poincare's boundary cycles ("cycles limites") for a differential equation of first order and first degree of the form dy (3) dx = Q P where P, Q are entire rational integral functions of the n-th degree in x, y ", cf. D. Hilbert, [3]. In our main result we consider equation (1) for a different class of coefficients than in [5], and we show that the maximal number of closed solutions (limit cycles for the periodic case) for this class is bounded by n. Notice that the number n is optimal;

indeed, it is easy to see that there exist constant coefficients an, ... , al, ao = f such that the equation (1) has exactly n (constant) solutions. More precisely, we prove the following result:

Theorem 1. Suppose that n is odd. Assume that f, a1, ... , an : [0,1] -+ 1R are continuous, and that min(o.11 lan(t)I > a, for some a > 0. Then there exists some 03 > 0 such that if maxio,j] la=(t)I < /3,i = 1,...,n - 1, then (1) has at most n closed solutions, for all right-hand sides f (t) E C[0,1].

We also give a direct application of this result to the 16th Hilbert Problem: Theorem 2. Suppose that the polynomial vector field Y(x, y) = (P(x, y), Q(x, y))

has a unique singular point (0,0). Then Y can be written in polar coordinates Y = (Yr, Yo). If Yr and Ye satisfy, for some k E {0, 2,..., n - 11, with k even and n odd: 1) Ye(r, 0) = rk f (0), with f (9) > ' > 0 2) Yr(r.0) = rnan(9) + rn-lan_1(e) + + rkak(9) with 2a) an(9) > a > 0

2b) Iaj(9)I < 0, k + 1 < j < n - 1 (on ak(O) no further restrictions are required).

Hilbert Type Numbers for Polynomial ODE's

55

Then, if $ > 0 is sufficiently small, the number H(k, n) of limit cycles of the vector field Y is bounded by

H(k, n) < n - k.

Idea of the proof of Theorem 1 To study the number of closed solutions of equation (1) we make use of a Lyapunov-

Schmidt reduction. We consider the Banach spaces E = {u E C' [0,1] : u(0) = u(1)} and F = C[0,1]. We split the space F into the direct sum F = F ® IR, where F = {u E F : f f u dt = 0}, and introduce the projections Q : F - IR and P : F --+ F defined respectively by r1

Qu=J udt and Pu=u-Qu, 0

and we write u E E as

u=s+y=Qu+Pu.

Then u is a closed solution of equation (1) if and only if u = s + y solves the following system of equations

(Pl)

y+Pg(s+y) = Pf = fl

Qg(s + y) (Q1) We solve the first equation for fixed s E IR:

= Qf

(4)

Proposition 1. For every fixed s E IR there exists a unique solution y = y(s) of equation (P1). Proof. (Idea) To prove existence for (Pl) we apply the Leray-Schauder principle (cfr. [2], [6]) to the equivalent integral equation.

y(t) - y(0)

JO g(s + y)dr + t

J0

g(s + y)dr + J fld-r =: K(y).

(5)

0

Since fo y(t)dt = 0 we have

- y(0) = Thus, equation (Pl) becomes

j

K(y)dt.

y = K(y) - I K(y)dt := k(y), 1

0

(6)

where k : F -+ F is a compact mapping. In [1] the following a priori estimate is shown: there exists a number R > 0 such that

y = \k(y) , A E (0,1)

(7)

implies IIyI I.< R. Hence the existence of a solution follows by the Leray-Schauder principle.

M. Calanchi and B. Ruf

56

To prove the uniqueness, we assume that for some s E IR fixed there are two solutions y, z E E of (PI), and we prove that since y, z satisfy the same Cauchy problem, then z y. By a standard application of the Implicit Function Theorem (see [101), we can prove that the function

sEIR"y(s)EE

(8)

is real analytic; in particular it is C'°. For s E IR, let now y(s) = y(s, fl) denote the unique solution of equation (P1). We insert this solution into equation (Q1) and obtain an equation in one dimension:

L

g(s + y(s))dt =

f(t)dt

(9)

Jo

called the "reduced equation". To study equation (9), we introduce the following function r : lli. - IR.

r(s) = f 1 g(s + y(s))dt - J f (t)dt. 1

a

(10)

u

We want to prove that equation (10) has at most n zeros. Clearly, this is sufficient to show that equation (1) has at most n solutions. First, we study the local behavior of I'. Suppose that r(so) = 0. If r(so) 54 0, then I' is locally invertible and r(s) = 0 has a unique solution in a neighborhood of all.

Suppose now that I "(so,) = 0; then uo = so + y(so) is a singular point of the mapping it + g(u), i.e., i, + g'(u(,)v = 0 , for some v E E \ {0}. (11) Indeed, differentiating equation (Pl) with respect to s, we have y,.(s) + P(g'(s + y(s))(1 + y.(s))1 = 0 , V s E IR.

(12)

Since

V(s) = f

1

g'(s + y.)(1 + y.(s))dt

(13)

u

we get by adding (12) and (13) r'(s) = II. + g'(s + y(s))(l + y,(s)) = v + g'(u)v.

(14)

Equation (14) shows that if r(so) = 0, then v = 1 + y.(so,) is the unique solution of (11) with fo vdt = 1, and vice versa. Indeed, a general solution of (11) has the form

ce-.f,'s'(uo)dr. Thus,

e-fu s (uo)dr yx

fo1 e- fo s'(LO)drdt

Also, for the corresponding adjoint equation we have

-i'' + 9 (uo)v' = 0 ,

with v' = d efit 9 (uo)dr

(15)

Hilbert Type Numbers for Polynomial ODE's

57

Choosing d = fo e- fo g'(uo)dT dt we have v' = 1/v. The crucial step consists now in showing that the stationary points of r are

degenerate to at most degree n - 1, that is we prove

Proposition 2. Suppose that the assumptions of Theorem 1 hold. If r(so) = 0, then r(n)(so) > C > 0. Proof (Idea) Differentiating equation (14) repeatedly with respect to s we get by induction

= y(k) + y

r(k) (s)

aiiui-ly(k)+

jj

+ ui==2 aiu'-2 LgEQk(k-1)

p(q)vg1 (y(2))g2 ... (y(k-1))gk-j

aiu'-(k-1)

+ Ei==k-1 + Ei==k aiu°-k2(t

ugEQk(2)

(16) p(q)vq, (y(2))Q2

- 1) ... (i - (k - 1))vk,

where v = 1 + y Qk(m) = {q = (ql,... , qm) E 10,

... , k}', E'j iqi = k}, and

p(q) > 0 are some integer coefficients.

We now multiply equation (16) by v" = 1/v and integrate. Setting a* _ f1 v'dt and noting that

J0

1 y(k)v'dt +

J0

1 Eaiiu'-ly(k)v'dt = 1 1 y(k)(-v' +g'(uo)v')dt = 0 i=1

0

we obtain aiui-2EgEQk(k-1)p(q)v41-1 ...(y(k-1))qk-,dt+

1'(k)(s)a* =E 2f0

Ek-1 fo ai7L'-(k-1) F +> k f0 ai7Li-ki(i - 1) ... (i - (k - 1))vk-Idt.

gEQk(z)p(q)vq,-1(y(2))Q2

(17)

In particular, for k = n we have

r(n)(s)a.

=E 2fo +E n-1 f0

a'ui-2L+gEQn(n-I)p(q)vq,-1 ...(y(k-1))qk-,dt

a.-(n-1)

EgEQk(2)p(q)v9,-1(y(z))mdt

+ fo ann!vn-1dt. Note that the last term in equation (18) can be estimated from below by fa' ann!vn-1dt > ma! fo 1 vn-1dt > an!(J 1 vdt)"-1 = an!

(19)

0

Therefore, we can conclude the proof of Proposition 2 if we can show that all the other terms in (18) can be bounded in terms of /3, where Q is given in the hypothesis of Theorem 1. The central part to obtain these estimates is contained in the following proposition (for the proof we refer to [1]).

M. Calanchi and B. Ruf

58

Proposition 3. Under the assumptions of Theorem 1 and supposing that C,6 < 1/2, where en = "i 2 1 (1 + 7), with ry = max Ian(t)I, one has:

1) 1/2 < v(t) < 2, 1/2 < v* (t) < 2, and a' = fo v'dt < 2 2) fo u"-'dt < to(n - 1) 3) IJy(k)I1«, < c(3, k = 1,2,...,n. Using these estimates and repeated applications of the Holder inequality, one

then proves for r(")(s) that

r(n) (s) > a (an! - c(n) (i) > 2(an! - c(n)(3).

(20)

Clearly, the last term is positive for (3 > 0 sufficiently small, and thus Proposition 2 is proved. 0 By Proposition 2 we therefore have: if r'(so) = 0

,

then r(")(so) > 0.

It is easy to see that this implies that r(s) = 0 has locally at most n solutions. Indeed, for s new so we can write by Taylor's theorem ro) (so) (s 1'so)' + ri"l(so + 0) (s _' o)"

r(s) _ =7

Since r(n) (so + 0) > 0 for s near so, we conclude that there exists a neighborhood

U of so such that r(s) = 0 has at most n solutions in U. To obtain a global result, we employ the following proposition, which is proved in [9]:

Proposition 4. Suppose that k : IR -4 IR is a smooth function satisfying

(i) k'(x) > -8, for all x E 1R (ii) for any y E IR with k'(y) = 0 holds I0)(y)I < c, i = 2,...,n - 1 (iii) let IQ = {x E IR : k'(x) < a}, and suppose that Ik"(x)I < b and k(n)(x) >

d>O, dxEIQ.

Then, if 6 > 0 is sufficiently small (for fixed positive constants a, b, c, d), the equation k(x) = t: has for any f E IR at most n solutions. Thus, to conclude the proof of Theorem 1 it remains to show that the function r satisfies the hypotheses of Proposition 4, for (3 > 0 sufficiently small. For the proof of this we refer to [1].

Remarks 1) We suspect that the oddness assumption on the degree n of equation (1) is not necessary, in view of the result of Yu. Ilyashenko in [5). The main problem for even

degrees n is that the estimate in Proposition 3, 2) is not sufficient, since u may change sign.

Hilbert Type Numbers for Polynomial ODE's

59

2) Note that the estimate of Yu. Ilyashenko (2) yields for Jai(t)I < c with c -+ 0: A(n, c) < 8 exp (2 exp(3°+1/2))

.

To compare with our result: for example, for n = 3 Ilyashenko's result gives for Iai(t)I < c, i = 0, ... , n - 1, with c small A(3, c) < exp(2 exp(40.5))

107-10'7

while our result gives for I ai (t) I < c, i = 1,. .. , n - 1, and no restriction on ao(t) = f (t) A(3, c) < 3 .

As already mentioned before, 3 is optimal for polynomials of degree 3.

Proof of Theorem 2 Assumptions 1) and 2) imply dr Y r

= - = rn_k _a (B) +... + &k(O)

where a, (O) = aj(B) If (0). The result follows by Theorem 1, since an(B)

> mIf(B)I > 0 ,

and provided that

Iai(0)I <-

Q ,

k+1<j
with f(3/y sufficiently small.

Example. Suppose that a vector field Y = (P, Q) in the plane is given as follows: P(x, y) Q(__, y)

= -y + x'' +xy4 + Eil=1 (bix` + cixy'-1) + dl y = x +x4y + y5 + E4=1 (biyxi-1 + c;y`) + d2x

where Ibi I, Ici I, Idi I <)3 sufficiently small. Then some easy calculations yield

Ye(r, 8) = cos2(0) + sin2(0) + d2 cos2(0) - dl sin2(0) = f (O) > 1 - 2,3 > 0 4

and

Y,

= rs a;,(0) +

ri ai(B) i=1

with a5(0) _ [cos4 (B) + sin4(0)] > a > 0 and Iai (B) I < /3,

i = 1,...,4.

Hence the vector field (PQ) satisfies the hypotheses of Theorem 1, with k = 0, and thus, if /3 > 0 is sufficiently small, the vector field (P, Q) has at most 5 closed limit cycles. We may view the example as follows: consider the vector field of the harmonic oscillator (Po, Qo) = (-y, x). This is the degenerate situation, in which all solutions are closed. If we add the term of order five (x5 + xy4, x4y + y5), then all closed

60

M. Calanchi and B. Ruf

solutions disappear and the point (0,) becomes a degenerate singular point of order 5. If we now also add a small perturbation, then Theorem 2 says that there can be at most 5 closed orbits for the resulting vector field.

References [1] Calanchi, M., Ruf, B., On the number of closed solutions for polynomial ODE'S and a special case of Hilbert's 16`h problem, Adv. Diff. Equ. 7, 2002, p. 197-216 [2] Deimling, K., Nonlinear Functional Analysis, Springer, 1985 [3] Hilbert, D., Mathematical Problems, Bulletin AMS, 8, 1902 (4) Ilyashenko, Yu., Finiteness theorems for limit cycles, Amer. Math. Soc., Providence, RI, 1991 [5] Ilyashenko, Yu., Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity 13, 2000, p. 1337-1342 [6] Krasnosel'ski, M.A., Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 1963, p. 123-140 [7] Lloyd, N.G., The number of periodic solutions of the equation .z = zN +p1(t)ZN-1 + + pN(t), Proc. London Math. Soc. 27, 1973, p. 667-700 [8] Neto, A.L., On the number of solutions of the equation AE = E, o a; (t)x', 0 < t < 1, for which x(0) = x(1), Inventions Math., 59, 1980, p. 67-76 [9] Ruf, B., Bounds on the number of solutions for elliptic problems with polynomial nonlinearities, J. Diff. Equ. 151, 1999, p. 111-133 [10] Tarantello, G., On the number of solutions for the forced pendulum equation, J. Diff. Equ. 80, 1989, p. 79-93

Marta Calanchi, Bernhard Ruf Dipartimento di Matematica Universith degli Studi Via Saldini 50 1-20133 Milano, Italy

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 61-77 © 2003 Birkhauser Verlag Basel/Switzerland

S2-type Parametric Surfaces with Prescribed

Mean Curvature and Minimal Energy Paolo Caldiroli and Roberta Musing Abstract. Given a function H E C'(R3) asymptotic to a constant at infinity, we investigate the existence of nontrivial, conformal surfaces parametrized by the sphere, with mean curvature H and minimal energy.

1. Introduction In this paper we deal with two-dimensional parametric surfaces in 1R3 of prescribed mean curvature. In particular we are interested in the existence of surfaces parametrized by the sphere S2 in case the prescribed mean curvature is nonconstant. This problem is motivated both for its geometrical interest, and by the fact that it arises as a mathematical model in capillarity theory to describe interface surfaces in presence of external forces (see [4], [6]).

The problem admits an analytical formulation, as follows: given a smooth function H: H3 -+ R, find a nonconstant, conformal function w: R2 -+ 1R3, smooth as a map on 92, satisfying:

Aw=2H(w)wxnwy in R2

fIt, Ivw12 < +00

Here subscripts mean partial differentiation, the symbols 0 and V represent respectively the Laplacian (in R2) and the gradient, and A denotes the exterior product in R3. A solution w of (1.1) satisfying the above listed requirements will be called shortly an H-bubble. Note that problem (1.1) is invariant with respect to the conformal group. This means that we deal with a problem on the image of w, rather than on the mapping w itself. When the prescribed mean curvature is a nonzero constant H(u) = Ho, Brezis

and Coron [2] proved that the only nonconstant solutions to (1.1) are spheres of radius 1Hol-' anywhere placed in R3. Work supported by M.U.R.S.T. progetto di ricerca "Metodi Variazionali ed Equationi Differensiall Nonlinearl" (cofin. 2001/2002).

P. Caldiroli and R Musina

62

Only recently, the case in which H is nonconstant has been investigated (see [31). Here we are interested in a class of curvature functions which are asymptotic

to a constant at infinity. This case has been already considered in [3], but in the present work we follow a different strategy that allows us to carry out some improvements, to get more general results and to exhibit new features in the problem. The first important remark on problem (1.1) concerns its variational nature.

That is, solutions to (1.1) can be detected as critical points of a suitable energy functional. More precisely, the energy functional associated to problem (1.1) is defined by

EH(u) =

2

JR2 IVU12 + 2L QH(u) uA u,, ,

w here Q11: R3 -p R3 is any vector field such that div QH = H. For several reasons

it is meaningful not only to look for H-bubbles, but also to study the existence of H-bubbles with minimal energy. Hence, we are lead to consider the following problems:

1) calling BJl the set of H-bubbles, find conditions on H ensuring that BH 54 0; 2) assuming that BH 0 0, study the minimization problem: PH = of EH. Bil

Having in mind a curvature function H which is asymptotic to a constant at infinity, we start with the simplest case in which H is assumed to be constant far out.

Theorem 1.1. Let HE C' (R3) satisfy (hi) H(u) = H. E R \ {0} as Jul > R, for some R > 0, I H(u)u - 3QH(u)I < z (h2) Then there exists w E BH such that EH (w) = PH. Moreover PH < 33 yam.

The assumption (h2), roughly speaking, measures how far H differs from the constant value Ham. In principle, it depends on the choice of the vector field QH which is not uniquely defined. In fact, we can replace the hypothesis (ha) with a weaker, but less explicit condition which actually depends just on H or, more precisely, on the radial component of VH (see Section 3). Since, by (hi) one immediately has that BH is nonempty, the problem reduces to investigate the semicontinuity of the energy functional along a sequence of H-bubbles. As shown by Wente [10], in general EH is not globally semicontinuous, even if H is constant. However, along a sequence of solutions of (1.1), thanks to the condition (h2), semicontinuity holds true. As a next step we want to give up the condition (hi ), by considering the more general case of a prescribed mean curvature which is just asymptotic to a constant at infinity. Clearly, now it is not known if BH is nonempty. The strategy consists in approximating H with a sequence of functions which are constant far out and for which Theorem 1.1 can be applied, and then, passing to the limit.

S'-type Parametric Surfaces

63

Theorem 1.2. Let H E C'(R3) satisfy (h2) and (h3) H(u) H. as Jul oo, for some H. E R, (h4) SUPUER IVH(u) ul < +00.

Then there exists a sequence (Hn) C C' (RI) such that H -p H uniformly on R3 and, for every n E N, BH is nonempty and EH = 1Hn is attained. If, in addition

(h5) lim inf IAH < 3 then BH is nonempty, µH < lim inf pH.. and µH is attained.

Actually, the condition (h5) in the above statement is not truly satisfying, since it seems to depend on the approximating sequence (Hn), and it is hard to check. Therefore, as a third step, we want to express the value µH (defined provided

that BH 0 0) as a minimax level for the energy functional, in order to be free of any information on BH. To do this, we need to study the geometrical properties of the energy functional. First of all, we note that, since H is bounded, EH (u) turns out to be well defined (by continuous extension) on the Sobolev space Ho (D, R3), where D is the unit open disc in 1R2. Furthermore, the functional fRs QH(u) u., A uy has the meaning of H-weighted algebraic volume, it is essentially cubic in u, it depends just on H (and not on the choice of QH), and it satisfies a generalized isoperimetric inequality. Therefore EH is expected to admit a mountain pass type geometry on Ho (D, R3), with u = 0 as a local minimum point. Let us introduce the value: CH =

inf

sup EH(su),

uEHa(D,R3) s>0

(1.2)

uO0

which represents the mountain pass level along radial paths. Now, the existence of minimal H-bubbles can be stated as follows. Theorem 1.3. (see [3]) Let H E C'(R3) satisfy (h3), (h6) SuPUER3 IVH(u) u ul < 1, (h7) CH <

Then there exists an H-bubble w with Ell (w) = cH = 1H

The assumption (h6) is a stronger version of the condition (h2) and it is essentially used to guarantee that the value cH is an admissible minimax level. The key point is that, if H is constant far out and verifies (he), then cH = µH. But CH is well defined even without information on BH. Hence, when (he) holds true, the inequality (h7) is the natural equivalent version of the condition (h5). We point out that, although the assumptions in Theorem 1.3 are stronger than in Theorem 1.2, they are easier to check, in general. For instance, (h7) is verified whenever H > H,o on a suitably large set. Moreover, the condition (he) allows us to find also a lower bound for the minimal energy µH and precisely µH >

47r

3IIHII2

P. Caldiroli and It Musina

64

We finally remark that all the previous theorems give no information about the position of the minimal H-bubble, but only the information on its energy is available. Hence, the same results hold true if in all the statements the assumptions are fulfilled by H( + p) for some p E 1R3. The work is organized as follows: in Section 2 we fix the notation and we state some preliminaries in view of a variational approach to problem (1.1). Section 3 contains the proof of a weaker version of Theorem 1.1. Moreover a semicontinuity

result is discussed. Section 4 concerns the case H asymptotic to a constant at infinity. In particular we show a generalization of Theorem 1.2 and we make some remarks about Theorem 1.3. For the complete proof of Theorem 1.3 we refer to [3].

2. Preliminaries In this section we introduce a variational setting suited to study problem (1.1).

We note that all the statements of this section hold true assuming just H E C' (R3) fl L

or, sometimes, even H E L°°(1R3)

Firstly, as a variational space we will take the Hilbert space of functions u: JR2 R3 with finite Dirichlet integral, which is isomorphic to H'(S2,1R3). It S2 given by

can be defined as follows. Consider the mapping w°: JR2

µx w°(z) =

JA1!

1-µ

µ = µ(z) =

,

2

1 + IZI2

(2.1)

being z = (x, y) and Iz12 = x2 + y2. Observe that w° is the inverse of the standard stereographic projection, it is a conformal parametrization of the unit sphere (centered at 0). According to the definition given in the Introduction, w° is an H-bubble, with H =- 1. Then, set X = {cp o w°: 1p

E H' (S2,1R3)}

.

The space X naturally inherits a Hilbertian structure from HI (S2, 1R3). In particular, the inner product in X can be defined as (u,v) = (u0Tr,voar)H1(sx Ra)

where Ir: S2 -+ 1R2 is the standard stereographic projection. Explicitly, one has

(u,v) =

f (Vu. Vv+µ2u.v) R

,

with µ given by (2.1). Hence the space X can be equivalently defined as X = {u E H11o,(R2, R3) :

f

(IDuI2 + c2IuI2) < +oo} Z

X turns out to be a Hilbert space, endowed with the norm IIuII = (u,u)1/2. For

every uEXset u(z)=u(z),where z= I .Then, fEXandllull=IIull.In

S2-type Parametric Surfaces

65

particular, for every R > 0

r

1vu12 = J

=I
IDu12

(2.2)

.

=> R

For every u E X we denote the Dirichlet integral by

D(u) = 1 f IVu12 . 2

R2

Now, given H E C' (R3), we define the H-volume functional VH : X nL°° - JR by setting V11(u) =

QH(u) - us n uy fU2

where QH: R3 -+ R3 is any vector field such that div QH = H. Note that VH can be defined just assuming H E L. In case H(u) - 1, one has QH(u) = 3u, and the functional VH reduces to the classical volume functional which satisfies the standard isoperimetric inequality. In fact the following generalization holds, as proved by Steffen in [9]. Lemma 2.1. If H E LO°(R3) is bounded on J3 then there exists CH > 0 such that IVH(u)1213 < CHD(u) for every u E X n L°°

.

(2.3)

Furthermore, Steffen in [9] proves that the functional VH admits a continuous extension on X and then, (2.3) holds true also for every u E X. As far as concerns the differentiability of VH, the following result is useful. This result is also known and we refer to [7] for a proof.

Lemma 2.2. If HE C'(R3) n L°°, then for every u E X and for h E the directional derivative of VH at u along h exists, and it is given by

R3)

dVH(u)h=f2 R

Moreover, for every u E X n L°° one has

dVH(u)u= f R2

Remark 2.3. By Lemma 2.2, the functional VH does not depend on the choice of the

vector field QH, but only on H. Moreover, if H is constant, then VH E C' (X, R), while if H is nonconstant, in general the functional VH is not even Gateauxdifferentiable at every u E X. The following result will be also useful in the sequel.

Lemma 2.4. If H E C'(R3) is such that H(u) = 0 as Jul > R, then for every

uEXnL°° one has VH(u) = VH(7rR O U) ,

where 7rR: R3 -+ BR is the retraction on the ball BR = {u E R3 : Jul < R}.

P. Caldiroli and R. Musina

66

Proof. Take Qy(u) = my(u)u, with mH(u) = f0 H(su)s2ds. Setting uR = 7rRou, one has

VH(UR) =

IRa

J

(u') FU R

J

z:Iu(z)IGR }

3

d

r(mH(u) - MH (uR) = VH(u) z:Iu(z)I>R) \

R

Fu-3

u ux A uy

.

But one can easily check that if Jul > R then

my(u) = MH (iR(u)) RI3 Hence VH (uR) = VH (U).

The energy functional £H : X n L°°

R is defined by

£H(u) = D(u) + 2VH(u)

.

By virtue of the above discussed properties for VH, the functional £H admits a continuous extension on X. Remark 2.5. The failure of lower semicontinuity of Vy (and then of £H) can be shown by the following example, essentially due to Wente [10]. Take, for simplicity, H =_ 1, let w°: R2 --+ S' be given by (2.1) and set wE(z) = w°(ez). Then for every

A E R one has D(Aw') = A2D(w°) = 4aa2, VH( However wE

E) = A3Vy(w°) = -A3.

w°(0) = e3 weakly in X, and £(Ae3) = 0.

Thanks to Lemma 2.2, critical points of £H correspond to weak solutions to problem (1.1). In the following statement we collect some facts about (weak) solutions to (1.1).

Lemma 2.6. Let H E C'(R3) fl L. If w E X is a weak solution to (1.1), i.e., d£H(w)h = 0 for every h E Cc°(R2,R3), then: (i) w E C3(R2,R3) is a classical solution to (1.1), (ii) w is conformal, and smooth as a map on S2, (iii) diam w < C(1 + D(w)) where C is a constant depending only on IIHII,,.

All the above statements are known in the literature and we refer to [5J for a proof (see also [1] Theorem 4.10).

3. The case H constant far out The goal of this section is to prove Theorem 1.1. Actually, the main result here is a semicontinuity property for the energy functional along a sequence of critical points (Theorem 3.1). This result can be stated in a more general form that will

S2-type Parametric Surfaces

67

be useful in the sequel and that allows us to recover a weaker version of Theorem 1.1 (see Theorem 3.3). Firstly, given H E C'(P3) n L°°, let

JH(u) = VH(u) u and assume (h4), that is, JH E L°O (RI). Let us introduce the functional WH : X n L°° -+ llt defined by

WH(u) = j (3QH(u) - H(u)u) ux A uy . Z

According to Remark 2.3, since div(H(u)u - 3QH(u)) = JH(u), the functional WH depends just on JH, and not on QH. In particular, WH = WH, if H and H' differ by an additive constant. Let us set

D(u)

inf } - uEXnL°° IWH(u)I It is also convenient to introduce for every R > 0 the value D(u) inf Art.R = "EX

11U11-
IWH(u)I

Clearly, the mapping R --+ AH,R is decreasing and AH,R - AH as R +oo. Since WH(u) = 3VH(u) - dVH(u)u on X n LOO, if u E X and IIuII,,.S R, then

I 1-

2 AH.R

) D(u) < 3EH(u) - dEH(u)u .

(3.1)

When AH > 2 (or AH.R > 2), (3.1) is useful to get a bound of the Dirichlet norm in terms of the energy. Note that, if vH = 2 IH(u)u - 3Q,r(u)I, then AH > 1/vH. In particular, (h2) implies AH > 2. The role of AH (or AH.R) is made clear by the following fact, which states a semicontinuity property along a sequence of solutions.

Theorem 3.1. Let (H,,) C C'(R3), H E C'(R3) and R > 0 be such that: (i) H,, -. H uniformly on BR = {u E R3 : Jul < R}, (ii) AH,,.R > 2 for every n E N, (iii) for every n E N there exists an Hn-bubble w" with IIw"II°O 5 R, IOw"(0)I = Ilow"II. = 1, and Supn IIVw"II2 < +00. Then there exists an H-bubble w such that, for a subsequence, w" - w weakly in X and strongly in C/., (R2, R3). Moreover EH (w) < lim inf EH (w")

Proof. We split the proof into some steps.

Step 1. There exists w E X n C' (R2, R3) such that, for a subsequence, w" - w weakly in X and strongly in C11"°(R2, R3). From the assumption (iii), there exists w E X such that, for a subsequence, w" - w

weakly in X. Now, we show that for every p > 0 and for every p > 1 the sequence

P. Caldiroli and R. Musina

68

(w") is bounded in H2'p(Dp, R3). To this extent, we will use the following regularity estimate (which is a special case of Lemma A.4 in (3]).

Lemma 3.2. Let H E CI(R3) n L°°. Then there exists e = e(IIHII,,) > 0 and, for every p E (1, +oo) a constant Cp = C,,(II HII.) > 0, such that if u: Sl: -. 1R3 is a weak solution to Du = 2H(u)u= A u, in 1 (with 11 open domain in 1R2), then 11VU11L2(DR(z)) <_ E

11VU11H"P(DR/2(z)) s CpR°-2IIou1IL2(DR(z))

for every disc DR(z) C fl, with R E (0,1]. Since by (iii) we are interested in the convergence of a sequence in the region BR, using (i), we may assume that H,, -' H uniformly on 1R3. Hence, by Lemma 3.2, for every n E N there exists en > 0 and Cp,n > 0 for which IIVwnIIL2(DR(z)) < En = IIV .' IlHI.v(DR/z(=)) < Cp.nRP-2llownl1L2(DR(z))

for every z E R2 and for every R E (0,1]. By (i), one has en > e > 0 and Cp,n < Cp for every n E N. Fix p > 0. Since II Vwn II ,. = 1, there exists R > 0 and a finite covering {DR/2(z;)}:EJ of DP such that IIVwnIIL2(DR(z,)) < e for every

n E N and i E I. Since Ilwnllo° < R, we have that IIwn1IH1,P(DR12(z,)) C Cp.R for some constant Cp,R > 0 independent of i E I and n E N. Then the sequence (w") is bounded in H2'p(Dp,R3). For p > 2 the space H2.p(Dp,R3) is compactly embedded into C' (Up, R3). Hence wn -+ w strongly in C1(Dp, R3). By a standard diagonal argument, one concludes that wn -, w strongly in Step 2. w is an H-bubble. For every n E N one has that if h E Cco°(R2,R3) then C110C(R2,R3).

=0.

IV VwnVh+2 fR2

By Step 1, passing to the limit, one immediately infers that w is a weak solution to (1.1). According to Lemma 2.6, w is a classical, conformal solution to (1.1). In addition, w is nonconstant, since IVw(0)I = lira IVwn(0)I = 1. Hence w is an H-bubble. Step 3. SH(w) S By Step 1, for every r > 0, one has £H, (wn, Dr) - Cm (w, Dr)

(3.2)

where we denote

jlVuI2 +2

J

(and similarly for .6H (u, 12)). Now, fixing e > 0, let r > 0 be such that

8H(w,1R2\Dr) <E IVw12<E.

(3.3) (3.4)

S2-type Parametric Surfaces

69

By (3.3) and (3.2) we have

< CH(w,Dr)+ = EH.(W",Dr)+E+0(1)

CH (w)

= EH. (w") - EH. (wn, R2 \ Dr) + E + o(1)

(3.5)

with o(1) , 0 as n - +oo. Now, by (iii) we have

f

Jwn

BD,.

aWnn

(IVwn12 +wn .

av

R2Dr

Nn)

J 2\Dr Ivw2+2J

R2\D,

= 3EH (wn, R 2 \ Dr) -

-2

r

fR2\Dr

1

2 R2\D,

wIVn12 (3.6)

Gn(wn) wi n Y,

where we set Gn (u) = 3QH, (u) - Hn (u)u. Hence (3.5)-(3.6) give 1

EH(w)

3 2

an

8D,.

wn

-av

1

f

6J R2\Dr

1vWn12

Gn(Wn) W2 / Wyn + E + 0(1)

3 R2\Dr Since wn --' w strongly in Co"(R2,1R3) and since w is an H-bubble, we have n n

JBD,. w

8wn

al/

8Dr

(3.7)

W aw -V

R2\Dr JR2\Dr

(W. AW + Ivw12)

(2H(w)w w= A wy + I VW I2)

(11w11.IIH11L=(BR) + 1)

fR2\D, (11w11.IIHI1t=(aR) + 1) E

IVW12 (3.8)

because of (3.4). Now we estimate

1J

1vwn12-2J

n

3 R2\Dr 6 R2\Dr For every n E N, let hn E H1(Dr,R3) be the harmonic extension of wn18Dr, and set

un(z) =

{hui(z)

as Iz, < r wn(z) as Izl > r .

P. Caldiroli and R. Musing

70

Note that un E X, and IIu"ll. <- R, because IIw"Il, 5 R and IIh"IIL-(D,.) 5 IlwnllL-(8D,). Therefore, using (ii) we obtain (u")I < D(un) and then 2 1

6

1,2\D,

IQW"l2

3 R2\D,

G. (w") w= A wy

-13 D(u") -

2 WH. (Un) 3

+'I IVh"I2+?J 6ID,

3D D I Gn(h") h" A by IVhnI2 + ,

3

(6 +

3IIHnllLoo(BR)R) Lr IVhnI2

(3.9)

Since, by Step 1, w' I aD,. -+ W aD, in C', one gets I

IVhnI2

Lr

IOhl2

./ D,

where h E H'(D,., R'3) is the harmonic extension of wI8D, Setting v(z) = w (57 ) as I z I < r, one has that vIOD,. = W I BD, and by (2.2)

I

lVh12

Dr IovIZ = 5

IVW12 J2\Dr

Therefore, by (3.4) and (3.9) we obtain that

-6 f,2\Dr Ivw"I2-?

3 R2\D, In conclusion, (3.7), (3.8), and (3.10) yield


(3.10)

EH(W) 5 EHn (Wn) + CE + o(1)

with o(1) - 0 as n - +oo. By the arbitrariness of f > 0, the thesis follows.

0

As a consequence of Theorem 3.1 we infer the following result which, in fact, is a generalized version of Theorem 1.1.

Theorem 3.3. Let H E C' (R3) satisfy (hl) and AH > 2. Then there existsW E BH such that Ell (w) = pj. Moreover µH < . 37MT

Proof. The assumption (hi) guarantees that BH is nonempty, since the spheres of radius I H,,.I - ' placed in the region Jul > R are H-bubbles. In particular, this implies that µH < . Now, take a sequence (W") C BH With EH(w") - PH. Since the problem (1.1) is invariant with respect to the conformal group, we may assume that IIVwnll,o = IVwn(0)l = 1. Since )H > 2, (3.1) yields that sup IIow"II2 < +00 .

(3.11)

S2-type Parametric Surfaces

71

We may also assume that

sup IJ""II, < +00 .

(3.12)

Indeed, by Lemma 2.6 and by (3.11), there exists p > 0 such that diam w" < p for every n E N. If lbw" ll < R + p, set w" = w". If lIw" II«, > R + p, then by the assumption (hi), w" solves Du = 2H,,.u= A us. Let pn E range w" be such that

Ipn1 =lIw"II<.Setgn=(1-

)pnandw"=w"-qn. ThenQw"11.
and 1w" (z) I > R for every z E R2. Hence, also (D" E BH, and EH (w") = EH- (w") _ EH (w"). The new sequence (w") satisfies (3.11), (3.12) and EH(w") --+ µy. Hence,

we are in the position to apply Theorem 3.1 (with H" = H for every n E N) and

0

then the conclusion follows.

4. The case H asymptotic to a constant at infinity In this section, we will prove the following result.

Theorem 4.1. Let H E C1 (R3) satisfy (hs), (h4) and aH > 2. Then there exists a sequence (H") C C'(R3) converging to H uniformly on R3 and such that for every If, in addition, (h6) holds, n E N there exists w" E BH with EH (w") = then there exists w E BH with EH (w) = µ1r

Since the condition AH > 2 implies (h2), Theorem 1.2 follows as a special case of the above result. The first step in order to prove Theorem 4.1 is given by the following result. Lemma 4.2. Let H E C' (R3) satisfy (h4) and H(u) -+ 0 as Jul exists a sequence (H") C Cc (R3) such that

+oo. Then there

H" H uniformly on R3, AH -+ AH .

(4.1) (4.2)

Proof. Let (Rn) and (bn) be two sequences in (0,+00) such that Rn -+ +00 and b" -+ 0. For every n E N let Xn E C' (R, [0,11) satisfy Xn (r) = 1 as r < R,,, Xn (r) =

0 as r > Rn + bn, and IX;,(r)I < 26n- 1. Let J, Jn: R5

1R be defined by J(u) =

VH(u) u and JA(u) = Xn(IuI)J(u). Noting that H(u) _ u

f+oo J(su)s-'ds (for

0), let us define for u E R3 \ {0} +00

Hn(u) = -J

Jn(su)s-' ds

.

We will see below that actually Hn can be extended continuously at u = 0. Step 1. Hn E C, (R3) and Hn -* H uniformly on R3.

P. Caldiroli and R. Musina

72

We have

j

X(slul)OH(su) u ds +00

Xn(lul)H(u) +

X;,(slul)IuIH(su) ds

+ 1

+00

Xn(lul)H(u) + I

X;,(t)H

i

dt .

Ct

IuI I

Hence, if Jul > Rn + bn then Hn(u) = 0. If Jul < Rn + bn then

f

+00

ul

Xn(t)H

(tj!)

R.,+6.,

dt

<

.L

IXn(t)H

\tlul/

< 2 sup IH(u)l . jul>Rn

Therefore, since H(u) -. 0 as Jul - +oo, the conclusion follows. AH. Thanks to the definitions given at the beginning of the proof, Step 2. X Hn we have WH (u) = Vi (u) and WHn (u) = Vrn (u) for every u E X n LOD. Hence, for

every u E X with hull. S Rn one has WHn(u) = WH(u), and consequently AH,Rn = XHn.Rn

Since XH.Rn - AH, in order to prove the thesis it is enough to show that XHn,Rn - AHn - 0 .

To this aim, fix e > 0 and let u" E X fl L°° be such that D(u") < WHn + E .

(4.3)

(4 . 4)

I WHn (un) I

Let 7r": R3 --+ BRn+sn be the retraction on the ball BRn+an By Lemma 2.4, since Jn(u) = 0 as Jul > Rn + bn, WHn (u") = WHn (1r" o u"). Moreover D(ir" o u") < D(un). Hence also a" o u" satisfies (4.4). In other words, we may assume Ilunllo < Rn + bn .

(4.5)

Now, set

v"=Tnu", with in=mini1,Ilu U. Note that Ilvnll. 5 Rn and thus D(v") > AH ,RnJWHn(v")I. If Tn < 1 then

yn + E > J

1

D(v")

Tn IWHn(u")I Xyn.Rn IWHn(yn)I

IWH,(un)l

Xyn.Rn Cl

IWHn(un)ll

C (4.6)

S2-type Parametric Surfaces

73

where

P. = IWH,,(v') -

)I

Hence there exists 0,, E [Tn,11 such that (1 - Tn) IdVJJR2 , (Bnun)unl

Pn

(1 - Tn)0n I

Jn( 0nun)uu/ up

G

(1 - rn)(R, +

<

(1 - T,)(R, + bn)IIJIIOo(AH., +E)IWHn(un)I .

bn)IIJII,D(un)

(4.7)

By (4.5) and by the definition of rn, one has that (1- Tn)(Rn +bn) < bn and then, (4.6) and (4.7) imply AH + E > AHn,R., (1 - bnhIJII°°(AH + E))

Hence, by the arbitrariness of f > 0, and since AH-R ? AH, one obtains 0 < AHn,R,, - AH < b,AH, IIJIIoo .

(4.8)

WH(u) for every u E X n L°°. This plainly Now, we observe that WH,(u) implies that limsupAH < AH. Then, since bn -+ 0, (4.8) implies (4.3). As a consequence of Lemma 4.2 we obtain the next result.

Corollary 4.3. Let H E CI(R3) satisfy (h3) and AH > 2. Then there exists a sequence (Hn) C C'(R3) satisfying (4.1), (4.2) and (hi).

Proof. Let H°(u) = H(u) - H. By Lemma 4.2 there exists a sequence (Hn) C H° uniformly on R3 and AHOn AH°. Take a sequence C, (R3) such that Hn (H°°) C IR \ {0} converging to H.f and set H,, (u) = Hn(u) + HHO. Then (Hn) satisfies the required properties, since AHo = AH and AHO = AH. In the sequel we will also need the following property. Lemma 4.4. Let (Hn) C C' (R3) be a sequence converging to some constant H° E R uniformly on BR. Then AH,,,R ' +00.

Proof. For every n E N set mn(u) = fo H,, (su)s2ds and Qn(u) = mn(u)u. Then div Qn = Hn, and for u E X, IIuII,,,. < R one has II3mn - HnIIL-(BR)R D(u) Since for every u E BR I3mn(u) - H°I <_ 11H- - H°II L-(BR)

the thesis immediately follows.

Now we study the behavior of a sequence of minimal Hn-bubbles when the curvatures sequence (Hn) approximates a given H.

P. Caldiroli and R. Musina

74

Theorem 4.5. Let H E C' (1R3) satisfy (h3) and let (Hn) C C' (1R3) be such that (i) Hn -+ H uniformly on 1R3, (ii) AH > A > 2 for every n E N,

(iii) for every n E N there exists wn E BH such that E.. (,n) < µH + En, With En-+0. If Ilwn I I

-' +oo then lim inf µH > 3r . If lim inf µH < 3

then there exists

W E BH with EH (W) = µH . Moreover µH < lim inf µH . Proof. Suppose that II W n II OO - +oo and lim inf µH < +oo (otherwise the result trivially holds). By the assumption (ii) and by (3.1), one has (for a subsequence)

sup IIVW"II2 < +00 .

(4.9)

Let pn = n(0) and set Hn(u) = Hn(u+pn) a n d wn = n-pn Then Dn is an Hn bubble, with wn(0) = 0. Because of the conformal invariance, we may also suppose Ilow"II,,. = 1. Moreover, thanks to Lemma 2.6, part (iii), the IIw"IIO' < R. sequence of diam wn is bounded, that is, there exists R > 0 such that

that Ivwn(0)I =

Furthermore, H,, -' H.. uniformly on BR and, by Lemma 4.4, ay R -+ +oo. Then, an application of Theorem 3.1 gives that lim inf E y (Wn) ?

But, since

3H2

pH +En, one gets 4a lim inf µH ? 3H2

Now, suppose lim inf µH < 3r- and let (wn) be the sequence given by (iii). Hence, up to a subsequence, one has sup I1w"II0, < +00 .

(4.10)

Moreover, as before, (4.9) holds and one can assume I Vw" (0) I = II Vwn 1I. = 1.

Therefore Theorem 3.1 can be applied again to infer that BH 36 0 and µH < lim inf µH . In particular µH < 3- yam. Finally we prove that µH is achieved. Note that (i) implies AH > lim sup AH (see the proof of Lemma 4.2). Hence, by (ii), AH > 2. Let (wn) C BH be such that CH(wn) -+ µH. We can apply the first part

of this Theorem (with H,, = H for every n E N), to deduce that (wn) satisfies (4.10). Moreover, since AH > 2, also (4.9) holds and a new application of Theorem 0 3.1 gives the conclusion.

Proof of Theorem 4.1 It is a consequence of Lemma 4.2, Theorems 3.3 and 4.5. 0 Now we discuss Theorem 1.3. We do not give the complete proof of Theorem

1.3 and we refer to [3] for all the details. Here we limit ourselves to sketch the procedure followed. Firstly one proves the result under the additional condition that H is constant far out. In this step, the main difficulty is to show that cH = µH, being CH defined

S2-type Parametric Surfaces

75

by (1.2). To do this, one introduces a family of approximating compact problems given by

div((1 + (P)Q

IDu12)°-10u)

1U=0

= 2H(u)ux n uy in D on 8D,

where D is the unit open disc in R2 and a > 1 (close to 1). This kind of approximation goes in the spirit of a well-known paper by Sacks and Uhlenbeck [8] and it turns out to be particularly helpful in order to get uniform estimates. More precisely, by variational methods, one finds that for every a > 1 (close to 1) problem (P)° admits a nontrivial solution u' E Ho'2°(D, R3). The family of solutions (u°) turn out to satisfy the following uniform estimates:

infllVu°112>0, ° SUP (11u°II + 11ou°112) < +00, a

EH(0°)

CH as a

1.

The limit procedure as a -, 1 is a delicate step. Indeed the weak limit u of (0°) is a solution of (P) JDu = 2H(u)ux n uy in D on 8D. 1,u=0 A nonexistence result by Wente [11] implies that u = 0. Hence a lack of compactness occurs by a blow up phenomenon. One introduces the functions

v°(z) = u°(za +e°Z) with z° E !R2 and e° > 0 chosen in order that 11vv°11. = IOv°(0)I = 1. One can

prove that there exists w E X such that v° -+ w weakly in X and strongly in C'°°(R2,iit3), and w is a nonconstant solution of Ow = 2,\H(w)w., n wy on R2 ,

for some A E (0, 1]. The assumption (ha) enters in an essential way in order to conclude that A = 1 and CH (W) < CH. Again the condition (he) gives that if BH 0 0 then µH > CH. Hence one concludes that EH(w) = µH. As a last step, one removes the additional assumption H constant far out following an argument similar to the proof of Theorem 4.1. We conclude this section by showing a lower bound for µH under the condition

(h6).

Proposition 4.6. Let H E Cl(R3)nL°° satisfy (ha). If BH 0 0, then µH >

3

H

.

The proof of Proposition 4.6 is based on the following argument. Since H E L°O(R3), by Lemma 2.1, the value 3H

u EX IVH(u)Il

(4.11)

76

P. Caldiroli and R. Musina

is a well-defined, positive number (apart from the trivial case H - 0). By the definition (1.2) of CH, and by (4.11) one easily obtains that

CH(S3) 4 As already mentioned, the condition (he) implies that if 8H # 0 then µH > cH. Hence the conclusion of Proposition 4.6 follows by the next estimate. Lemma 4.7. If H E C°(R3) n L°° then

=

0-67r

IH 11213 00

1

Proof. As proved by Steffen in (91, for every u E X there exists a measurable function i : R3 - R satisfying the following properties: (i) i takes integer values; (ii) i E L3/2(R3) and ,fR3 (36a)- D(u) ; (iii) VII (u) = fR;, iuH. Note that (i) and (ii) imply that i E L'(1R3) and then fRa iuH is well defined whenever H E L°°(R3). Using the properties (i)-(iii), we have IVH(u)I < IIHII,,IIiuIILI < (36rr)-112IIHIIaV(u)3/2, and then SH > ' 36nIIHII0013. Now let us prove the opposite inequality. For every p E R'3 and 6 > 0 let u6,p = 6w° + p, with w° defined by (2.1). Note that D(u6.p)

Vti(u°'p)

= 4x62

= J 6(p) H() d.

Hence

SH < lim

D(u6'p)

6_° IVH(u6,p)I2/3

_

0-6-7r

IH(p)I2/3

By the arbitrariness of p E !R3, one infers that Sit S 3 36,rflHII0

2/3-D

11

References (11 F. Bethuel and O. Rey, Multiple solutions to the Plateau problem for nonconstant mean curvature, Duke Math. J. 73 (1994), 593-646. [21 H. Brezis and J.M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Rat. Mech. Anal. 89 (1985), 21-56. [31 P. Caldiroli and R. Musina, Existence of minimal H-bubbles, Comm. Contemp. Math. 4 (2002), 177-209. [41 P.R. Garabedian, On the shape of electrified droplets, Comm. Pure Appl. Math. 18 (1965), 31-34. (51 M. Grater, Regularity of weak H-surfaces, J. Reine Angew. Math. 329 (1981), 1-15.

S2-type Parametric Surfaces

77

[6[ A. Gyemant, Kapillaritat, in Handbuch der Physik, Bd. 7., Springer, Berlin (1927). [7[ S. Hildebrandt and H. Kau], Two-Dimensional Variational Problems with Obstructions, and Plateau's Problem for H-Surfaces in a Riemannian Manifold, Comm. Pure Appl. Math. 25 (1972), 187-223. [8[ J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. Math. 113 (1981), 1-24. [9[ K. Steffen, Isoperimetric inequalities and the problem of Plateau, Math. Ann. 222 (1976), 97-144. [10[ H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318-344. [11] H. Wente, The differential equation Ax = 2(x Ax,) with vanishing boundary values, Proc. Amer. Math. Soc. 50 (1975), 113-137.

Paolo Caldiroli Dipartimento di Matematica Universit> di Torino via Carlo Alberto, 10 1-10123 Torino, Italy

E-mail address: [email protected]. it Roberta Musina Dipartimento di Matematica ed Informatica University di Udine via delle Scienze, 206 1-33100 Udine, Italy E-mail address: [email protected]

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 79-90 © 2003 Birkhiiuser Verlag Basel/Switzerland

Representations of Solutions of Hamilton-Jacobi Equations Italo Capuzzo Dolcetta

1. Introduction In this paper we report on some classical and more recent results about representation formulas for generalized solutions of the evolution partial differential equation (1.1) ut + H(x, Du) = 0 , (x, t) E RN x (0, +oo) . We consider here only the case where H = H(x, p) is a convex function with respect to the p variable. In this setting, representation formulas can be obtained by exploiting the well-known connection existing via convex duality between the Hamilton-Jacobi equation (1.1) with Calculus of Variations or, more generally, Optimal Control problems. In Section 2, the Hopf formula of Section 1.1 is revisited from a quite different point of view, pointing out some links with the classical vanishing viscosity method and with the closely related large deviation problem for the underlying stochastic processes. Some indications to the connection with the Maslov's idempotent analysis approach to Hamilton-Jacobi equations are also given. The final part of the paper comprises a description of a recent generalization, due

to H. Ishii and the author, of the Hopf representation formula for the solution of (1.1) in the case H = H(p). This generalization covers some cases of state dependent equations, including possible degeneracies in the x dependence.

2. The Cauchy problem in the viscosity sense We consider first order nonlinear evolution equations of the Hamilton-Jacobi type

ut + H(x, Du) = 0 , (x, t) E IRN x (0, +oo) equipped with the initial condition u(x, 0) = g(x) , x E IRN .

(2.1) (2.2)

Here, H is a given continuous scalar function of the variables (x, p) E 1R2N which

we will always assume to be convex in the p variable and the initial datum g is Work partially supported by the TMR Network "Viscosity Solutions and Applications".

Italo Capuzzo Dolcetta

80

given on IRN. The notations ut and Du stand, respectively, for the time derivative and the spatial gradient of the real-valued unknown function u = u(x, t). It is well known that problem (2.1), (2.2) does not have, in general, global

classical solutions, even for smooth data. The notion of viscosity solutions has proved to be appropriate for the analysis of the well-posedness of such nonlinear problems in a nondifferentiable framework, see [10], [14], [4], [3]. Let us recall for the convenience of the reader the Barron-Jensen [5] definition of

lower semicontinuous viscosity solution (or bilateral supersolution in the terminology of [3]) of problem (2.1), (2.2). The definition below extends the classical Crandall-Lions definition to the lower semicontinuous case and coincides with it for continuous solutions, provided the Hamiltonian H is convex with respect to p. A lower semicontinuous function u is a viscosity solution of (2.1), (2.2) if

A + H(x, ti) = 0 b(q, A) E D-u(x, t) ,

(2.3)

where D- u(x, t) is the subdifferential of u at (x, t), that is the closed, convex, possibly empty set whose elements are the vectors (rt, A) E RN x IR such that lim inf (V'S)-.(X.0

u(y's) - u(x, t) - t) (y - x) - A(s - t) > 0 . ly - xI + Is - tl

We refer to [5], [4], [3] for a through discussion of this notion of solution and for existence, comparison and stability results.

We briefly report now on three different methods which produce existence results together with representation formulas for the viscosity solution of problem (2.1), (2.2). 2.1. The method of characteristics The method of characteristics is the classical approach to construct local solutions to nonlinear first-order partial differential equations, see [11] for a recent presentation. Consider for simplicity the Cauchy problem

ut + H(Du) = 0 , (x, t) E IRN x (0, +oo) , u(x, 0) = 9(x) , x E IRN . The associated characteristic system is

x'(t) = aH (p(t))

(2.4) (2.5)

,

p'(t) = - 8H (p(t)) = 0

(2.6)

,

p(O) = D9(x) .

(2.7)

with the initial conditions

x(0) = x

The solution of (2.6), (2.7) is of course

x(t; x) = x + t

(Dg(x))

W

,

p(t)

Dg(x) ,

Representations of Solutions of Hamilton-Jacobi Equations

81

and so the candidate solution produced by the method of characteristics is

u(x, t) = g (x-1(t; x)) + t ( ap . Dg - H(Dg) f (x-' (t; x))

.

(2.8)

Here, x-1 is the inverse of the map x x(t; x), which is defined, in general, only for small t > 0. As a consequence, the function u is not globally defined by (2.8). However, under some restrictive assumptions on H and g, the map x x(t; x) is globally invertible and the above formula defines u as a global solution of (2.4), (2.5).

A model global existence result taken from [14] is as follows:

Theorem 2.1. Assume that H and g are C2(IRN) and convex. Then, the function u given by (2.8) is a classical and, a fortiori, viscosity solution of (2.4), (2.5). Under the assumptions of the theorem, we have indeed det

(I + t a

(Dg(x)) D2g(x) > 1

for all x E IRN and t > 0; so u is globally defined and the fact that it is a viscosity solution is a simple, direct verification.

2.2. The optimal control method We assume here that the convex function p - H(x, p) can be expressed as the envelope of a family of affine functions of p, namely

H(x, p) = sup [-F(x, a) p - L(x, a)[

(2.9)

aEA

where A is a closed subset of IRM, F : IRN x A -i IRN and L : IRN x A Lipschitz continuous in the first variable, uniformly in a.

IR are

This the typical situation in optimal control theory; note, however, that quite general functions H can be represented in this way, see [13[. Let us associate to H the autonomous control system

(t) = F(y(t), a(t)) , y(O) = x ,

(2.10)

where the control a is any measurable function of t E [0, +oo) valued in A, and the functional J(x, t; a) =

o 0

L(y(s), a(s))ds + g(y(t)) ,

(2.11)

where y(s) = y(x, a; a) is the solution trajectory of (2.10). The minimization of J with respect to all possible controls defines the value function of the above Bolza optimal control problem as the function u given by

u(x, t) = inf J(x, t; a) 0

.

(2.12)

Formula (2.12) provides a representation for the solution of (2.1), (2.2). Indeed, we have the following

Italo Capuzzo Dolcetta

82

Theorem 2.2. Assume H as in (2.9) with F and L as above. Assume also that L is bounded and that g is bounded and uniformly continuous. Then, the value function (2.12) is a viscosity solution of (2.1), (2.2). Moreover, u is the unique bounded and continuous function on RN x [0, +oo) solving (2.1), (2.2) in the viscosity sense. The proof of this existence and uniqueness result (and of some of its generalizations) can be found in [3]. Let us only mention here that the basic ingredient to prove that u is a viscosity solution of (2.1) is the Dynamic Programming Principle, a consequence of the nonlinear semigroup property of the control system (2.10): the value function of the optimal control problem at hand satisfies the identity u(x, t) = inf

J

ao

r L(y(s), a(s))ds + u(y(r), t - r)

for all x c RN andall0
We assume again here that H does not depend on x and, moreover, that H is superlinear at infinity, namely, him IPI +oo

H(p)

= +oo .

1PI

By a classical duality result in convex analysis we have then

H(p) = sup [a p - H*(a)] aE IRN

where H*(a) = supgERN [q a - H(q)] is the Legendre-Fenchel transform of H. Therefore, the representation (2.9) holds in this case with A = IRN, F(x,a) = a and L(x,a) = H*(a). The optimal control problem in the previous section becomes then the classical Bolza problem in the Calculus of Variations inf J(x, t; a) = inf

Ja

t

H* (y(s))ds + g(y(t)) ,

where a is any measurable functions taking unrestricted values in RN and y(s) = x +

J0

t a(s)ds

is the solution of the control system (2.10) in the case under consideration. The representation formula (2.12) has a simplified expression in the present setting, namely

u(x, t) = i of f H' (y(s))ds + 9(y(t)) N

[9(y)

+ tH'

(i)]

(2.13)

Representations of Solutions of Hamilton-Jacobi Equations

83

The proof of the equivalence between (2.12) and (2.13) relies on one side on the fact that the geodesics of the variational problem are just straight lines; this easily gives the inequality

(_!)]

rt

inf f H`(y(s))ds +9(y(t)) <

[9(y) + tH` inf The reverse inequality follows by an application of the classical Jensen's convexity inequality, see [11]. As a particular case of Theorem 2.2, the function J

u(x,t)

vinfN

[9(y)+tH* (x t

ll

(2.14)

usually called the Hopf (or Hopf-Lax) function, is a viscosity solution of the Cauchy problem (2.15) u= + H(Du) = 0, (x, t) E RN x (0, +oo) (2.16) u(x,0) = 9(x) , x E IRN . This result has been proved in [2], extending to the viscosity setting the

original result of Hopf [12] and generalized later in several directions, see [1], [6].

The main issue to be pointed out here is that the Hopf solution given by formula (2.14) expresses the solution u of the state-independent Cauchy problem (2.15), (2.16) as the optimal value of the following family of unconstrained static, finite-dimensional optimization problems i

y

nf N

9(y) + tH'

x- y

= inf sup [g(y) + q (x - y) - tH(q)]

t

YEIItN gEIRN

parametrized by (x, t). An alternative way, which makes no explicit reference to the associated variational problem, of deriving the Hopf function can be found in [12]. Since in Section 3 below we will exploit similar ideas in order to deal with the state-dependent case, let us describe briefly Hopf's construction. This starts from the simple observations that for affine initial datum the smooth solution of (2.15), (2.16) is

v(x, t) = g(x) - tH(Dg(x)) and that, for general g, the affine functions v"g(x, t) = 9(y) + q - (x - y) - tH(q)

solve (2.15) for any choice of (y, q) E RI x 1RN but do not satisfy (2.16). The procedure proposed in [12] is then to build a significant solution of the Cauchy problem by means of the following envelope inf

sup v&'e(x,t)

VERN gEltN

Italo Capuzzo Dolcetta

84

of the family

It is easy to check that sup vb"q(x, t) = g(y) + tH' qE IRN

(x t y)

so that the above-defined function coincides indeed with u in (2.14). Hopf's original result illustrating the connection between function (2.14) and Hamilton-Jacobi equations is next:

Theorem 2.3. Assume that H is convex and superlinear at infinity and that g is Lipschitz continuous. Then, the function

u(x, t) = inRfN [g(y) + tH Y

(i)]

(2.17)

is Lipschitz continuous on IRN x (0, +oo), satisfies equation (2.15) almost everywhere and lim u(x, t) = g(x)

t0+

at any x E IRN .

3. Hopf's formula and convolutions Let us recall that the inf-convolution gt (sometimes also called Yosida-Moreau transform) of a function g : RN - IR is defined for t > 0 by zJ 1

gt(x) = inf, [g(y) +

YER

Ix

- yI 2t

.

(3.1)

This is a well-known regularization procedure in convex and in nonsmooth analysis.

Indeed, if g is continuous then gt is Lipschitz continuous and also semiconcave, that is

gt(x + h) - 2gt(x) + gt(x - h) < (C + t) Ih12

(3.2)

holds for some constant C = C(t) > 0 and any (x, h) E 1R21' and t > 0. Moreover, functions gt converge to g locally uniformly as t - 0+. We refer to [3] for additional information on this topic.

In the special case when H(p) =

2IpI2

= H*(p), Hopf's function (2.14)

becomes

u(x,t)

inf [g(y) + Ix 2tyI2J N

= yE

(3.3)

which is precisely the inf-convolution of the initial datum g. An interesting but not evident relationship exists between inf-convolution and another standard regularization method, namely the classical integral convolution

Representations of Solutions of Hamilton-Jacobi Equations

85

procedure. Let us illustrate this with reference to the Cauchy problem

= 0, (x, t) E RN x (0, +oo) = g(x) , x E RN

ut + 2 IDuI2 u(x,0)

(3.4) (3.5)

whose solution is given by (3.3) by the results reported in Section 2.

Assume that g is continuous and bounded and consider the parabolic regularization of the Cauchy problem (3.4), (3.5), that is IDu'12=0,

ui - EAU' + I

u`(x, 0) = 9(x) ,

(3.6)

where a is a positive parameter. A direct computation shows that if u` is a smooth solution of the above, then its Hopf-Cole transform

wt = e-w satisfies the linear heat problem (3.7) wi - eAw` = 0 , wE(x,0) = 9`(x) = eBy classical linear theory, see [11] for example, its solution w` can be expressed as the convolution w` = r * g` where IF is the fundamental solution of the heat

equation. More explicitly, N

e-

w`(x, t) _ (41ret)-T

i

2

dy ;

1RN

hence, by inverting the Hopf-Cole transform,

u` (x, t) = -2e log ((4iretY'l

J

e-

i

e-

dy

\

(3.8)

)

turns out to be a solution of the quasilinear problem (3.6).

It is natural to expect that the solutions u` of (3.6) should converge, as e -- 0+, to the solution of ut + 1 IDuI2 = 0 , u(x, 0) = 9(x) , or, which is the same, to Hopf's function (3.3).

We have indeed the following result which shows, in particular, how the infconvolution can be regarded, roughly speaking, as a singular limit of integral convolutions:

Theorem 3.1. Assume that g is bounded. Then,

lim -2e log ((4lret)-f

J

tN

e-

e-dyl/ =

"

r I

VE1R

() +

Ix

111

(3.9)

The proof can be obtained by a direct application of a general large deviations result by S.N. Varadhan. Consider the family of probability measures Pi.t defined

Italo Capuzzo Dolcetta

86

on Borel subsets of IRN by

P(B) = (4rret) Ix.t(Y) =

Le-u1dy, ix

-4t

Y12

It is not hard to check that, for all fixed x and t, the family Pz.i satisfies the large deviation principle, see Definition 2.1 in [19], with rate function Ix,t . By Theorem 2.2 in [19], then E

lli e log (JIRN YEIRN

+

[F(y) - I(y)]

for any bounded continuous function F. The choice F = - i in the above shows then the validity of the limit relation (3.9). The same convergence result can be proved also by purely PDE methods. Uniform estimates for the solutions of (3.6) and compactness arguments show the existence of a limit function u solving (3.4), (3.5) in the viscosity sense. Uniqueness results for viscosity solutions allow then to identify the limit u as Hopf's function, see [14], [3].

The way of deriving Hopf's function via the Hopf-Cole transform and the large deviations principle is closely related to the Maslov's approach [16], [17] to Hamilton-Jacobi equations based on idempotent analysis. In that approach, the base field 1R of ordinary calculus is replaced by the Bemiring 1R' = IR U {oo} with operations a ® b = min{a, b}, a O b = a + b. A more extended description of this relationship is beyond the scope of this paper; let us only observe at this purpose that the non-smooth operation a ® b has the smooth approximation

a®b= lim -elog(e- a + e-4). 0+ A final remark is that the Hopf-Cole transform can be also used to deal with the parabolic regularization of more general Hamilton-Jacobi equations such as ut + 2 ka(x)DuI2 = 0 where a is a given M x N matrix, provided the regularizing second order operator is chosen appropriately. Indeed, if one looks at the regularized problem 1

ui - ediv (o` (x)a(x)Du`) + la(x)Du' I2 = 0 , then the Hopf-Cole transform w` = e-

2

solves the linear equation

w' - e div (a* (x)a(x)Dw`) = 0

This observation will be developed in the forthcoming work [8].

Representations of Solutions of Hamilton-Jacobi Equations

87

4. A Hopf formula for state-dependent Hamiltonians As described in Section 1.2 and 1.3, the value function representation and Hopf's function coincide when the Hamiltonian does not depend on the variable x. In this section we present a new Hopf type formula, obtained in collaboration with H. Ishii, see [9], for the viscosity solution of the state-dependent Cauchy problem

ut + H(x, Du) = 0 , (x, t) E IRN x (0, +oo) ,

(4.1)

u(x, 0) = g(x) , x E IRN

(4.2)

.

It is not hard to realize that the Hopf envelope method of Section 2 does not work if H depends on x. Nonetheless, a Hopf type formula can be proved even in this

more general case under the basic structural assumption that the Hamiltonian H : IR2A H IR is of the form

H(x,p) =

(H.(x,p))

(4.3)

where H,, is a continuous function on IR2N satisfying the following conditions

p H H0(x, p) is convex, H0(x, )tp) = .\H.(x, p)

(4.4)

H0(x,p)>_0, IH0(x,p)-H0(y,p)I :5 W(Ix-yl(1+IPI))

(4.5)

for all x, y, p, for all \ > 0 and for some modulus w such that lim,..o+ m(s) = 0 Concerning function t we assume

t : [0, +oo) -4 [0, +oo) , convex, non decreasing, 4'(O) = 0 .

.

(4.6)

The next result shows that the validity of a Hopf type formula for the solution of problem (4.1), (4.2) is guaranteed if the associated stationary eikonal problem

H0(x,Dd)=1, xERN\{y} , d(y) = 0

(4.7)

has a solution d(x) = d(x; y) for any value of the parameter y E IRN. We shall describe below a setting in which this condition can be enforced. Theorem 4.1. Assume (4.3), (4.4), (4.5), (4.6) and

g lower semicontinuous, g(x) > -C(1 + Ixi) for some C > 0.

(4.8)

Assume also that problem (4.7) has a unique continuous viscosity solution d(x) = d(x; y) for each y E IRN . Then, the function u(x, t)

JEjN

[gy+ t(d(ty)) J

(4.9)

is the unique lower semicontinuous viscosity solution of (4.1) which is bounded below by a function of linear growth and such that lim inf

u(y, t) = g(x)

Italo Capuzzo Dolcetta

88

In order to understand why the Hopf function (4.9) solves (4.1), let us proceed heuristically by assuming that (4.7) has a smooth solution d(x) and look for special solutions (4.1) of the form v5(x, t) = 9(y) + OF

(d(x; y) ) t

l

where y E IRN plays the role of a parameter and 4 is a smooth function to be appropriately selected. Set now r = e i' > 0 and compute

i = 41(T) +

V-'

11(r) - T4i'(r)

;

DvY = t4i'(r) Dt d = W'(r)D=d .

Imposing that v1' solves (4.1) gives

4<(r) - r4i'(r) + 4'(Ho(x,

0;

therefore, if 4' is strictly increasing, the positive homogeneity of H. and the fact that d solves the eikonal equation yield 41(T) - r41'(r) + 4'(W'(T)) = 0

.

Since the solution of this Clairaut's differential equation is 4< = V, the above heuristics leads then to formula (4.9). These formal arguments can be made rigorous by some duality arguments in convex analysis and by using the apparatus of comparison and stability methods of the theory of viscosity solutions. We refer to [9] for details.

The assumption that the eikonal equation has a unique continuous viscosity solution made in Theorem 4.1 is trivially satisfied with d(x; y) = Ix - Ill for the simplest case p) = IpI and, more generally, when the Hamiltonian H,, is of the form H0(x,p) = IA(x)pl where A(x) is a symmetric positive definite N x N matrix. The associated eikonal equations are solved in this case by Riemannian metrics, see [14], [18] at this purpose. In the examples mentioned above, the coercivity condition lim 1PI-+00

H0(x, p) = +oc

(4.10)

obviously holds true. Let us briefly discuss now the issue of finding sufficient conditions for the validity of the eikonal assumption in Theorem 4.1 even for degenerate situations when (4.10) may fail. Consider for example the homogeneous Hamiltonian H0(x, p) = I o(x)pl

where o(x) is an M x N (with M < N) matrix such that x - o(x) is C-(RN) and satisfies the Chow-Hormander rank condition of order k, see [3], [7]. Consider then the differential inclusion

X(t) E 8H,,(x(t),0)

(4.11)

Representations of Solutions of Hamilton-Jacobi Equations

89

and, for x, y E RN, the set Fy,y of all trajectories X(.) of (4.11) such that

X(0)=x, X(T) = y for some T = T (X

0 . By the well-known Chow's Connectivity Theorem, see [7], the set Fx 1, is non-empty and, consequently, the function d(x; y) = inf T (X (4.12) is finite for all x, y. Moreover, d is a sub-Riemannian metric of Carnot-Caratheodory type which compares locally with the Euclidean distance Ix - yJ on IRN as

Ci Ix - yI < d(x; y) < C2Ix - yI I .

The proof that x

d(x, y) actually solves equation (4.7) makes use of several technical refinements to the standard dynamic programming argument mentioned in Section 2.2. We refer to [9] for a detailed proof. Let us observe that in the present non-coercive setting, the function d is not, in general, differentiable almost everywhere; the notion of viscosity solution seems therefore to be essential to interpret d as a solution of (4.7). An interesting particular case (here N = 3 to simplify notations) is H. (x, p) = v(p,

- 2 1>3)2 + (p2 + 2 p3)2

arising in connection with Carrot-Caratheodory on the Heisenberg group H1. Our Hopf formula (4.9) coincides in this case with the one recently found for this example by Manfredi-Stroffolini [15].

References [1)

[21

[3]

[41

[51

[61

0. Alvarez, E.N. Barron, H. Ishii, Hopf formulas for semicontinuous data, Indiana University Mathematical Journal, Vol. 48, No. 3 (1999). M. Bardi, L.C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., TMA 8 (1984). M. Bardi, I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of HamiltonJacobi-Bellman Equations, Systems & Control: Foundations and Applications, Birkhiiuser Verlag (1997). G. Barles, Solutions de Viscosity des Equations de Hamilton-Jacobi, Mathematiques et Applications, 17, Springer-Verlag (1994).

E.N. Barron, R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. PDE 15 (1990). E.N. Barron, R. Jensen, W. Liu, Hopf-Lax formula for ut + H(u, Du) = 0, J. Diff. Eqs. 126 (1996).

[71

[81

A. Bellaiche, The tangent space in sub-Riemannian geometry, in A. Bellaiche, J.J. Risler eds., Sub-Riemannian Geometry, Progress in Mathematics, Vol. 144, Birkhauser Verlag (1997). V. Cafagna, I. Capuzzo Dolcetta, work in progress.

halo Capuzzo Dolcetta

90

I. Capuzzo Dolcetta, H. Ishii, Hopf formulas for state-dependent Hamilton-Jacobi equations, to appear. [10] M.G. Crandall, P: L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983). [11] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence RI (1988). [12] E. Hopf, Generalized solutions of nonlinear equations of first order, J. Math. Mech. [9]

14 (1965).

[13] H. Ishii, Representation of solutions of Hamilton-Jacobi equations, Nonlinear Anal., 12 (1988).

[14] P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics 69, Pitman (1982). [151 J.J. Manfredi, B. Stroffolini, A version of the Hopf-Lax formula in the Heisenberg group, to appear in Comm. PDE. [16] V.P. Maslov, On a new principle of superposition for optimization problems, Russian Math. Surveys, no. 3, 42 (1987). [17] V.P. Maslov, S.N. Samborskii, Editors, Idempotent Analysis, Advances in Soviet Mathematics, Volume 13, American Mathematical Society, Providence RI (1991). [18] A. Siconolfi, Metric aspects of Hamilton-Jacobi equations, to appear in Transactions of the AMS. [19] S.R.S. Varadhan, Large Deviations and Applications, Society for Industrial and Applied Mathematics, Philadelphia PA (1984).

Italo Capuzzo Dolcetta Dipartimento di Matematica University di Roma 1 P.le. A. Moro 2 1-00185 Roma, Italy

E-mail address: capuzzo@mat. uniromal. it

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 91-105 © 2003 Birkhauser Verlag Basel/Switzerland

Nonexistence of Global Solutions of Higher Order Evolution Inequalities in RN Gabriella Caristi

1. Introduction This paper is devoted to the study of the nonexistence of global nontrivial weak solutions of higher order evolution inequalities in RN of the form: 8i u - IxIoAmu > lu19,

(1) (2)

&u(x, 0) = uj(x), j = 0, k - 1 where we assume that m and k are integers greater or equal to 1, q > 1 and or < 2m. The idea is to find a set of conditions which imply the nonexistence of global weak solutions and to see how this set depends on the order of the operator. In particular, we will define an exponent q' such that if q < q', then (1)-(2) does not admit any nontrivial solution in a weak sense to be specified later, (see Definition 2.1). We will address to q* as to the critical exponent of problem (1)-(2) . This kind of research, which was originated by the celebrated paper by Fujita

[4], has been extensively pursued in the last years. Here, we quote the survey papers by Levine [8] and Deng and Levine [3] and refer the interested reader to the bibliography contained therein. In particular, for problem (1)-(2) when k = 1 and k = 2 and under suitable positivity conditions on the initial data, Mitidieri and Pohozaev have shown respectively in [10] and [9], that the following phenomenon occurs:

if tr < 2m (subcritical degeneracy), then, the critical exponent q* depends on the dimension N, while if o = 2m (critical degeneracy), then, q' is independent on N and in some dimensions (1)-(2) does not have any weak nontrivial solution for any q > 1. Here, we will show that a similar result holds also if k > 2 and extend it to the case of systems. In the proofs, we will apply the test function method as it has been developed by Mitidieri and Pohozaev (see [11]) to deal with quasilinear elliptic, parabolic and hyperbolic partial differential inequalities. This method allows a unified approach for the limit case q = q' and does not require any sign condition on u nor uses any comparison argument. For these reasons, the Partially supported by the Italian Minister delta Universitn e delta Ricerca Scientifica e Tecnologica, funds 60% and 40%.

Gabriella Caristi

92

results can be easily extended to more general problems, containing more general differential operators than the polyharmonic.

In this paper we will not be concerned with the question of the sharpness

of q'. To this regard, we recall that in the special case k = 1, m = 1, Q = 2, our approach allows to reach the sharp critical exponent. We refer to Giacomoni [5] for the case of the equation, to Mitidieri and Pohozaev [10] for the case of an inequality and to Caristi [2] for systems of inequalities. In [10] and [2] it is not assumed any pointwise sign condition on the solutions. Finally, we wish to mention that higher order in time problems not only are natural generalizations of first- and second-order problems, but also appear, for instance, in mathematical models of quasi-steady processes in anisotropic continuum electrodynamics, see Korpusov et al. [6]. In the context of higher order evolution inequalities we mention also a recent paper by Laptev [7], where the nonexistence of solutions is studied for some classes of problems in unbounded cone domains when in = 1. The paper is organized as follows: in Section 2 we consider problem (P) with critical degeneracy, i.e., o = 2m. Theorem 2.1 extends to the case k > 2 Theorem 3.1 of [9]. Theorem 2.2 is the corresponding result for systems and extends to the higher order case and for m > 1 Theorem 3.1 of [2]. In Section 3 we deal with problems with subcritical degeneracies. We consider also inhomogeneous problems, see Theorem 3.1, partially extending a result of Bandle, Levine and Zhang [1]. This means that we obtain a higher order generalization of Theorem 2.1 part (a) of [1], assuming that the initial data satisfy some weak positivity condition, (see assumption (31) below). Throughout the paper, C will denote a positive constant which may vary from line to line. Moreover, if p > 1, p' will denote the conjugate exponent of p, that is, , + ,' = 1.

2. Problems with critical degeneracies 2.1. Scalar problems Set D = (R' \ {0}) x (0, +oo). We consider the following initial value problem

r Oku -

IxI2nQmu > July, (x, t) E D,

l c?; u(x, 0) = u j(x), x E R'` \ {0}, j = 0, k - 1

where we assume that in, k are integers greater or equal to 1, q > 1 and u3 E LI.(RN \ {0}) for j = 0, k - 1 Definition 2.1. We say that. u is a weak solution of (P) if the following assumptions are satisfied:

(i) u E Ll ,_(R-"' \ {0} x (0, oc)),

Nonexistence of Global Solutions

93

and for any nonnegative ' E Co (RN \ {0} x R) the following inequality holds: ID

uIIxIdxdt f u((-1)klxlNam -

m(IxI2m-N) dxdt

D

k-1

+r(-1)j+IJRN

j1(x)IxIa(x,0)dx

uk

(3)

Theorem 2.1. Assume that k > 2, uk_1IxI-N E L1(RN) and that JRN Uk-

1(x)IXIN dx > 0.

(4)

If one of the following assumptions is satisfied:

(i) N#2(j+1) forj=0,m-1 and1 1, then, the problem (P) has no weak nontrivial solution.

Proof. Let (u, v) be a weak nontrivial solution of (P). Let 0 E Co (RN \ {0} x R) be nonnegative, radial, that is, ¢(x, t) = 4)(I xI , t), and such that ac.o('0)

= 0, ...,ai-14)(',0) ° 0.

From (3) we obtain that if u is a weak solution of (P), then it satisfies ID

Iuj9I xI -N¢dxdt <

JRN

u k-I (x)I xI

f

(-1)klxl-Na 4) -

IuI I

Qm(IxI2tn-N¢)I dxdt

N4)(x, 0)dx

(5)

Applying Young's inequality to the right-hand side of (5) we get

(1D uIQIxI_NOdxdt) ID lUlglXI-NOdxdt < 1 IxI3

I

I(-1)klxl-"a+On`(IxI2m-"m)I4

+q CID

-f

4)I

_y dxdt I

uk-I (x)I xI -NO(x, 0)dx.

(6)

Introducing the change of variable

P=log1xI, x00, from (6) we obtain ITLI9IxI-NO dxdt+q ID 00 0

00

JftN

21k-1(x)IxI-NO(x,0)dx

eN4 P I(-1)ke-NPat m - e-2mpP(aP)(e(2m-N)po)

9

-oo 00

01_9 dpdt <

fo

00

I(-1)kai

-

01-° dPdt,

(7)

Gabriella Caristi

94

where

m-!

P(ap) = 2mArn

= jj (ap - 2j) o (a,, + N - 2 - 2j).

(8)

0

Remark that P(.) is a polynomial of degree 2m and that the coefficient of its first-order term is given by: M-1

1)! jj (N - 2 - 2j).

Cl

(9)

0

In (7) we have used the following property of P: P(ap)(e(2m-N)p4)) = e(2m-N)pP(-ap)(4))

We refer to [91 for its proof.

Now, suppose that (i) holds. From (9) we deduce that Cl 0 0. Let us choose the test function 0 as follows: O(P,t) = 4)o(R +

(10)

where Q > 0 and 4o : R --+ R is a nonnegative smooth function such that 0 < 00 < 1 and Lo(s)1, 0<s<1

O, s>2. Introducing the following change of variables

=R-ap, j T = R-!t, it follows that

I(-1)kai 4)(P, t) - P(-ap)m(P, t)I I(-1)kR-kar4)(S2

=

+ r) - CiR-aae0(S2 +T)

2m

+E

CSR-'FO4)(t2

+ r)I

2

c < R-klar.O(t2 + r)I + CR-"3Iato(`2 + r)I

2m

+R-24 E

r)I.

Nonexistence of Global Solutions

95

Taking /3 = k and substituting (11) into (7), we obtain

LR

-

<

IQII-^' ddt <

ff

-q' IR^' RN

IuIQIxI-Ndxdt ID

I(-1)ka o + P(-8P)(0)I Q

m'-9

dpdt

I xI -"b(x, 0)dx

tt + r)IQ + < CR-kQ +1+k j (IarW(S2

I

01-q' dddr

I0{0(C2 + T)IQ 01-9 l dT

2

-q JRN uk-1(x)I xI -"1(x, 0)dx,

(12)

where BR = {(x, t) E D : R-1t+R-2k(log IxI)2 < 1} and A = {(£,T) E Rx (0, oo) :

1
-kq'+l+k<0,

(13)

then, taking the limit as R tends to oo into (12) we get

I

IuIQIxI-N dxdt = 0,

D

against the assumption that is is a nontrivial solution. Since (13) is equivalent to

q<1+k, the theorem is proved when (i) holds with the strict inequality. If q = 1 + k, from (12) it follows that LR IuIQIxI-N dxdt < C - q' fftN uk-1(x)IXI-"4,(x, 0)dx.

(14)

From the definition of 0 and the assumption (4), taking the limit as R tends to infinity on both sides of (14), we get L IuIQIxI-^' dxdt < oo.

(15)

On the other hand from (5) taking into account of the definition of 0, we get that L I uI QjxI -"0dxdt <

f

IuI

fftN u k-1(x)IXI"1(x,0)dX,

I(-1)dIxI-"at o - A-(IxI2--",)I dxdt (16)

Gabriella Caristi

96

where Cx = {(x, t) : 1 < R-'t +R-2kIxI2 < 2}. Now, applying Holder's inequality to (16) we obtain

ID

juIyIxI-~0dxdt < r rcFjuI4Ix!-vbdxdt l

C CR I(_l)kIxl-N8'J (1

-f Now,

-

A'n(IxI2m-N41)I9 ,#1-9 dxdt)

(17)

0)dx.

Uk-1(x)I XI

from (4) we can assume that JRN

uk -1(x)Ix{N(x0)dx

> 0,

for any R > 0 sufficiently large. Recall that ¢ depends on R, see (10). Applying estimate (11) we conclude that

f

lUlglXI-Ndxdt < f IuIQIxI-N0dxdt

D

BR


(18)

CR

On the other hand from (15) it follows that

fe"

0, R -+ oo,

IuJ9IxI-N4'dxdt

and hence, from (18) we get a contradiction. This concludes the proof in case (i). Next, we assume that (ii) holds. In this case it follows that C1 = 0 and by direct inspection of the expression of P we have that its second order coefficient is different from 0. Now, we choose 4'(p, t) as follows: // t 0(P> t) _ 00 (RQ +

R2

(19)

2ry

where -y > 0 and Q > 0 and ¢o is as above. Proceeding as in the proof of case (i), we get the following estimate: fn

<

Iuj1jxj-'0dxdt CR-k7Q'+7+t3

j I0(k)4' 2m

2

+.T)19'01 -q'

/

+CR-239+1+l3 `f Ia 'm(t2 + 2

4dT

T)I401-9 ddr

A

-9 JRN uk-1(x)IXI-N4'(x,0)dx.

(20)

Nonexistence of Global Solutions

97

Choosing 3 = 1 and -y = 2/k, we obtain that

-2/3q'+ry+/3= -kryq'+-y+0=-2q'+

2

k

+1,

and

-2q'+ k2 +1<0, which is equivalent to

2 2-k k q<1+_k.

Since this condition is satisfied by any q > 1 the statement follows.

2.2. Systems with critical degeneracies In this section we shall consider the system of critically degenerate evolution inequalities of the form:

- Ixl2momu > IvI9, (x, t) E D, ai 2v - IxI2mQmv > IuV', (x, t) E D, .9,1-1 u

(S) =

i

u(x, 0) = u.i(x), x E RN, j = 0, k1 - 1,

8,v(x,0)=Vh(x), xERN, h=0,k2-1, where ki and m are integers greater or equal to 1, p, q > 1 and uj, vh E LI OC (RN \

{0})forj=0,k1-l and h=0,k2-1.

In the following theorem it is understood that the concept of weak solution of system (S) naturally extends the definition of weak solution of problem (P), that is, Definition 2.1.

Theorem 2.2. Assume that 2 < k1 < k2, uk1_IIxI-N, vk,-IIxI-N E LI(RN) and that Jf

JRN

/ u k1_IxI-Ndx>0,

f

vkz-l(x)IxI-Ndx>0.

(21)

N

If one of the following assumptions is satisfied:

(i) N # 2(j + 1) for j = 0, m - 1 and pq <

max{ k2k2g-+klk2++l1

I

kiq+kl +1, kip+k2+1

(ii) N=2(j+1) with j E {0,...,m-1} and p, q> 1, then, the problem (S) has no nontrivial weak solution. Proof. Let (u, v) be a nontrivial weak solution of (S). In the course of the proof we will use the same method employed in Theorem 2.1. For this reason we will omit some details.

Gabriella Caristi

98

First assume that (21) and (i) holds. Let 0 = 46R be chosen as in the first part of the proof of Theorem 2.1, see (10), and remark that (21) implies that for any R > 0 sufficiently large we have:

J ak,-I(x)IxI-`vO(x,0)dx > 0,

> 0.

!RN

(22)

After the usual change of variables and the application of Holder's inequality, we get the estimates:

CJ Iv191XI-N0dxdt b

jujVjxj-NOdxdt)

(R-kip '+I+;9 + R-Br'+1+R) P

CL jujyjxj-'vOdxdt 1

(Lj,jqjXj-NOdXdt)

iC

(R-k2q +1+d + R-sq'+'+s

` Substituting (24) into (23), we obtain :

J

,q

IvI9IxI-odxdtlrv

< R,1(3) +Rr2(9) + Rr3(a) +R*(O)

v

(23)

(24)

(25)

and analogously substituting (23) into (24), we obtain:

C

J

n

JuIPIx'-' Odxdt f

/

r

< R,i(o) + R52(v) + R"3(B) + Rs4(0),

where we have set: r1(3) = (pq)-'(pq(1 + 3 - k1) - k2q - 1 -)3), r2(a) = (pq)-'(pq(1 + 3 - ki) - aq - 1 - 3), r:1(0) = (pq)-'(pq - k2q - 1 -0),

r.i (3) = (pq)-' (pq - )3q - 1 - a), 81(3) = (pq)-'(pq(1 + a - k2) - kip - 1 -)3), 82(3) = (pq)-'(pq(1 + 3 - k2) - 3p - 1 - a), s3(3) = (pq)-'(pq - kip - 1 - 3),

84 (0) = (pq)-' (pq - ap - 1 - a).

(26)

99

Nonexistence of Global Solutions

It is easy to check that since by hypothesis k1 < k2, we have:

(pq)-'(pq(1 + k2 - ki) - k2Q - 1 - k2) = ri(k2) = r2(k2) > r3(k2) = r4(k2), (pq)-'(pq - kiq - 1 - k1) = r2(ki) = r4(ki) ? ri(ki) = r3(ki), (pq)-'(pq - kip - 1 - ki) = 33(k1) = s4(k1) > sl(kl) = s2(ki), (pq)-1(pq - kip - 1 - k2) = si(k2) = s:3(k2) > 32(k2) = 34(k2)

If (i) holds, we conclude the proof following the procedure of Theorem 2.1. Now, assume that (ii) holds. Let 46 = OR be chosen as in the second part of

the proof of Theorem 2.1, (see (19)), and remark that (21) implies that for any R > 0 sufficiently large (22) holds. Instead of (23) and (24) we obtain: CJ lvlQlxl-NO dxdt n 2P

(R-k,YY +7+34 + R-20p'+7+A) P

\JnlulPIxI-N0dxdtl

27)

C r luIPlxI-N4dxdt JD

o

lvl 9 lxl -N Odxd /

< Substituting (28) into (27) we get: C

lvI Qlxl-N0dxdt)

(R -k277 +-y+Q +

< R1l(-r,O) +

R-2a4 +1+,9 1

a

Rf3(,,s) +

(

28 )

(29)

where

fl ('y, Q) = (pq)-' (pq(1 + y - ki -t) - k2g7 - 1 - y), f2(y,)3) = (pq)-'(pq(1 + -y - kip) - 2q - 1 - y),

f3(y,0) = (pq)-'(pq(y - 1) - yk2Q - 1 - y),

f4('Y,16)=(pq)(pq('Y-1)-2q-1-y). It is easy to check that

2ki'(pq - 1) - (pq+2q+ 1) = f2(2ki',1) = f4(2k1',1) f1(2k11,1) = f3(2ki 1, 1). Since k1 > 2 it follows that for any p, q > 1 we have

2ki '(pq - 1) < pq + 2q + 1. Hence, from (29) the statement follows.

0

Remark 2.1. Theorem 2.2 can be extended to deal with systems containing two different polyharmonic operators. For the sake of brevity, we leave to the interested reader the statement of the corresponding result.

Gabriella Caristi

100

3. Problems with subcritical degeneracies In this section we study the nonexistence of solutions of higher order evolution problems containing subcritical degeneracies. As we will see below, in this case the critical exponent depends on the dimension N.

3.1. Scalar problems Let us consider the following problem: a, u - IxI°A"`u > luN9, (x, t) E RN x (0, oo),

(PI)

8,' u x 0 = u3(x) x E RN

where we assume that m and k are integers greater or equal to 1, q > 1, or < 2m

and ujlxl-° E Li,,.(RN) for j = 0, k - 1. First of all, we define the concept of weak solution of problem (Pl):

Definition 3.1. Set D' = RN x (0, 00). We say that u is a weak solution of (P1) if the following assumptions are satisfied:

(i) lul°lxl-° E LI (D`), (ii) ulxl-O E Li (D'), (iii) u E 10 and for any nonnegative 0 E CC°(RN x R) the following inequality holds:

f lUlglxI -°cbdxdt < L.

u((-1)kIxI-na`

m - A-0) dxdt

k-1

+E(-1)j+l j=0

JRN

uk_i-I (x)Ixl-'&t'O(x,0)dx.

(30)

Theorem 3.1. Assume that k > 2, q > 1, or < 2m and ujIxI-° E LLC(RN) for j = 0, k-2, uk-llxl-a E LI (RN). Farther, let the following condition be satisfied:

J

(Idx > 0.

(31)

RN

If q(k(N - 2m) + 2m - a) < Nk + 2m - o(k + 1),

(32)

then, the problem (PI) has no weak nontrivial solution.

Proof Let (u, v) be a weak nontrivial solution of (P1). Here, we use the same method employed in the proof of Theorem 2.1 with a different choice of the test function q5. More precisely, given )3 > 0, R > 0 and 0i E C$°(R) (i=1,2) such that

0<¢<<1and

ti(s) =

1, 0<s<1

0 s>2

we take (33)

Nonexistence of Global Solutions

101

It is always possible to choose 01 in order that

0, ...,ai-1.01(0) =0.

at4)i(0)

By Young's inequality from (30) we obtain that there exists C > 0 such that

CJ

U.

IuI"IxI-°cbdxdt

(34)

lai olq Oi-v'jxj-° dxdt + J CR

IA"`01q'01 -v'Ixlo,(v'-I)dxdt

R

- f 'k-1(x)IXI-°O(x,0)dX,

(35)

where CR={(x,t)ED:Rs
I

R-ix,

r = R-Ot,

we deduce the estimates:

fri Iae -OIq'O1-q'lxl-O dxdt <

f",

CR-gky +n+a-o

Io,n0Iq'01-q IxI'(q'-1) dxdt <

CR-2mq'+o(q'-1)+N+a

(36)

(37)

Set Q = (2m - a)/k. It follows that the exponents on the right-hand side of (36) and (37) are equal to

q(k(N - 2m) + 2m - a) - Nk - 2m + a(k + 1) k(q - 1) If q(k(N - 2m) + 2m - a) < Nk + 2 - a(k + 1), then taking the limit as R tends to infinity in (35), we obtain that

Ir (ulglxl-° dxdt < 0, which contradicts our assumption.

If q(k(N - 2m) + 2m - o) = Nk + 2m - a(k + 1), from (35) we get that

fD.

lulg lxl-° dxdt < oo.

Proceeding as in the second part of the proof of Theorem 2.1, we reach again a contradiction. 0

Remark 3.1. We remark that if a = 0 and N < 2m, then (32) is satisfied for any

q>1.

Gabriella Caristi

102

The method used in this paper allows also to find the critical exponent of inhomogeneous problems, of the form:

8k u - IxI°0'"u > Jul'? + w(x), (x, t) E RN x (0, oc),

(P2) _

u(x,0) = ui(x), x E RN, j = 1,k - 1,

see Theorem 2.1 of [1] for the case k = 1, m = 1 and or = 0.

Theorem 3.2. Assume that m and k are integers greater or equal to 1, q > 1, or < 2m, wlxI-° E L'(R') andujIxl-O E LioC(RN) for j =0, k-2, uk_IIxl-O E LI (RN). Further, suppose that (31) holds and that: JRN

w(x )IxIdx > 0.

(38)

If q(N - 2m) < N - or,

(39)

then, the problem (P2) has no weak nontrivial solution.

Proof. The proof is similar to the proof of Theorem 3.1. Here, we show only the crucial point. Using the same test function of Theorem 3.1 and taking Q = (2m - o)/k, we get the following estimate:

_I)

Jv. w(x)IxI-°O(x,t)dxdt+.luI'IxI-°Odxdt ID CR Moreover, we have: J

q-1

w(x)lxl -°O(x, t)dxdt =URN w(x)lxl-°O1

y

=R

(40)

a) ¢2 (iii) dxdt

(ji(r)dr) f

w(x)I xI

2

(iii) dx.

RN

(41)

From (40) and (41) we get: 9(k(N-2m )f2m-e)-Nk-2m f e(k }1) 2rn-n CRS- J w(x)IxI-°102 (1.t) dxdt < R

R

N

(42)

if

2m.-a

q(k(N - 2m) + 2m - o) -Nk-2m+o(k+1)

k

k(q - 1)

- N+2mq-Nq-or >0 q-1

then, (42) yields to a contradiction, and hence, no nontrivial weak solution of (P2) 0 exists.

Nonexistence of Global Solutions

103

Remark 3.2. Theorem 3.2 extends to problem (P2) Theorem 2.1 part (a) of [1] which concerns the case k = 1, m = 1, o = 0, under the weak positivity condition on the initial data (31). Remark that the exponent

N-o N - 2m' which appears in (39) is the critical exponent of the corresponding elliptic problem:

-i mu >

IxI-°IuI",

in RN,

(see Theorem 1 of [10] for a proof of this fact).

3.2. Systems with subcritical degeneracies Consider now the system of subcritically degenerate evolution inequalities

8L'u- IxI°'0,°u> IvI", (x,t) E D, (Si) =

- IxI°y 0nv > IuIP, (x, t) E D, &u(x, 0) = uj (x), x E RN, j = 0, kl - 1, 8i v(x, 0) = vh (x), x E RN, h = 0, k2 - 1, at 2 V

where k, and m, n are integers greater or equal to 1, p, q > 1, of < 2m, 02 < 2n

andu,i,vhELIOC(RN\{0}) for 0=1,k1-1 and h=0,k2-1. Theorem 3.3. Assume that 2 < ki < k2, that uk,_IIxI-C, vk2_1IxI-0'2 E L1(RN) and that the following condition holds:

,a, dx > 0, JRN

Vk2-i(x)Ix1°2 dx

> 0.

(43)

IRN

Moreover, let the following conditions be satisfied:

k2(2m - c1) > kl(2n - 02), (o21p - ol)p' + N > 0,

(ol/q - 02)9 + N > 0,

and

min{k2N(pq - 1) + (2n - 0`2)(pq(1 - k1) - k2q - 1) + k2oi (1 - pq), kl N(pq - 1) + pq(2m(1 - k1) - c1) - 2m + of + k1o1 + ki q(o2 - 2n),

k1N(pq-1)+a,(1-pq)-2m(l+kip-pq)-pgal +kl (pal + 02 - 2npq), k2N(pq - 1) + k2o2 + 02(1 + kip - pq) -2n(l + kip - pq + k2pq)} < 0

(44)

then, the problem (S1) has no nontrivial weak solution.

Proof. Let (u, v) be a nontrivial weak solution of (Si ). Here, the concept of weak

solution naturally extends that one given in the case of single inequalities, see Definition 3.1.

Gabriella Caristi

104

Proceeding as in the proof of Theorem 3.1 with the obvious modifications due to the fact that we are dealing with a system (see the proof of Theorem 2.2) and choosing the test function according to (33) we get the following estimates: Ip

CJ ItI

Odxdt < (

D

(o21P-0i)

\R-3k,p'+N+i3+P

C1

IuIPIxI-124dxdt I

D

vlglxl-°'40dxdt

JuIPIrI-°20dxdt <

UD

(45)

R-2nnp'+N+3+ol(1-p') 1 a

+

14

I

(R-3k2q'+N+3+q'(t7,/9-c2)

+ R-2nq'+N+3+e2(1-9 )) i

.

(46)

Substituting (46) into (45) and (45) into (46), and then, evaluating the exponents for

2n-Q2

, and Q =

k2

2m-al k1

we get the result.

Remark 3.3. We point out that the results remain true under more general hypotheses on the initial data. For instance, in Theorem 2.1 we can assume that uk_1 satisfies in place of (4) the following condition: uk-1IxI-N

E Lj,,,(RN) and there exists a sequence {Rh} of positive numbers

diverging to oc such that for any h the following inequality holds: (x)IXI-'OR,. JRN Uk- I

(x, 0) dx > 0,

where ¢R,, is given by (10) with R = Rh. A similar remark applies to the term w(.) in Theorem 3.2. Remark 3.4. The results of this section can be extended to deal the case of porous media type differential inequalities as for instance:

8i u -

IXIvOntup > Iul',

&t' u(x,0)=uj(x), j=0,k-1 References [1] C. Bandle-H. A. Levine-Q. Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl., 251 (2000), 624-648. [2] G. Caristi, Existence and nonexistence of global solutions of degenerate and singular parabolic systems, Abstract and Applied Analysis, 5 (2000), 265-284. [3] K. Deng-H. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000), 85-126. [4] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = Au+ul+a J. Fac. Sci. Univ. Tokio Sec. 1A Math. 16 (1966), 105-113.

Nonexistence of Global Solutions

105

[5] J. Giacomoni, Some results about blow-up and global existence to a semilinear degenerate heat equation, Rev. Mat. Complut. 11 (1998), 325-351. [6] M.O. Korpusov, Yu.D. Peletier, A.G. Sveshnikov, Unsteady waves in anisotropic dispersive media, Comput. Math. Math. Phys. 39 (1999), 968-984. [7] G.G. Laptev, Some nonexistence results for higher order evolution inequalities in cone-like domains, Electronic Research Announcements of AMS, 7 (2001), 87-93 . [8] H. Levine, The role of critical exponents in blow-up theorems, SIAM Reviews, 32 (1990), 262-288.

[9] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for degenerate and singular hyperbolic problems on R^ , Proc. Steklov Inst. Math., 232 (2000), 1-19. [10] E. Mitidieri, S.I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on RN , J. Evolution Equations, 1 (2001), 189-220. [11] E. Mitidieri, S.I. Pohozaev, A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234, (2001).

Gabriella Caristi Dipartimento di Scienze Matematiche University di Trieste 1-34100 Trieste, Italy

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 107-116 © 2003 Birkhauser Verlag Basel/Switzerland

Morse Index Computations for a Class of Functionals Defined in Banach Spaces S. Cingolani and G. Vannella

1. Introduction In Morse theory the behavior of a C2 Euler functional. F, defined on a Hilbert space H, near its critical points can be described by the estimates of the critical groups in the critical points. For convenience of the reader we recall the definition of critical group. For any a E R, we denote F° = {v E H : F(v) < a}. Moreover let u be a critical point of F, at level c = F(u). We call Cq(F, u) = HI (F', F` \ {u}) the qth critical group of F at u, q = 0, 1, 2, ... , where Hq(A, B) stands for the qth Alexander-Spanier cohomology group of the pair (A, B) with coefficients in K (cf. [2]).

In a Hilbert setting the estimates of critical groups are a quite clear fact and classical results based on Morse Lemma (see Theorem 2.5) allow to compute the critical groups via Morse index, namely the supremum of the dimensions of the subspaces on which the second derivative F"(u) of F in a critical point u is negative definite. For quasilinear elliptic equations, containing the p-Laplacian (p > 2), it happens that the Euler functionals are defined in Banach spaces. In such a case, the attempt of extending Morse theory to Banach spaces presents some conceptual difficulties. The first one is, in general, the lack of Fredholm properties of the second derivative of the functionals in the critical points. So the classical Morse Lemma does not hold and also generalized versions of Morse Lemma, due to GromollMeyer, fail. Another difficulty in a Banach (not Hilbert) setting is the lack of a proper and reasonable definition of a nondegenerate critical point (see Section 2 below). Furthermore if X is a Banach space and f : X R is a C2 functional,

the existence of a nondegenerate critical point u E X of f having finite Morse index, which is the most interesting case in Morse theory, implies the existence of an equivalent Hilbert structure (see for details [13]). In literature, Uhlenbeck et al. [17, 16, 1] have given a different weaker definition of a nondegenerate critical point in a Banach setting. Such definition involves the existence of an hyperbolic operator which commutes with the second derivative of f"(u) in the critical point u and it does not seem very easy to prove the existence of such operator.

S. Cingolani and G. Vannella

108

In this paper we focus our attention on the estimates of critical groups for some functionals associated to a class of quasilinear equations, involving pLaplacian. Precisely, let A be an open subset of W "(O), where fl is a bounded domain of RN, (N > 1), with sufficiently regular boundary On. We shall consider the following class E(A) of functionals

E(A) _ {Fa : A -41R : F,\(u) = f,,(u) + h (Pf,(u - uo))}

(1)

where A > 0, fa : W,"(52) -+ R is defined by setting

fa(u) =

p Js

IVuI" dx + 2

f

J:

IVUI2 dx + / t G(u) dx

(2)

where 2 < p < oc, G(t) _ o g(s) ds and g E C' (R, IR) satisfies the following assumption:

(g) Ig'(t)I S C1 ItIy + c2 with cl, c2 positive constants and 0 < q < p - 2, p' _ Npl (N - p) if N > p, while q is any positive number, if N = p. Otherwise, if N < p, no restrictive assumption on the growth of g is required. Moreover h : V IR is a C2 function, V is a finite-dimensional subspace of IVo'P(11)nL-(Cl), Pl.: W1 "(fl) -' V is a continuous projection on V and uo E A.

We emphasize that any functional F,, belonging to the class (1) is defined on an open subset of a Banach (not Hilbert) space. We notice that, for each functional F,, E E(A), the second derivative F, \"(u), in any critical point u, is not a Fredholm operator. Moreover in our case, any critical point of Fa is degenerate, in the classical sense introduced in a Hilbert space, since W,l"(52), with (p > 2), is not isomorphic to the dual space W-"D (52) (where 1/p+ 1/p' = 1). In a recent paper [3] we have introduced a new definition of a nondegenerate critical point in the setting of the functional fa introduced in (2) and we have computed the critical groups of the functional fa in each nondegenerate critical point via Morse index in the case A > 0. Here we extend the definition of nondegeneracy introduced in [3], to the admissible functionals belonging to the class E(A), for an arbitrary open subset A of 14."P(c1).

Definition 1.1. Let A be an open subset of WW'P(Ii) and F,, E E(A). A critical point u of F, is said to be nondegenerate if Fa (u) = 0 and Fa (u) : W,"P(52) -- W - i.d (52) is injective.

We notice that the above definition of a nondegenerate critical point is weaker than the usual nondegeneracy condition in an Hilbert setting. Furthermore, it is possible to show that a nondegenerate critical point, in the sense of Definition 1.1, is isolated.

Morse Index Computations

109

In what follows, we fix an open subset A of W0"(S1), an admissible functional

F,, E E(A), and denote the Morse index of F,, in u by m(Fa, u) and furthermore by m'(F,\ u) the sum of m(Fa, u) and the dimension of the kernel of F, \"(u) in

W"(1). At this stage we state our main result.

Theorem 1.2. Let Fa E E(A) with A > 0. Let u E A be a nondegenerate critical point of Fa (in the sense of Definition 1.1). Then m(F,, u) is finite and Cq(Fa, u)

= K,

CC(F,, u) = {O},

if q = m(F,, u), if q # m(F,,, u).

We mention that Uhlenbeck in [16], wrote that in an unpublished article Smale conjectured that the nondegeneracy condition, "the second derivative of the Euler functional in the critical point has no kernel", could be sufficient to develop Morse theory in a Banach Sobolev space. Theorem 1.2 shows that the critical groups of Fa E E(A) in each nondegenerate critical point u E A depend only upon an Hessian type notion, that is the Morse index of F,, in u and in some sense it proves the conjecture by Smale in the setting of class E(A). Furthermore according to the definition of nondegeneracy introduced in (1.1), we can say that Theorem 1.2, in the setting of the admissible functionals Fa E E(A), extends a classical result in Hilbert spaces (see Theorem 2.5 in Section 2). Moreover, for an isolated, possible degenerate critical point, we obtain the following result.

Theorem 1.3. Let F,, E E(A) with A > 0. Let u E A be an isolated critical point of Fa. Then m`(F,,, u) is finite and

Cq(Fa,u)_{0}, ifq<m(F,,,u)-landq>m"(Fa,u)+l. In this paper we also announce that in a forthcoming paper [5] Theorem 1.2

will be useful to prove a perturbation result in the spirit of the Marino-Prodi Theorem [12]. Precisely, we show that any functional F,, belonging to the class (1) with A > 0 is near in C2 norm to a functional, also belonging to the class (1) having nondegenerate critical points in the sense of Definition (1.1). Precisely we announce the following result.

Theorem 1.4. Let us fix A > 0 and A an open subset of Wo''(1). Let u E A be an isolated critical point of F.\ E E(A). Then for any e > 0 and 6 > 0 sufficiently small there exist A > 0, A open set of Wo"(0) and Fi, E CZ(A) such that 1) A C {v E W1 "(!) : 11v - uli < 6} C A and Fa[A E E(A); 2) JJD'Fj(v) - D`F,,(v)II,o < e for i = 0,1, 2 if v E A, 11v - ull > d; F. (v) = FA(v) 3) the critical points of FF, if any, are in A, they are finitely many and nondegenerate, according to Definition 1.1.

110

S. Cingolani and G. Vannella

We remark that if A # 0, then we can take A = A. The case A = 0, instead, is not covered by the critical groups estimates of Theorem 1.2. Nevertheless Theorem 1.4 keeps holding for FO. This assures that near a functional only involving

p-Laplacian (p > 2) there is a functional involving also a semilinear term, for which we are able to compute the critical groups. This perturbation result will allow us to apply Morse relation in order to obtain multiplicity result of solutions for problems merely containing p-Laplacian.

2. Recalls and classical results in Morse theory In this section we recall some notions and results in Morse Theory.

Firstly we need some notations. Let X be a Banach space and f be a C2 real function on X. For any a E 1R, we denote by f' the sublevel {x E X : f (x) < a}. Furthermore let us introduce the notations:

K = {x E X : f'(x) = 0},

K. = {x E K : f (x) = a}. Definition 2.1. Let K be a field. Let u be a critical point of f , and let c = f (u). We call

CC(f, u) = HI (f', f` \ {u}) the qth critical group of f at u, q = 0, 1, 2, ... where Hq(A, B) stands for the qth Alexander-Spanier cohomology group of the pair (A, B) with coefficients in K (Cf. [1]).

Remark 2.2. By excision property, we have also that, if U is a neighborhood of u, then Cq(f, u) = HI (fl n U, (f` \ {u}) n U). Now let us fix u E X a critical point of f. We recall the following definition.

Definition 2.3. The Morse index of f in u is the supremum of the dimensions of the subspaces of X on which f"(u) is negative definite. It is denoted by m(f,u). Moreover, the large Morse index off in u is the sum of m(f, u) and the dimension of the kernel of f"(u). It is denoted by m*(f,u).

Definition 2.4. Let H be a Hilbert space and f be a C2 real functional on H. A critical point u E H of f is said nondegenerate, if f"(u) has a bounded inverse. As showed in the following result, it is possible to compute the critical groups of a nondegenerate critical point via its Morse index, using the Morse Lemma (see for example Theorem 4.1 in 111).

Theorem 2.5. Let H be a Hilbert space and f E C2(H, R). Let u be a nondegenerate critical point off with Morse index in. Then

Cq(f,u)=III

ifq=m, Cq(f,u)=(0)

Nevertheless, if m = +oo, we always have Cq(f,u) = {O}.

ifg0m.

Morse Index Computations

111

These ideas are extended by Gromoll and Meyer for computing the critical groups of isolated critical points, possibly degenerate, of functionals defined on a Hilbert space (see Theorem 5.1 in Ill).

3. The finite-dimensional reduction the scalar product in RN, by Firstly we need some notations. We denote by II the usual norm in Wd"(Sl). Let us define Br(u) = {v E W01''(S1) : Ilv - ull < ' (St) x r}, where u E WW "(1) and r > 0. Moreover we denote by (,) : Wo'p(cl) -, R the duality pairing. In what follows, we fix A > 0 and for convenience let us denote by F the functional FA. Standard arguments prove that F is a C2 functional on A and it is easy to prove that the second order differential of F in u E A is given by II

(F"(u) v, w) =

f(A + IVul2)(VvlVw) dx

+ f(p-2) I oulp-4(VuIVv)(VuiVw) dx + j9'(u)vwdx

+ (h"(PV(u - u°)) Pvv, Pvw) for any v, w E Wo'p(fl). Let us fix an isolated critical point u E Wo'p(fZ) of F and set c = F(u). Since V is a finite-dimensional subspace of WW''(fl) fl L°°(fl), by [14, 151, we can infer

that u E C'(Si). Let b(x) =

Ivu(x)I(p-4)/2 Vu(x) E L°°(11). Let Hb be the closure

of CO '(11) under the scalar product (v, w)b = fn ( A + Jb12)(VvlVw) dx + (p - 2)(biVv)(biVw) dx

and let Hb x Hb -+ IR denote the duality pairing in Hb. We notice that the space Hb is in some sense suggested by F"(u) itself. It is obvious that Hb is W01'2(Sl) equipped by an equivalent Hilbert structure, which depends on the critical point u and Wo'p(f1) C Hb continuously. Furthermore F"(u) can be extended to a Fredholm operator Lb : Hb -, Hb defined by setting (LbV,W)b = (v,w)b + (Kv,W)b where (Kv, w)b =

f g'(u)vw dx + (h"(Pc, (u - u°)) Pvv, Pvw) n

for any v, w E Hb. As g'(u) E L' (0) and V has a finite-dimensional subspace of Wo'p(12), we infer that Lb is a compact perturbation of the Riesz isomorphism from

Hb to Hb and thus Lb is a Fredholm operator with index zero. We can consider the natural splitting

Hb=H-®H°®H+

S. Cingolani and G. Vannella

112

where H-, H°, H+ are, respectively, the negative, null, and positive spaces, according to the spectral decomposition of Lb in L2((2). We remark that H- and H° have finite dimensions and

(Lbv, w) = 0 dv E H- ®H°, dw E H+. Furthermore, denoting by II - IIb the norm induced by

(3)

it is obvious that

there exists c > 0 such that

(Lbv,V)b+c f v2dx>IIvII2 t/vEHb 0

and thus we can infer 3fc > 0 s.t. (Lbv, v)b > µIIvIIb

Vv E H+.

(4)

Since u E C' (Sl), we can deduce from standard regularity theory [9]

H- ®H° C C' (Sl).

Consequently, denoted by W = H+ n Wo'p(Sl) and V = H- e H°, we get the splitting Wj,n(cl) = V E) W

(5)

(F"(u)v,v) > µIIvIIb Vv E W.

(6)

and, by (4) we infer

In particular m' (F, u) = dimV is finite. Furthermore by (3), V and W are also orthogonal with respect to F"(u). We notice that (4) does not assure that F is convex in u along the direction of W, as II ' IIb is weaker than the norm of W,'°((2). Moreover (4) does not guarantee a "uniform convexity" of F near u along the direction of W, with respect to the weak - IIb. Therefore it seems not easy to obtain a finite-dimensional reduction of our problem. In order to get a finite-dimensional reduction, we shall follow the arguments developed in a recent paper by [3). Next two lemmas are necessary to prove the central result given in Lemma 3.3. Indeed, using arguments related to Lemma 4.4 in [3], we shall prove a sort of local convexity of F in the bounded sets of A n L°°(fl) along the direction of W. Firstly we need the following regularity result, which can be deduced using similar arguments in Lemma 4.3 in [3], taking into account that V is a finite-dimensional subspace of L°°((2) n Wj'"(ul). norm II

Lemma 3.1. Let a > 0 such that BQ(u) n A. If z E Ba(u) is a solution of

(F' (Z), w) = 0

(7)

for any w E W, then z E L°°(1). Moreover there exists K > 0 such that IIzfl,,. < K, with K depending on a.

Morse Index Computations

113

Prof. Let {e2 i ... , em. } bean L2-orthonormal basis in V, where m* = m'(F, u). For any v E Wo'p(f ), we can choose v - >_ z e;v dx) e; E W as test function in (7). Therefore we get

(F'(z),v) _

(

jeivdx) (F'(z),ej)

(8)

Let us define r(x) _

(F'(z),e;)e1

+

n

IVzlp-2)

(VzlVe;) + 9(z)ej dy + (h' (Pv(z - uo)) , Pie;) l e;.

By (8), z(x) solves the equation

(F'(z), v) = in

(9)

for any v E Wo''(1). Since V C L°O(fl), we have r(x) E L°°(Sl). Moreover IIrII. C where C is a positive constant depending on a. For j E N let us consider the real functions Xj defined by

Xj(z)=

Iz+j ifz<-j, if -jj.

We can choose v = Xj(z) E W"() as test function in (9). Since I(h'(PV(z-uo)), PvXj(z))I < IIh'(Pv(z-uo))II IIPvXj(z)II < c(a) IIXj(z)IIL1, (10)

we get by (9) and (10)

j

n,

IVzV'dx
2dx+c2(a) j Izldx s2;

where nj = {x E SZ : Iz(x)I > j}. Applying inequality (3.4) in [13], we conclude z E L°°(1) and that there exists K > 0 such that IIzII < K, with K = K(a). Now we get the following crucial lemma.

Lemma 3.2. For any M > 0 there exist ro > 0 and C > 0 such that B,.°(u) C A and for any z E B,.°(u) n L°O(SZ), with IIzII. < M, we have (F"(z)v, v) > CIIvIIb `dv E W.

(11)

S. Cingolani and G. Vannella

114

Proof. By contradiction, we assume that there exist M > 0 and two sequences z E An L'(S1) and v E W, such that IIznLIL < M, IIVmIIw 2 = 1, IIZn - ulI + 0 and lim innf(F"(z )vn,

lnm of ((f"(zn)vn., v,) + (h"(Pf,(zn - uo)) PPv., Pt.vn)) (12)

0,

where f stands for fa. There exists v E W such that v converges to v weakly in Hb and strongly in L'(0), up to subsequences. Since { zn } is bounded in LOO(S) and g E C2 (R, R), we derive

lim J g'(Zn)vn = J g'(u)v'.

n-.00

t

(13)

2

Moreover as Pc, has finite rank, we have that

lim (h"(Pt.(zn - u,)) PP v,,, PPvn) = (h"(PP(u - uo)) PPv, PPv).

ii r )(i

(14)

Ifv=Owe get f0g'(u)v'=0and (h"(PP (u - u(,)) PPv, PPv) = 0. Hence by (13) and (14) we obtain the following contradiction

0> liminfn-z(I'"(zn)vn,vn) 2 A+liminfn_. ,fSg'(zn)vn +liminfn-.oa(h"(P1.(zn - to)) Pl vn, PPVn) > A > 0.

(15)

Then v 34 0. Now, recalling (4), we deduce

0>

liminfn_.OO(f"(zn)vn,Vn)

+liminf,,..-.x,(h"(Pi (z,, -uo))PL'en,PPv.) 2 (L,v,V) 2 PIIVIIb which is a contradiction.

(16)

0

Arguing as in Lemma 4.5 in [3], it is easy to show that u is a strict minimum point in the direction of W and this allows to show a finite-dimensional reduction. More precisely we get the following result. Lemma 3.3. There exist r > 0 and p E ]0, r[ such that

u+(V n B,(0))n(W n B,.(0))c A and, for each v in V n BP(0), there exists one and only one w E W n B,. (0) n L1 (0)

such that for any z E W n 9,.(0) we have

F(v+w+u) < F(v+z+u). Moreover u, is the only element of 14' n Br(0) such that

(F'(u + v + w), z) = 0

dz E W.

(17)

Morse Index Computations

115

The proof of Lemma 3.3 follows by similar arguments to Lemma 4.6 in [3], tak-

ing into account that the functional F E E(A) is sum of two functionals which are sequentially lower semicontinuous with respect to the weak topology of Wo'P(Sl) and then F is also sequentially lower semicontinuous with respect to the weak topology of Wo "(1). So for any v E V the functional F(u + v + ) attains a minimum zu along the direction of W. Using Lemma 3.1, it is possible to show that, if v is near to 0, this minimum w belongs to L°°(f1). Then by Lemma 3.2 one can prove that it is unique and solves equation (17).

So we can introduce the map

0:VnBP(0)- WnBr(O)

(18)

where i,b(v) = w is the unique minimum point of the function w E W n B,.(0) H F(u + v + w), and the function

0:VnBp(0)-'IIt defined by ¢(v) = F(u + v + t(v) ). It is not difficult to show that the maps 0 are continuous. Now let us introduce the set

(19)

and

Y = {u+v+lk(v) : v E VnBP(0)}. Using a suitable pseudogradient flow, as in Section 5 in [3], it can be proved that

C3(F,u) ^--Cj(F1y,u). Moreover as 0 is continuous map, we can deduce that

C3(0,0) ^'Ci(Fr,u), so finally

Cj (F, u) ^-C;(¢,0). In particular, if F"(u) is injective, it can be deduced that 0 is a local maximum

of 0 in v n BP(0), so that Theorem 1.2 comes.

More generally, not requiring the injectivity of F"(u), it is clear that C3(0,0) = {0} when j > m'(F,u) + 1 = dimV + 1. Finally Theorem 2.6 of [6] assures that C3 (0, 0) _ {0} if j < m(F, u) - 1 and thus Theorem 1.3 derives.

References [1] K.C. Chang, Morse Theory on Banach space and its applications to partial differential equations, Chin. Ann. of Math. 4B (1983), 381-399. [2] K. Chang, Morse theory in nonlinear analysis, in Nonlinear Functional Analysis and Applications to Differential Equations, A. Ambrosetti, K.C. Chang, I. Ekeland Eds., World Scientific Singapore, 1998. [31 S. Cingolani, G. Vannella, Critical groups computations on a class of Sobolev Banach spaces via Morse index, to appear on Annales Inst. Henri Poincare: analyse non-

linaire

S. Cingolani and G. Vannella

116

[4] S. Cingolani, G. Vannella, Some results on critical groups estimates for a class of functionals defined on Sobolev Banach spaces, Rend. Acc. Naz. Lincei 12 (2001), 1-5.

[5] S. Cingolani, G. Vannella, A Marino-Prodi perturbation type result for a class of quasilinear elliptic equations, to appear. [6] J.N. Corvellec, M. Degiovanni, Nontrivial solutions of quasilinear equations via nonsmooth Morse theory, J. Diff. Eqs. 136 (1997), 268-293. [7] E. Dibenedetto, C"' local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis T. M. A. 7 (1983), 827-850.

[8] H. Egnell, Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. Rational. Mech. Anal. 104 (1988), 57-77. [9] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1998.

[10] O.A. Ladyzhenskaya, N.N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York London, 1968. [11] S. Lancelotti, Morse index estimates for continuous functionals associated with quasilinear elliptic equations, Advances Dif. Eqs. 7 (2002), 99-128. (12] A. Marino, C. Prodi, Metodi perturbativi nella teoria di Morse, Boll. U.M.I. (4) 11 Suppl. face 3 (1975), 1-32. (13] F. Mercuri, G. Palmieri, Problems in extending Morse theory to Banach spaces, Boll. U.M.I. (4) 12 (1975), 397-401. [141 P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eqs. 51 (1984), 126-150. [15] P. Tolksdorf, On the Dirichlet problem for a quasilinear equation in domains with conical boundary points, Comm. Part. Diff. Eqs. 8 (1983), 773-817.

(16] A.J. Tromba, A general approach to Morse theory, J. Dif. Geometry 12 (1977), 47-85.

[17] K. Uhlenbeck, Morse theory on Banach manifolds, J. F'unct. Anal. 10 (1972), 430445.

S. Cingolani and G. Vannella Dip. Inter. Matematica. Politecnico di Bari via E. Orabona 4 1-70125 Bari, Italy

E-mail address: [email protected] E-mail address: vannella@dm. uniba. it

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 117-126 © 2003 Birkhauser Verlag Basel/Switzerland

A Global Compactness Result for Elliptic Problems with Critical Nonlinearity on Symmetric Domains Monica Clapp Abstract. We give a precise description of all G-invariant Palais-Smale sequences for the variational problem associated with an elliptic Dirichlet problem at critical growth on a bounded domain which is invariant under the action of a group G of orthogonal transformations.

1. Introduction and statement of results Lack of compactness in elliptic problems which are invariant under translations or dilations has been extensively studied. It is known to produce quite interesting phenomena. In particular, it gives rise to an effect of the topology of the domain on the number of solutions of suitable perturbations of such problems (for a detailed discussion see for example [2], [11], [13]). If the domain is invariant under the action

of some group of orthogonal transformations of R', then there is an influence of these symmetries as well. The purpose of this note is to give a precise description of the way the symmetries of the domain affect the lack of compactness. We consider the problem

(P)

1 u= Q(x) I U I2*-2 U

u=0

in S2

on09

where Sl is a bounded smooth domain in RN, N > 3, 2* = NN2 is the critical Sobolev exponent, Q is continuous and strictly positive in Q. M. Struwe [10] gave a global compactness result for this problem when Q - 1. That is, he gave a complete description of the Palais-Smale sequences of the associated variational problem. He showed that the lack of compactness is produced by solutions of the limiting problem Au =1 U I2*-2 u

(P-)

U(X) -+ 0

in 111 ;N

as IxI -+ u

concentrating at points of the domain. Partially supported by CONACyT, Mexico, under Research Project 28031-E, and by INdAM, Italy.

Monica Clapp

118

Here we shall consider domains 1 which are invariant under the action of some closed subgroup G of the group O(N) of orthogonal transformations of RN, that is, domains it such that gx E 11 for each x E Sl, g E G. We also assume that

the function Q is G-invariant, that is, Q(gx) = Q(x) for each x E 0, g E G, and consider the problem

-Du = Q(x) I U I2'-2 u

in 11

U(X) = 0

on aft

u(gx) = u(x)

for all g E G.

Our aim is to give a global compactness result for this problem. Roughly speaking, we will show that lack of compactness is produced by solutions of limiting problems of the form 12.-2 -Du =l u u in RN (Pc,) u(x) 0 as jxj -' 0

u(gx) = u(x) for all g E r concentrating at G-orbits of Sl with orbit type G/r for some closed subgroup r of finite index in G. Before giving a precise statement we recall some basic notions. The G-orbit of a point y E RN is the set

Gy:={gyERN:9EG} and the G-isotropy group of y is the subgroup

Gy:={gEG:gy=y} of G. The G-orbit Gy is G-homeomorphic to the homogeneous G-space of right cosets G/Cy. Observe that isotropy subgroups satisfy that Cy = gGyg-1. Therefore, the set of isotropy subgroups of the points of the C-orbit Gy is the whole conjugacy class of the subgroup G. in G. The C-isomorphism class of GIG, is called the G-orbit type of the G-orbit Gy [7). We denote by IG/Gyp the index of G. in G, that is, IG/Gyj = #Gy is the cardinality of Gy. The action of G on St induces an orthogonal G-action on the Sobolev space Ho (f2) given by

(gu)(x) := u(g-lx) The energy functional

E(u)

2

I

IVu12

jQIul2 .

defined on H(; (St) is G-invariant, that is, E(gu) = E(u) for every u E HH (St), g E G.

The weak solutions of problem (PG) are the critical points of the restriction of E to the subspace of G-fixed points Hii(S1)0 :_ {u E HH(S1) : u(gx) = u(x) for all g E G}

A Global Compactness Result for Elliptic Problems

119

of Ho (12). A sequence (Uk) such that

E(uk) - c, and

uk E HH(1l)G,

IIDE(uk)II -' 0 in H-'(SZ)

will be called a G-invariant Palais-Smale sequence for E, or a C-PS-sequence for short. We shall prove the following. Theorem 1. Let (Uk) be a G-PS-sequence for E. Then, replacing (Uk) by a subsequence if necessary, there exist a solution u of problem (PG), m closed subgroups r, i ... , r, of finite index in G and, for each i = 1, ... , m, a sequence (yi,k) in S1, a sequence (Ei,k) in (0, oo), and a ri-invariant solution (iii) of the limiting problem (P,i ), such that,

(i) G,, , =ri forallk> 1, and yi,k -'yi ask --+oo, (ii) Eik dist(yi.k, O1?) - oo and Ei k IgF/i.k - g'1/i.k I - oo as [9] 0 [g'] E G/ri, ZN sk t.k 11

(iii)

Iluk - u- F_

E Ei' Q(yi) '

iii ( g -1 t

[gIEG/r; in Dl.2(IRN) as k -+ oo, i=1

(iv) E(uk) -> E(u)+

i=1

(_Gird) E(iii) Q(y.)

\

\

JI

k - oo for all 0

as k - coo.

Let us look at some consequences of this result. We write G := min µQ

#Gx

1

N_2 xEfd Q(x)rN

N

S 2 < 00

where #Gx is the cardinality of the G-orbit of x, and S is the best Sobolev constant L2. for the embedding of Ha (fl) in (9). So N S' is the least energy of a nontrivial solution of the limiting problem (P,,.).

We say that E satisfies the G-Palais-Smale condition (PS)" at c if every G-PS-sequence for E such that E(uk) - c has a convergent subsequence. An immediate consequence of Theorem 1 is the following.

Corollary 2. E satisfies (PS)G at every c < µQ. In particular, if every G-orbit in SI is infinite, then E satisfies (PS)G at every c E R. For Q =_ 1 this result is due to P.L. Lions [8]. For arbitrary Q this has been shown in [4]. Corollary 2 says, in particular, that lack of compactness can only occur if 12 contains some finite G-orbit. It implies that problem (PG) has infinitely many solutions if every G-orbit of SZ is infinite. Let us take a closer look at the level MG. The nontrivial least energy solutions

of (P,,.), up to sign, are the instantons N-2

U' Ax) = aN

1-2

E2 + Ix - z 12

aN = [N(N - 2)J a', 6 > 0, z E IIt N.

Monica Clapp

120

cf. [1],[12]. They satisfy

f ' Ivv_,LI = S' = J

N

IU .2I

.

Theorem 1 implies, in particular, the following.

Corollary 3. Let (Uk) be a G-PS-sequence for E such that E(uk) - 4. Then a subsequence of (Uk) either converges strongly to a nontrivial solution of u of problem (PG), or there exist v = ±1, and sequences (yk) in Il and ek in (0, oo), such that y E Sl as k oo, Gyk = Gy for all k, and (i) Yk

(ii) eA 1 dist(yk. 81)

#Gy_, =min #Gx v, < oo, Q(y) 2 xES? Q(x)' r oc and eA I911k - 9'ykI -' oc ask - oo for all [g] # 1

[9'] E GIG, (iii)

Q(7/)T_UEk.9ykll Uk

-' 0

in Dl'2(RN) as k - oo.

19)EC/Cy

Corollary 3 says that least energy nonconvergent G-PS-sequences must concentrate at C-orbits of the set

l :=

y E Ti : Q(Y)N-2

min

#Gx

xE12 Q(x)

N-z

This hints towards the fact that the topology of M must have an effect on the number of low energy solutions of suitable perturbations of problem (pr). This is, in fact, true. Examples of this behavior can be found in [4] and [6]. But it says

more than that: It says that concentration occurs along G-orbits Gyk with the same orbit type as Gy. This has proved to be quite useful in applications, see for example [5], [6].

The main step in the proof of Theorem 1 consists in showing that concentration occurs along G-orbits with the same orbit type. This allows us to proceed inductively to obtain a global compactness result for problem (P°), that is, to give a description of all C-invariant PS-sequences, and not only of C-minimizing sequences as was done in [4].

2. Proof of Theorem 1 As in the non-symmetric case [11], [13], Theorem 1 follows inductively from the following proposition.

Proposition 4. Let (uk) be a G-PS-sequence for E such that uk - 0 weakly in H01(S1)' and E(uk) -+ c > 0. Then, replacing (Uk) by a subsequence if necessary, there exist a closed subgroup I' of finite index in G, a sequence (yk) in 0, a sequence

A Global Compactness Result for Elliptic Problems

121

Ek in (0, oo), a r-invariant solution u` of the limiting problem (P.), and a G-PSsequence (vk) for E such that (i) Gvk = r for all k, and yk y as k oo, (ii) Ek' Igyk - g'yk I -b oo and Ek 1 dist(yk, OSt) -+ oo as k - oo for all [g] # [g'] E G/r, (111) Vk

= Uk-

Q(y), 4Nu

Ek

+o(1) in

(9- 1

D1,2(RN),

Ig]EGG/r

(iv) E(Vk)

C - (_iciri

ask - oo.

I E(u)

Q(Y) r

Proof. The proof will follow in several steps:

1) Since PS-sequences for E are bounded in H,()),

Q(x) Iuk12 dx = NE(uk) - 2 DE(Uk)uk

Let b := min{ 2(maxi Q)

2-11

Nc > 0.

N

(2) 3 } where S is the best Sobolev constant for the embedding of Ho (S2) in (1). Let B(x, r) denote the closed ball in RN with center x and radius r. The Levy concentration function L2.

IUk12'

'Ik(r) := SUP XERN

J

x,r)

Q

satisfies that 4k(0) = 0 and (Dk(oo) > d for k large enough. Hence we may choose lk E S2 and Ck > 0 such that

sup f (x,Ek) Q Iuk12 = f ( k,Ek) Q Iuk12 = a,

xERN

(1)

Observe that, since SZ is bounded, the sequence (Ek) is bounded.

2) Let V = RN and, for each closed subgroup H of G, let

VH={xEV:gx=x for aligE H} be its H-fixed point subspace. If C E V we write l:H for the orthogonal projection of t: onto V H .

We shall show that there is a closed subgroup r of G such that, up to a subsequence,

a) r has finite index in G,

b) Gtr = r for all k, c) Ek 1

I -+ oo as k -' oo for all [g] 54 [g'] E G/r,

d) Ek

1

I
for all k, then r = G has the desired properties.

M6nica Clapp

122

If £k' Ilk - ek I

00, let VI be the orthogonal complement of VC in V and 0 and k

write. Sk = A + k Since £k' Irk I -' 00, up to a subsequence, l

171 =

n1 E Vi. Ifk1

Let G' := Gn'. For every closed subgroup H of G', (qk)H

(171)H = i7'. Hence,

GO = G(E6)H = G(fk)H C G,11 = G'.

for k large enough and, in particular, G,k = G'. We now show that I GIG, I < 00.

For every finite subset {gi, ..., g,,,} of G representing distinct cosets EG/G' let p> 0 be such that

fori 34 j

I9i71' - 9ii7'I>2P Then, for every closed subgroup H of G1,

for i 36j and k large enough

I9()H - 9j (,1k) H I > p and, since Ek' Irk I - 00,

[ - 9i [ £k' l9ibk

£k' I&kI P < £k'

I

oo

fori

j.

In particular, B(g, k,£k) fl B(gAk,£k) = 0 for i 0 j and k large enough so, since Uk and Q are C-invariant,

.OA

MS

J=1

Q luk 1

2< f Q

Iuk12.

= Nc + o(1).

S2

A.EA.)

It follows that IG/G' I < oo and that

=£k' Igk

£k'

00

if [910 [9'1 E GIG'

for every closed subgroup H of G'. So, if £k' Ck' I < C < oo for all k, then l = C' has the desired properties. If not, we proceed inductively as above to obtain a set of closed subgroups G = G° G' J J G" = I', a set of linear subspaces V = Vo i V, j ... J V",

and a set of points l;k = £A, tk,..., k, such that IGi/Gi+' I < oo, Vii c V°i+', E V+1,

V V" W Vi+1, h+ . --

ek' Igk - 9 k I

and £k'

c'

- oo

for [g] 0 [g'] E G` /G`+' ,

oo, f o r e a c h i = 0, ... , n - 1, but £k' I{k

for all k. Therefore, r has the desired properties. 3) We write Yk := k and define x-z Uk(Z) := £k

uk(EkZ + Ilk)

and Qk(Z)

-

Q(£kz + 1/k)

I < C < 00

A Global Compactness Result for Elliptic Problems

123

Thus, Uk and Qk are F-invariant, JQkIlZkI2

and

f IDuk12 = f IDukI2

= JQItkI2

In particular, (uk) is a bounded sequence in D1,2(RN)r. Hence, up to a subsequence, Uk - u weakly in D' 2(RN)I', Uk --+ u a.e. on RN and ilk -+ u in L aC(RN)

If u = 0 then, for every z E RN and every h E C°°(B(z,1)),

/r \ S1JIhkl21 J

2

JIV(hik)l2

Vk - O(h2ilk) +

J

IVhI2 Lk

f h2Qk IUkI2' - DE(uk) (h2

(.-Ek k) uk) +0(1) IV

Q),

z

-29-

( maxft

(Qk

< (Q)b (f -z

2

fB(z,l)Iukl2')

2

(JIhkI2)

2

+0(1)

24

+0(1)

(fiiJ +0(1),

where the first inequality is Sobolev's inequality, the second one follows from the fact that (Uk) is a PS-sequence and from Holder's inequality, and the third one uses (1). It follows that Uk - 0 in Ll '(RN). On the other hand, since ek 1 Itk - ykI < C < oo for all k,

b= J

f

Iuk12.

Q

Q IukI2* < B(Yk.£k(C:+1))

( 4k,Ek)

(O.C+1)

Qk IukI2* <_ (maxQ) J 3i

IukI2'

(O,C+1)

This is a contradiction. Therefore, i 0- 0. 4) Since St is bounded and uk -s 0 weakly in Ho (1l), up to a subsequence,

yk -+yEandek - 0. If (ek 1dist(yk,80)) is bounded, we may assume that

lim ek ldist(yk,8il) = d.

k-»oo

It is then easy to verify that, up to a rotation of RN, the sets ilk := {z E RN CkZ + Yk E il} satisfy 00

n=1

-

(kUn Qk) = HN

{(z1, ... , 1N) E RN : ZN

-d}.

Monica Clapp

124

Hence, u is a solution of the autonomous equation U

-Du = Q(y)

in lil
and, by Pohozaev's identity [9], u =- 0. This is a contradiction. Therefore, Ek'dist(yk,On) -+ 00

and u := Q(y)

u is a nontrivial solution of the limiting problem (PP) in RN. yk I < C < oo for all k, it follows also that yk E Cl.

Moreover, since ek'

5) We define vk E Ho (Cl)G as follows: Let V E C, (RN) be radially symmetric

and such that 0 <

Thus, Ek'Pk - oo. Since Gy,, = I and since u is I'-invariant, the function 2-N

EQ(y)

wk :=

2

u(Ek'9-'(' - 9yk))'P(P, 1(' - 0k)) E H'(0)

]g]EG/r

does not depend on the choice of g in [g], and it is G-invariant. We define Vk := Uk - Wk E Hp(Q)G.

Property iii) above follows from the fact that ek'Pk

00. We verify property

iv). We write IIuII2 :=1 IVu12.

Let G/t = { [gl] , ... , [g,,]}. Since E.-' I9iyk - 9iykI -+ oo ask -+ 00 for each i # j, u = Q(y) u' weakly in D1 2(RN), and Uk is G-invariant, it follows that Uk 2

2-N

2- V

9tyk

uk- \L' Ek Q(y) ugi 1 i=1 n

Ek `

N-2

Ek uk(Ek

n -

+91yk)-

Q(y)T-

u -1 9`

i=1

(,

2

+ 91yk -9iyk Ek

J ,2

uk91 ' - Q(y) 2

u9i 1-

Q(y) i#1

ii9i-1 l(

91Yk - 9iyk Ek

J

A Global Compactness Result for Elliptic Problems

125

2

-1 uk91

4 u9i Q(y) 2-N

+91yk-9iyk/

1

Ek

i#1

a9-1

-11Q(y)"

u91 1112+0(1)

-

- 9i yk

Q(y)2-N

2

Ek

- Q(y) -II ull2+o(1)

and, inductively, n

=

IIuk1I2

z Q(y) 2 °N

Ek

Uk

2-N

i=1

+

-1 u9i

- 9iyk

2

Ek

"N_2 IIuI12 +o(1)

Q(y) z

IIvk112 +

IG/rl 2

11112 +o(1)

Q(y) N2

Similarly, using the Brezis-Lieb Lemma [3], we obtain IuI2.

f Q luk l2. =

fQ

Ivkl2*

+ Q(Y) (y))

and property iv) follows. A similar argument using Lemma 8.9 in [13] shows that IIDE(vk)IIH

-' 0.

As in [10], [11], [13], Theorem 1 follows inductively from this proposition. We sketch the proof for the reader's convenience.

Proof of Theorem 1. Since PS-sequences for E are bounded in Ho (St),

I

10uk12dx = NE(Uk)

- N2

2DE(uk)uk

Nc.

Therefore c > 0. We may assume that Uk - u weakly in Ha (SZ)G and Uk -+ u a.e.

in n. It is easy to see that DE(u) = 0 and that uk := Uk - u is a G-PS-sequence such that uk - 0 weakly in H] (SZ)' and E(uk) = E(uk) - E(u) = c- E(u) +0(1). The result now follows inductively from Proposition 4.

References [1] Th. Aubin, Problemes isoperimr triques et espaces de Sobolev, J. Diff. Geom. 11 (1976), 573-598. [2] H. Elliptic equations with limiting Sobolev exponents - The impact of topology,

Comm. Pure Appl. Math. 39 (1986), S17-S39.

Monica Clapp

126

[3]

Brtzis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490. M. Clapp, On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem, Non]. Anal. 42 (2000), 405-422. M. Clapp, G. Izquierdo, Multiple positive symmetric solutions of a singularly perturbed elliptic equation, Topol. Meth. in Nonl. Anal., to appear. M. Clapp, 0. Kavian, and B. Ruf, Multiple solutions of nonhomogeneous elliptic equations with critical nonlinearity on symmetric domains, preprint. T. tom Dieck, Transformation groups, de Gruyter Studies in Math. 8, de Gruyter, Berlin-New York 1987.

P.L. Lions, Symmetries and the concentration compactness method, in Nonlinear variational problems, Research Notes in Math. 127, Pitman, London 1985, 47-56. S. Pohoiaev, Eigenjunctions of the equation Du + of (u) = 0, Soviet Math. Dokl. 6 (1965), 1408-1411. M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), 511-517. M. Struwe, Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer, Berlin-Heidelberg 1990.

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353-372. M. Willem, Minimax theorems, PNLDE 24, Birkhauser, Boston-Basel-Berlin 1996.

Monica Clapp Instituto de Matema.ticas Universidad Nacional Autonoma de Mexico Circuito Exterior, Ciudad Universitaria 04510 Mexico, D.F., Mexico E-mail address: mclapp®math.unam.mx

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 127-139 © 2003 Birkhauser Verlag Basel/Switzerland

Variational Methods for Functionals with Lack of Strict Convexity Marco Degiovanni Abstract. We consider a class of functionals of the calculus of variations which

are not strictly convex in the gradient of the function. When critical point theory is applied, this causes some troubles with the Palais-Smale condition, which we overcome using a suitable nonsmooth approach.

1. Introduction Let St be a bounded open subset of R1. Consider a boundary value problem of the form

div(V'(Du)) = g(x,u) in 1),

(1.1)

u=0

onOSl. Problem (1.1) has a variational structure, being the Euler equation associated with the functional f (u) _ 11 IF (Du) dx t

Jn

G(x, u) dx,

G(x, s) = J s g(x, t) dt, 0

defined on a suitable functional space. The most studied model case is

1
Suppose also that 1 < p < n and that g : S2 x R

b,v>0. ]l8 is a Caratheodory function

satisfying

Ig(x,s)I 5 a(x)+blslp'-1,

a E L(p )'(S2),b E R,

where, as usual, p' :_ The research of the author was partially supported by Ministero dell'istruzione, dell'Universita e della Ricerca (COFIN 2001) and by Gruppo Nazionale per l'Analisi Matematica, la Probability e le loro Applicazioni (INdAM)..

Marco Degiovanni

128

Under these conditions, f is naturally defined on WW'"(12) and f : W, "(fl) - R is in fact of class C'. The main feature is that ' is not required to be strictly convex. For instance, one could take p = 2 and P : R' -. R defined as W(i;)=

ICI2

if IcI <-1,

2ICI-1

if l
I ICI2 + 1

if ICI > 2.

Then, for the nonlinearity g, one could consider two typical cases:

(a) suppose lim

g(x's)

= 0 uniformly in x and look for a minimum point u of

./

(b) suppose g(x, s) zt Is]q-2s with 2 < q < 2' and look for a mountain pass point u of f, as in the Ambrosetti-Rabinowitz theorem [1]. Case (a) is standard, because f is weakly lower semicontinuous and coercive in W01'2(S ). In case (b), we have the geometry of the mountain pass. However, the Palais-Smale condition does not hold (in both cases (a) and (b)). For instance, in

case (a), it may happen that the set of minima for f is not strongly compact in W;'2(S2).

Thus, with this choice of 41, the situation is absolutely standard when looking for minima, but not for critical points. The reason is that there is a gap between the two approaches: usual minimization techniques are based on the weak topology of Wi1'2(S2), while classical critical point theory uses the strong topology. In a certain

sense, minimization techniques follow a nonsmooth approach (the functional is discontinuous in the weak topology), while in classical critical point theory the functional is at least of class C' (see, e.g., [3, 15, 16, 17]). The purpose of the paper is to show that a suitable nonsmooth approach, described in Section 4, provides a unified setting which allows to treat both cases (a) and (b). The idea, in the case of the above example, is to extend the functional to L"(0) with value +oo outside W01'2(11). In this way we are led to consider the strong topology of L2 (S1), where enough compactness can be recovered. Of course, this is obtained at the expenses of the smoothness of the functional. A similar strategy has been used in [8, 13, 14], to deal with the area functional. From the abstract point of view, the reference setting is that of [18], recalled in Section 3. For the compactness properties, a crucial role is played by Theorem 2.1, which is an adaptation of [8, Theorem 4.101 and could be interesting in itself.

2. A compactness result Let Il be a bounded open subset of R", let I < p < n and let $ : continuous convex function. Assume that

r there exist b, v > 0 such that

Sl vKKI"-b<"(c)

b(II"+1)

R",N

-+ lR be a

Variational Methods for Functionals with Lack of Strict Convexity

and define a functional £ :

129

R by

£(u) = 1 T (Du) dx.

(2.2)

In the sequel, for every q E [1, +oo] we will denote by II IIq the usual norm in Lq(D; RN).

Theorem 2.1. If (uh) is weakly convergent to u in W1 "(SZ; RN) with £(uh) - £(u), then (uh) is strongly convergent to u in LP *(Q).

Theorem 2.2. Assume that Il has Lipschitz boundary. If (uh) is weakly convergent to u in Wl'P(1l;1RN) with E(Uh) - £(u), then (uh) is strongly convergent to u in LP (11).

The section will be devoted to the proof of these results. First of all, let us see that Theorem 2.1 follows from Theorem 2.2. Proof of Theorem 2.1.

Let B be an open ball in iir with Il C B. Let Vh, v E W0" (B) be the extensions of uh, u with value 0 outside Q. Since

limJ 41 e

JB

0

the assertion follows by Theorem 2.2.

To prove Theorem 2.2, some lemmas are required. Lemma 2.3. There exist t:o, a E 1R"v such that

VC#0: Proof. We give here a proof when IV is of class C. The general case can be easily deduced e.g., from [4, Theorem 1.4.1]. 1RnN By By Sard's theorem, there exists a regular value a of VI 1RnN minimizing {l a l;}, we find Co E RnN with VW(t:o) = a. Since a is a regular value, W"(t:o) is nondegenerate, hence positive definite by the convexity of T. Then the assertion easily follows. 0 Define ;P- : 1RnN --+ R by

Define also -u E Wl'p(Il;1RN) by u(x) = tox and £ : W',P(II) -+ R by

£(v) = I

dx.

n

Then, for every u E

we have

r

E(u-u)=£(u)-£(u)-J

I,

Marco Degiovanni

130

Therefore, to prove Theorem 2.2, we can assume without loss of generality that

f q'(0) = 0, lVV96 0: Lemma 2.4. The following facts hold:

(a) for every e > 0 there exists C > 0 such that CISI1'+e,

VF, E RnN :

(b) for every E > 0 there exists 5 > 0 such that

V E R'

:

q)

W

? S (I Ip - E) .

0

Proof. The assertions easily follow from (2.1) and (2.3).

Let now 19EC,!(RN)with 0 2. Define t9h : IRN --. R and Th, Rh : RN

)h(s)

99

\h/ ,

Rh(s) = (1

Th(s) = 6h(s)s,

RN by 99h(s))8.

Lemma 2.5. The following facts hold:

(a) for every E > 0 there exists C > 0 such that 6(7'k. o u) + E(Rk o u) < E(u) + e + C

J{zEft: k
whenever k > 1 and u E Wl,r(Cl;RN); 0such that >(in

(b) for every e > 0 there exists 5

E(u) > b

IDuI' dx -

E\ I

,

whenever u E W',N(S2;1RN).

Proof. Since

D[19(u) u]

0`u)Du+ u0 .1.(u)Du]

we have

ID(Tk o u)I 5 5IDul ,

ID(Rk o u)I < 5IDul .

Given e > 0, by (a) of Lemma 2.4 there exists C > 0 such that V t E IR"` N :

'P(t) 5

C 5P 2

Itip + 2C' 1)

Variational Methods for Functionals with Lack of Strict Convexity

131

Since

I IV (D(Tk o u)) dx +

J

IF (D(Rk o u)) dx

(D(Tkou))dx

- f1jul:5k) iP (Du)dx+J{k
f

4 (D(Rk o u)) dx + k
J{IuI>2k}

41 (Du) dx,

the assertion (a) follows. The assertion (b) is an easy consequence of (b) of Lemma 2.4. Lemma 2.6. Let (uh) be a bounded sequence in Wi"P(11; RN). Then for every e > 0

and every k E N there exists k > k such that

I

liminf h-oo

J

IDuhI'dx < E.

{k
Proof. Let m > 1 be such that sup fn I Duh I" dx < h

Si

2

and let io E N with 2'0 > k. Then, since +o+m-i i=io

J

I Duh I' dx < 21
r

J

I Duh I P dx <

S

me 2

there exists ih between io and io + m - 1 such that IDuhlPdx < 2 . {2'h
Passing to a subsequence (ih; ), we can suppose ih, we get

i > io, and setting k = 2' g

IDuh,lPdx< 2.

`djEN: {k
Then the assertion follows.

Lemma 2.7. Let (uh) be a sequence weakly convergent to u in W',P(1l;RN) with ENO --. E(U)Then for every e > 0 and every k E N there exists k > k such that lim anf IID(Rk 0 uh)IIP < E.

Marco Degiovanni

132

Proof. Given e > 0, by Lemma 2.5 there exist b, C > 0 such that

r

E(u)>b

P\

JtIDuIPdx - 4 J

(2.4)

,

r 66P E(Tk o u) + E(Rk o u) < E(u) + -+C J 4

IDuIP dx ,

(2.5)

ff {xEf2: k<ju(i )I<2k}

whenever k > 1 and u E that

By Lemma 2.6, there exists k > k such

5P

E(Tk o u) > E(u) - 4 h-00

P

I

lfm inf

,

IDuhIP dx <

(k<ju.j<2k)

From (2.5) we deduce that E(Tk o u) + lim inf e(Rk o uh)

h-

4C

h--oo

< lim inf E(Tk o u,1) + lhm inf E(Rk 0 Uh)

<

lim inf (E(Tk o uh) + E(Rk o uh))

< E(u) +

4P

+ C lim inf

h -o

J

IDuh IP dx

Y

3 < E(u)+ 2 <E(Tkou)+4beP,

whence

liminfe(Rkouh) <

38ep.

h-.oo

From (2.4) the assertion follows. Proof of Theorem 2.2.

By contradiction, up to a subsequence we may assume that there exists e > 0 such that Iluh - uIIP. _> e. On the other hand, (uh) is strongly convergent to u in LP(fl;R"). Let c be a constant such that < e(IIDvhIP+IIvIIP) for any IIvIIP

v E W I -P(II; RN). According to Lemma 2.7, let k E N be such that I1Rkoulip <

2

, Iim inf I1RkouhIlp <- c liminf (IID(Rk 0 uhIIP + 11 R;. 0 uhIIP) <

h-.

h-oo

2

Then we have Iluh - uIIP' <- IIRk o uhIIP + IITk o uh - Tk o uIIP + IIRk o uIIP Since Tk o Uh

TA, o u in LP* (Il; RI) as h

oo, passing to the lower limit in (2.6)

we get

lim inf lluh - uIIP < e, h-co whence a contradiction.

(2.6)

Variational Methods for Functionals with Lack of Strict Convexity

133

3. On a particular class of functionals Let X be a Banach space and f : X

R U {+oc} a functional of the form f = fo + It, with fo : X - R U {+oe} convex, proper and lower semicontinuous and f, : X -+ R of class C. Variational methods have been first applied to this class of functionals in [18]. Further developments and refinements can be found in [8, 9, 13], as an application of the metric critical point theory of [6, 7, 11, 12].

Definition 3.1. We say that u E X is a critical point for f, if f(u) E R and -fi(u) E afo(u) It is readily seen that, if u is a local minimum for f, then u is a critical point for f.

Definition 3.2. Let c E R. We say that (uh) is a Palais-Smale sequence at level c ((PS)c-sequence, for short) for f, if f(uh) - c and there exist ah E afo(uh) with

(ah + fi(uh)) - 0 in X. We say that f satisfies the Palais-Smale condition at level c ((PS),, for short), if every (PS), -sequence admits a strongly convergent subsequence in X. Theorem 3.3. Assume that f is bounded from below and satisfies (PS), at the level

c=infx f. Then there exists a minimum point, in particular a critical point, for f. Proof. It easily follows from Ekeland's variational principle (see, e.g., [10]).

Theorem 3.4. Let f be even and assume that there exists a strictly increasing sequence (Vh) of finite-dimensional subspaces of X with the following properties: (a) there exist a closed subspace Z of X, _o > 0 and a > f (0) such that X = VomZ and

Vu E Z: IIull=9

f(u)>a;

(b) there exists a sequence (Rh) in ]p, +oo[ such that Vu E Vh :

Hull > Rh - f(u) < f(0);

(c) for every c > a, the function f satisfies (PS)c. Then there. exists a sequence (uh) of critical points off with f (uh) -' +z. Proof. In a weaker form, the result can be found in [18, Corollary 4.8]. The general case is contained in [5, Theorem 4.8] and in [13, Theorem 1.2].

4. A nonsmooth approach Let again fZ be a bounded open subset of R". 1 < p < n and T : R"^' -. R a convex function. Assume that T is of class C' and satisfies (2.1). As a consequence, we have that

there exists b > 0 such that

(4.1)

1VT(O] < b0cip-I + 1). Therefore, the functional E : Wo'P(1l; RN) -+ R defined by (2.2) is now of class C'.

Marco Degiovanni

134

Theorem 4.1. Let (Uh) be a sequence weakly convergent to u in Wo'P(Sl; RN) with (E'(uh)) strongly convergent in W-1,P'(SZ;1RN). Then (uh) is strongly convergent to u in LP* (0;1RN).

Proof. By the convexity of E we have Therefore E(uh)

E(u) 1 E(uh) + (E'(uh), u - uh) . E(u) and the assertion follows by Theorem 2.1.

Now consider also p E W-'-P'(11; RN) and a function C : Cl x RN --+ R such

that: for every s E iRN, G(., s) is measurable,

for a.e. x E Cl, G(x, ) is of class C' , G(x, 0) = 0 and there exist a E L(P* )' (St) and b E R such that

(4.2) (4.3)

JV C(x, s)J <- a(x) + blsw'-1 Then the functional E(u) - G(x, u) dx - (p, u) in

is well defined and of class C' on W,'P(C1;1RN). If we set g(x, s) :_ V9G(x, s), its critical points are the weak solutions u of the problem

div(VW(Du)) = g(x,u) +p in fl,

on M.

U=0

(4.4)

However, as we have observed in the Introduction, we cannot expect the PalaisSmale condition to hold in such a space, because of the lack of strict convexity for T. On the other hand, the general results of Section 3 and Theorem 4.1 suggest to try a nonsmooth approach, based on the space LP' (11; RN). More precisely, define f : LP* (Cl; RN) - 1R U {+oo} by f = fo + fl, where fo(u)

E(u) - (p,u) if u E WW'P(f1;RN), if u E LP'(ll;1RN) \ WO.P(Sl;R '), +cc

fn(u)=- rG(x,u)dx.

n Then fig : LP' (SZ; R') R U {+oo} is convex, proper and lower semicontinuous and f, : LP* (Sl; RN) --+ R is of class C', according to the setting of Section 3. First of all, let us describe the concrete counterpart of the abstract notions of Section 3.

Proposition 4.2. Let u E L p (52;1RN). Then u is a critical point off if and only if u E Wo'P(SI;RN) and u is a weak solution of (4.4).

Proof. Given w E L(P')'(fl;1RN), it is easily seen that w E Ofo(u) if and only if u E WW'P(1l;IRN) and

- div(Qlt(Du)) - it = w. Then the assertion is just a reformulation of Definition 3.1.

Variational Methods for F unctionals with Lack of Strict Convexity

135

Proposition 4.3. Let c E R and let (uh) be a sequence in LP(11;RN). Then (uh) is a (PS),-sequence for f if and only if uh E Wo'P(Sl; RN), f (uh) ' C, div(V%P(Duh)) +µ E L(P)'(Sl;RN) and

g(x, uh) + µ '0 strongly in L(P)'(S2; RN). Proof. The assertion is just a reformulation of Definition 3.2.

Now we can prove a first result based on minimization. Actually, assertion (c) of the next theorem follows from standard arguments based on weak lower semicontinuity. The interesting point is that, in spite of the lack of strict convexity for 'Y, it can be deduced also by Theorem 3.3. This is possible, as the "natural" domain Wo'P(Sl; RN) has been substituted by LP (Sl; RN), at the expenses of the smoothness of the functional. Let us also observe that a true subcritical growth for g is not needed. Assumptions (4.3) and (4.5) are enough also for an approach based on (a suitable adaptation of) the Palais-Smale condition. Theorem 4.4. Assume that for every e > 0, there exists aE E L' (Sl) such that

G(x,s)
(4.5)

Then the following facts hold:

(a) the functional f is bounded from below; (b) the functional f satisfies (PS), at the level c = inf f ; (c) there exists a minimum point u E Wo'P(il; RN) of f, hence a weak solution of (4.4).

Proof. From (4.5) it follows that f is bounded from below and coercive on Wo'P(Sl;RN) and that fl is sequentially lower semicontinuous with respect to the weak topology of Wo'P(Sl; RN). Therefore (a) holds true. On the other hand, we will show that (b) also holds, so that (c) can be deduced from Theorem 3.3. Thus, let (uh) be a (PS),-sequence for f. Since f is coercive, up to a subsequence (uh) is weakly convergent to some u in Wo'P(Sl;RN) and

limf(uh)=c< f(u). Since fo and f, are both sequentially lower semicontinuous with respect to the weak

topology of Wo'P(Sl;RN), we have fo(uh) --' fo(u), hence 6(uh) - 6(u). From Theorem 2.1 we conclude that (uh) is strongly convergent to u in LP' (Sl; RN ).

When g has subcritical growth, the Palais-Smale condition is equivalent, as usual, to an a priori bound. Proposition 4.5. Assume that for every e > 0, there exists aE E LiP)'(Sl) such that (4.6)

Ig(x,s)1 <_ a,(x) + elsIP*-'

Marco Degiovanni

136

Then, for every c E R, the following assertions are equivalent:

(a) the functional f satisfies (PS),; (b) each (PS),.-sequence for f is bounded in D" (S); RN ).

Proof. It is obvious that (a) implies (b). To prove the converse, consider a (PS)-sequence (uh) for f . We know that (uh) is bounded in LP* (S2;1R^' ). It follows that (f1 (Uh)) is bounded, hence (f f)(uh)) is bounded. By (2.1) we deduce that (Uh) is bounded in 4' ''(S1; R'), hence weakly convergent, up to a subsequence, to some u in R'). By assumption (4.6) we have that (g(x, uh)) is strongly convergent to g(x, u) in R"). This fact follows by [2, Theorem 2.2.7] in the case p = 2, but the same argument works in the general case. According to Proposition 4.3, we deduce that. (div(V%P(Duh))+p) is strongly convergent in L(p')'(St; R"). In particular, (div(V'P(Duh))) is strongly convergent in W-",n'(fl; RN). From Theorem 4.1

0

we conclude that (uh) is strongly convergent to u in .1/(S2;RN).

Finally, to give an example in which critical point theory is used in a more decisive way. we prove a result of Ambrosetti-Rabinowitz [1] type.

Theorem 4.6. Assume that g satisfies (4.6) and that there exist q > p and R, b > 0 such that R 0 < qG(x, s) 5 g(x, a) s, R . (q - 6)T(t)

(4.7)

Suppose also that and G(x, ) are both even. Then there exists a sequence (u,,.) in W,'F'(S2;R") of weak solutions of

-div(V%P(Du))=g(x,u)

lu=0 with E(u,,) -

Jif

inst. on 60,

G(x, uh) dx - +oo.

Proof. By (4.6), (4.7), (2.1) and (4.1) there exist a E L'(Sl) and b E R such that

G(x, s) > R-9G(x, R)JsJ9 - a(x),

(4.8)

qG(x, s) < g(x, s) s + a(x) ,

(4.9)

(q - 6)4'(t;) + b.

(4.10)

We want to apply Theorem 3.4. First of all, let us see that (PS), holds for any c E R. Let (uh.) be a (PS). -sequence for f. According to Proposition 4.3, let wh - 0 strongly in L(P )' (52; RN) with - div(VW(Duh)) - 9(x,uh) = wh .

Variational Methods for Functionals with Lack of Strict Convexity

137

By (4.9) and (4.10), it follows dx

(V f1

< f((q - b)W'(Duh) - qG(x, uh)) dx + J (a + b) dx

sFrom

n

-6 f W(Duh)dx+gf(u),)+ f(a+b)dx. (2.1) we conclude that (Uh) is bounded in Wo'p(Sl; RN) and (PS)c holds by Proposition 4.5. Arguing as in [13, Lemma 3.8], we can find a strictly increasing sequence (Wh)

of finite-dimensional subspaces of WW''(Sl; RN) n L°° (Sl; RN) and a strictly decreasing sequence (Zh) of closed subspaces of LP* (Sl; Rn') such that La (ft RN) _ 00

Wh 6) Zh and n Zh = {0}. h=0

Let us show that sup (inf {f (u) : u E Zh, IIuIIp = 1}) > f (O) h

By contradiction, let uh E Zh with Iluhllp = 1 and limsup f(uh) < f(0). From h

(4.6) it follows that (uh) is bounded in Wo'p(Sl;RN), hence weakly convergent up 00

to a subsequence to some u. Since u E n Z, , it must be u = 0. By (4.6) we also h=O

have lim f h

G(x,uh)dx = 0, f1

hence lirE(uh) = E(0). From Theorem 2.1 we deduce that (uh) is strongly conk

vergent to 0 in LP' (n; R'), whence a contradiction. Now, let h with inf if (u) : u E Zh, IIuIIp = 1) > f (0), let Z = Zh and let Vh = Wh+h Since q > p and Vh is finite-dimensional, from (2.1), (4.7) and (4.8) it follows that there exists Rh > 1 such that u E Vh, IIuIIp > Rh f(u) < f(0) Therefore Theorem 3.4 can be applied and the assertion follows by Proposition 4.2.

Marco Degiovanni

138

References [1] A. AMBROSETTI AND P.H. RABINOWITZ, Dual variational methods in critical point

theory and applications, J. Funct. Anal. 14 (1973), 349-381. [2] A. CANINO AND M. DEGIOVANNI, Nonsmooth critical point theory and quasilinear

elliptic equations, in Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), A. Granas, M. Frigon and G. Sabidussi, eds., 1-50, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 472, Kluwer Acad. Publ., Dordrecht, 1995. [3] K.C. CHANG, "Infinite-dimensional Morse theory and multiple solution problems", Progress in Nonlinear Differential Equations and their Applications, 8, Birkhauser Boston, Inc., Boston, MA, 1993. (41 F.H. CLARKE, Yu.S. LEDYAEV, R.J. STERN AND P.R. WOLENSKI, "Nonsmooth analysis and control theory", Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. (51 J.-N. CORVELLEC, A general approach to the min-max principle, Z. Anal. Anwendungen 16 (1997), 405-433. (6] J.-N. CORVELLEC, M. DEGIOVANNI AND M. MARZOCCHI, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal. 1 (1993), 151-171. [7] M. DEGIOVANNI AND M. MARZOCCHI, A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl. (4) 167 (1994), 73-100. [81 M. DEGIOVANNI. M. MARZOCCHI AND V.D. RADULESCU, Multiple solutions of hemi-

variational inequalities with area-type term, Calc. Var. Partial Differential Equations 10 (2000), 355-387. [9] M. DECIOVANNI AND F. SCHURICHT, Buckling of nonlinearly elastic rods in the presence of obstacles treated by nonsmooth critical point theory, Math. Ann. 311 (1998), 675-728. [10] I. EKELAND, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474. [11] A. IOFFE AND E. SCHWARTZMAN, Metric critical point theory 1. Morse regularity and homotopic stability of a minimum, J. Math. Pures Appl. 75 (1996), 125-153. (121 G. KATRIEL, Mountain pass theorems and global homeomorphism theorems, Ann. Inst. H. Poincarr Anal. Non Lindaire 11 (1994), 189-209. [131 M. MARZOCCHI, Multiple solutions of quasilinear equations involving an area-type term, J. Math. Anal. Appl. 196 (1995), 1093-1104. [141 M. MARZOCCHI, Nontrivial solutions of quasilinear equations in By, in Well-Posed

Problems and Stability in Optimization (Marseille, 1995), Y. Sonntag, ed., Serdica Math. J. 22 (1996), 451-470. [151 J. MAWHIN AND M. WILLEM, "Critical point theory and Hamiltonian systems", Applied Mathematical Sciences, 74, Springer-Verlag, New York-Berlin, 1989. [16] P.H. RABINOWITZ, "Minimax methods in critical point theory with applications to differential equations", CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986. [171 M. STRUWE, "Variational methods", Springer-Verlag, Berlin, 1990.

Variational Methods for Functionals with Lack of Strict Convexity

139

[18] A. SZULKIN, Minimax principles for lower semicontinuous functions and applications

to nonlinear boundary value problems, Ann. Inst. H. Poincarc Anal. Non Lineaire 3 (1986), 77-109.

Marco Degiovanni Dipartimento di Matematica e Fisica. University Cattolica del Sacro Cuore Via dei Musei 41 1-25121 Brescia, Italy

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 141-162 © 2003 Birkhauser Verlag Basel/Switzerland

Some Remarks on the Semilinear Wave Equation Vieri Benci and Donato Fortunato

1. General principles 1.1. The main equation We study the following equation

00+W'(rb) = 0

(1)

where a

R4-4C; 0 and W`(0) _ &

+ ice;

=c2at2-0

_ 0 + iO2i W : C

R. We assume that

W(e"'*) = W(Vi) ) E R

(2)

so that W'(e'''ii) = e"OW'(V)) and W'(t) is real for S real. Our equation is the simplest nonlinear equation which satisfies the two assumptions which, today, are shared by every fundamental theory in physics. These two assumptions are the following ones: A-1. It is variational. A-2. It is invariant for the Poincare group. Moreover (1) satisfies also the following assumption: A-3. It is invariant for the gauge S' . Next we will discuss these assumptions in some detail and we show, taking as model (1), that they are responsible for the main relativistic effects. 1.2. Variational principle Our equation is variational, namely it is the Euler-Lagrange equation of the functional

,(lp) = f f

IVV,12

2l;1

1

at

2

- W(rb) I dxdt.

Assumption A-1 states that the fundamental equations of physics are the Euler-Lagrange equations of a suitable functional.

Vieri Benci and Donato Fortunato

142

For example, the equations of motion of k particles whose positions at time

t are given by x j (t), xj E R', j = 1,. .. , k are obtained as the Euler Lagrange equations relative to the following functional

S=

j

2

2-V(t,xi,...,X)

dt

(3)

where mj is the mass of the j-th particle and V is the potential energy of the system. More generally, the equations of motion of a finite-dimensional system whose generalized coordinates are qj (t), j = 1, . . . , k are obtained as the Euler-Lagrange equations relative to the following functional

S= J L(t,gi,...,gk.,41,...,4k.)dt. The function L is called Lagrangian of the system and it determines the dynamics. Also the dynamics of fields can be determined by variational principles. The basic fields of physics can be regarded as a modification of an entity which, in the nineteenth century was called "ether" and which is now called "vacuum". From a mathematical point of view a field is a function 0 : Ra+i _+ Rk,

V,

(TGI.... ,00,

=

where R"+' is the space-time continuum and Rk is called the internal parameters space. The space and time coordinates will be denoted by x = (x1, X2, x3) and t respectively. The function 0(t,x) describes the internal state of the ether (or vacuum) at the point x and time t. From a mathematical point of view, assumption A-1 states that the field equations are obtained by the variation of the action functional defined as follows:

S Jf G (t,X,?P,V*IL I dxdt. The function G is called Lagrangian density but in the following, for simplicity, we will call it just Lagrangian.

1.3. The Poincark group Assumption A-2, is the real content of relativity. We recall that the Poincare group is a 10 parameter Lie group generated by the following one-parameter transformations:

Space translations in the directions x, y, z :

x'=x+xo y'=y z'=z

t'=t

x'=x YI

'

=Y+Yo

z'=z

t'=t

x'=x .

'

Y'=y Z'=Z+zo t,

t.

Some Remarks on the Semilinear Wave Equation

143

The invariance with respect to these transformations guarantees that space is homogeneous, namely that the laws of physics are independent of space: if an experiment is performed here or there, it gives the same results. Space rotations: X, = x

x' = xcosO2 - zsin02

x' = xcosO3 - ysinO3

t'=t

t'=t

t'=t.

y'=ycos9i-zsin0l y' = y y'=xsinO3+ycos03 z'=ysinOj+zcos9l ' z'=xsin02+zco502 ' z'=z

The invariance with respect to these transformations guarantees that space is isotropic, namely that the laws of physics are independent of orientation. Time translations: X, = X

y' =y

z' =z t'=t+ to. The invariance with respect to these transformations guarantees that time is homogeneous; namely that the laws of physics are independent of time: if an experiment is performed earlier or later, it gives the same results. Lorentz transformations:

x'='y(x-vit) y'=y z'=z

ti =7(t-jx)

x'=x '

y'=7(y-v2t) z'=z

t'=7(t-

y)

x'=x y'=y '

z'=y(z-vat)

t'=-r(t-

z)

where 1

Y =

1- r z

(4)

with v = v;, i = 1, 2, 3. The invariance with respect to these transformations is an empirical fact and, as it will be shown below (Subsections 2.3, 2.4, 2.5), it implies the remarkable facts of the theory of relativity. Here c is to be interpreted as a parameter having the dimension of velocity.

The Poincare group is the 10 parameters Lie group generated by the above transformations (plus the time inversion, t - -t, and the parity inversion

(x, y, z) -' (-x, -y, -z)). An equation for a scalar field t is said to be invariant for the Poincare group if the following happens: if ?li(t, x, y, z) is a solution of our equation, then Vi(t', x', y', z') is also a solution of our equation where (t', x', y', z') are obtained from (t, x, y, z) by applying a Poincare transformation. The simplest equation invariant for the Poincare group is the d'Alembert equation:

,0 = 0

(5)

Vieri Benci and Donato Fortunato

144

where

=820

2

D_

AiP and AV)

+8Z+

8

i -W2 2 y It is easy to check that if iI'(x, y, z, t) is a solution of this equation, then

7(x-vlt), y,z, -f t -

11 x

are also solutions of the equation. The d'Alembert equation is the simplest variational field equation invariant for the Poincare group. In fact it is obtained from the variation of the action 2

=

Jf

I

C2

ov, i9t

1

- IVeI I2J dxdt.

(6)

In this case, the Lagrangian is given by 12

G I

2

Ot

(7)

The equation (1) is the simplest nonlinear variational equation which is invariant under the Poincare group. Its Lagrangian is the following: 1_ IVV,12

G

2

c2

- W(VG)

(8)

1.4. Conservation laws

Noether's theorem states that any invariance for a one-parameter group of the Lagrangian implies the existence of an integral of motion, namely of a quantity which is preserved with time by the solutions (see, e.g., [9]). Thus Eq. (1) has 10 integrals. We describe the most important ones: energy, momentum and angular momentum. In the following we shall confine ourselves to consider only complex-valued fields y;.

Sometimes, it will be useful to write V) in polar form, namely

'(t, x) = 0, x)e{a(e,:).

(9)

Energy. Energy, by definition, is the quantity which is preserved by virtue of the time invariance of the Lagrangian; it has the following form (see, e.g., (91)

E-

J

I

e

G l dx,

/

where 7 denotes the complex conjugate of z. In particular, if we take the Lagrangian (8), we get

8t2+2IV+J1,2+W(w)J(lx. (10)

Some Remarks on the Semilinear Wave Equation

145

Using (9) we get:

e=

(a4)2 ()2+Dul1++Vj21uz+W(u)1dx. [

(11)

Momentum. Momentum, by definition, is the quantity which is preserved by virtue of the space invariance of the Lagrangian; the invariance for translations in the x; direction gives the following invariant (see, e.g., [9])

P, _ - Re J 8G 0-0 dx. ff a P ax;

-

In particular, if we take the Lagrangian (8), we get N/i

P; _ -Re f C280 at ax, 1

dx

and since P = (PI, Pi, P3) is a vecftor, we can write

P = - Re

J c ftVP- dx.

(12)

Using (9) we get:

r

P=-J 1 (V 0 ' U 2

'kU

)

dx

(13)

Angular momentum. The angular momentum, by definition, is the quantity which is preserved by virtue of the invariance under space rotations of the Lagrangian with respect to the origin (see, e.g., [9]) In particular, if we take the Lagrangian (8), we get

M = ReJ

x x VOLO dx.

(14)

Using (9) we get:

M= /

IxxV1, u2+xxVu

I

dx.

(15)

Charge. The Poincare group is a 10-parameter group which acts on the spacetime variables x, t. When we take into account a complex field b, we can consider the S' action on given by

eL*O, a E R.

(16)

A Lagrangian t =,C (t, x, O, v', V, ) is called St gauge invariant if it is invariant under the action (16). The charge, by definition, is the quantity which is preserved by virtue of the Sl gauge invariance of the Lagrangian C. The charge has the following expression (see, e.g., [8]) C = Im

f

as ipdx. ((a)

Vieri Benci and Donato Fortunato

146

In particular, if we take the Lagrangiian (8), we get

C=

dx.

Im f

Using (9) we get:

C=

J

(17)

u2dx.

2. Solitons

Roughly speaking a soliton is a solution of a field equation whose energy travels as a localized packet and which preserves this localization property under perturbations. In this respect solitons have a particle-like behavior and they occur in many questions of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics, plasma physics (see, e.g., [7], [8], [11], [13]).

In Subsection 2.2, we shall say precisely what we shall mean by soliton. Now we first examine the existence of a particular class of solutions of (1). 2.1. Existence of standing waves A standing wave is a finite energy solution of (1) having the following form

Vio(t, x) = u(x)e-"'0t, u real.

(18)

Substituting (18) in Eq. (1), we get

-Du + W'(u)

=

2

u.

(19)

Then, looking for standing waves, one is reduced to the nonlinear elliptic equation (19). The solutions u of (19) in the Sobolev space Hl give rise, by means

of (18), to standing waves of (1): in fact the HI norm of u is equal, up to some factors, to the energy E ('o) of 1'o (see (40)). Observe that the solutions of (19) are the critical points of the reduced action functional "2 r (20) f (u) = 2 (ivui2 u2) dx + J W(u)dx, u E H1.

J

-

The following existence result can be proved (see Theorems 1 and 2 in ([5]))

Theorem 1. Let W' satisfy the following assumptions 2

there exists C > 0 s. t. W (C) - 2 (2 < 0 m Ws s) > lim lim a00

WA(S)

s5

(21)

(22)

c2

= 0.

(23)

Some Remarks on the Semilinear Wave Equation

147

Then equation (19) has a positive, G`2 and radially symmetric solution u which, together with its derivatives up to order 2, has exponential decay at infinity. Moreover such u is a ground state, i.e., it minimizes the reduced action among all the nontrivial solutions.

Finally it will be useful to observe that if u(x) is a solution of (19) also u(x - a) is a solution of (19). From now on we will denote by uo(x) the solutions of (19) which are centered around the origin, namely the solutions of (19) with barycenter Q(uo) =

- 0. ff xuo(x)2dx uo(x)2dx

2.2. Travelling solitary waves and solitons By the Lorentz invariance of G, given a standing wave ipo(t, x), we can obtain a travelling solitary wave V), (t, x) = tlio(t', x') just making a Lorentz transformation. If we take the velocity v = (v, 0, 0), t' and x' are given by

t'

t-2xl xi(x1-vt),X2=x2,x3=X3-

If u(x) _u(xl, x2, x3) will denote a solution of Eq. (19) then y'v(t, Si, x2, x3) = u (7 (x1 - Vt) , x2 X3)

ei(kx-wt),

(24)

is a solution of Eq. (1) with 1

1-

2

w = Two; k = 7 Z v.

(25)

C:2

This solution represents a solitary wave which travels with velocity v = (v,0,0) in the x1 direction. It is well known that the expression -YC

[y v is a 4-vector in the Minkowsky space (called 4-velocity); then also [ ck

is a J

4-vector and we have by (25)

[ ck c° [ ' J Now we need to recall the notion of orbital stability. To this end we shall first introduce some notations. We denote by Sw the set of the standing waves (x)e iwot Y'wo (t, x) = uwo of (1) where uw is a ground state solution of (19) i.e. uwo minimizes the reduced action (20) among all the nontrivial solutions of (19).

Vieri Benci and Donato Fortunato

148

Clearly, if ?b,,° (t, x) E S,°, all its orbit {ti;,,. (t, x - a) : a E R3 } is contained in Srn,

Roughly speaking a standing wave ifi..(t,x) = u,,,, (x)e-'W°t E S,,,° is called orbitally stable if any solution ?P(t, x) of (1) which "starts near" iP,,, (t, x) "remains near" S,w,, for all t _> 0: this means that for all e > 0 there exists 6 > 0 such

that if Ik'(0,x) - u "(x)Ilt>> + 11

a1i(0, x) M

- (-iwou",°(x))I

< L2

then for all t > 0 we have

inf{

l

12±(t, )

(t,.) -

at

- (-iwo,P)

L.

In this paper we call solitons those solutions of (1) which are obtained by a Lorentz transformation from orbitally stable standing waves. In the following we recall some well-known results ([12]) on the existence of orbitally stable standing wave for (1). To this end consider the real map

d:wo - d(wo)=f(u ,)

(26)

where u,,,, is a ground state solution of (19) and f (u,,,,) is its reduced action (20). Observe that, by using equality (36), which will be proved in Subsection 2.4, it can be easily shown that d(wo) =

I VU,.

3J

12

dx.

(27)

In ([12]) Shatah proved the following stability result:

Theorem 2. If the map d in (26) is strictly convex in a neighborhood of a given real number wo, then u,,, (x)e-"11' is orbitally stable. Now, following ([12]), we consider a simple example in which we have stable standing waves. Take

W(() = IA1(12-pI(IT,(Ec, where A,p>0and 6>p>2.

(28)

In this case (19) becomes /

'l

-Du+1\- J

)U_.U,Ulp-2U=O. (29)

If

Uj2

>

(30)

then assumptions (21), (22), (23) are satisfied and (29) has a ground state solution

u;".

Some Remarks on the Semilinear Wave Equation

149

Now it is easy to see that the solution u,, can be obtained by rescaling the positive solution uo of (29) with wp = 0 uwo(x) = buo(Qx),

b=\1-ID zAc2

I

\\\

/

Then, by (27), we have

Ac'

.

(31)

2

/ IVue12dx.

d(wo)

(32)

Inserting (31) in (32), we get 2

d(wo) = 3

C1

- ace

J

2F" - JIvI2dx.

An elementary calculus shows that d is convex for those wo such that _

2

1>Ac2>8-p.

(33)

2
(34)

So such w's exist if

2.3. Space contraction and time dilation of solitons Some relativistic effects like the space contraction and the time dilation are consequence of our assumptions Al and A2. In fact, first of all observe that, by Eq. (24), the following theorem follows: Theorem S. Any moving material body experiences a contraction in the direction of its movement of a factor 1/y with -y = Now the standing waves of Eq. (1) can be considered as a clock. Let us denote

by q (t) the position of our clock at the time t. A mathematical definition of q is given by

fX q(t) = Q(V,(t, .)) =

dx II+I,(t, x)22

f IV,(t,x)I dx If we assume that at t = 0, q(0) = (0, 0, 0), the motion of the clock is given by q(t) = (vt, 0, 0);

Vieri Benci and Donato Fortunato

150

then our description of the clock can be made replacing x by q(t) in Eq. (24); taking into account Eq. (25), we get: ipu(t,q(t)) = 0 (t, vt, 0, 0) i(kq(f)--t) = u (0, 0, 0)

= u (0, 0, 0) e"*'

,, ., W

= u(0,0,0)e = u (0, 0, 0) e = u (0, 0, 0) e

This equation shows that our moving clock is vibrating with a frequency WO

ti

/ 1-

_

va

j.

Since the intervals of time measured by a clock are inversely proportional to the frequency of the vibrations, we get that

i - oz Then we get the following

Theorem 4. A moving clock moves slower than a resting clock by a factor

2.4. The energy momentum of a soliton as 4-vector Lemma 5. Let u be a solution of the Eq. (19); then 1

Iz

Wulz

dx+

2

J

(Ju2 -W (35)

Proof. Set

where u solves equation (19). So, since u is a stationary point of the reduced action f (see (20)), we have d

T. f (u.\) = 0 for A = 1

Then the conclusion directly follows from the above equation.

0

From this lemma, adding the terms of (35), it follows that 6

` z r jIVuI2dx - ,I 2 c2 uzdx + W (u)dx = 0.

J

(36)

Some Remarks on the Semilinear Wave Equation

151

Taking Eq. (35) for i=1,2, we get

1f au

12

12

au

dx

2

ax,

if

aI2dx-I axiI

2

1 I

au

axe

21

dx

fI

dx-2J

11,0u2

I2 dx +

rr1

3I2dx+

I

2 c2

J

5x3

- W (U) I dx = 0

u2-W(u)Idx=0

then

IT-, 12&-2-J

Z2dx=2J

S

2

2dx-2J ITX2

IT I2dx

and we get

au

I Thus in general we have that 8x,1

au

I

1

2

dx

2

dx.

ax2

I

2

au

dx =

ax,

=

2

dx,

axj

i , j = 1 , 2, 3

and so 2

11 axi

IDu12

dx =

J

I

dx, i = 1, 2, 3.

(37)

3

From (36) we get r

J

z

r

W(u)dx = 2

r

u2dx - 6

J

IDu12

J

dx.

(38)

Using (11), (24), (37) and (38), we get the following expression for the energy: E

2

f

I

2

2 2

c2

I

ax,

I

2

2

+

2

I ate, I +' I a I

2

+rW(u)dx=2ry2(j +11J

I2dx+2l

j2

5

+ k2 Iu12

+k2/J

Iul2dx

+2J I I dx+J W(u)dx=ry21 v2 +1I6J IvuI2 r

(,,j2

+ k2) 1 Iu12 dx + 3

=

2

+i

Y2

/

v2

O / IVU12 dx +

6J

(1++1

/

(k2 +

/

r IVuI2 dx

IVu12 +

+

l

2

J

W (u) dx

r

2

+

I

J

u12 dx

(i+) /+ 1) Jr

IuI2 dx.

dx

Vieri Benci and Donato Fortunato

152 Since

7

(v2

2

2

1±_3

+ 1) + 1 =

+1=

t,2

2 v2

= 272

1

we get

£(&) _ -Y2

2J

/p

Iou12dX +

[3 J

Iu12dx]

In the above expression the function u and its derivatives are computed at the point y = (y1,Y2,Y3) = ('Y(x1 -Vt),x2,x3) y, we get the following expression:

Making the change of variable x

IvuI2

£

7 [3 f

dx +

f Iul2 dx]

.

Thus £ (tit,) =1£ (00)

(39)

where

£(

0) _ 2 J

Iul2 dx + 3 / lvul2 .

(40)

Observe that £ (t/ip) is an intrinsic property of the soliton. In fact it is the energy of tPc (see (18)) which represents the soliton in the reference frame where it is at rest. So we have proved the following result Theorem 6. The energy of a soliton £ (tyt,) increases with velocity by the factor 1

71 77Now we recall that, by (13), the momentum of a soliton is given by P,

(t) _ _

1

at 614D

2

axi at

au 8u ax; at

and

P (tVv)

0t 5T u2 + Du -N 1 dx

where

'(t, x) = kx - wt.

Some Remarks on the Semilinear Wave Equation

153

Using (25), we get:

r

a uz + ay=

J (

P,

)

z f f flu1zdx+v T

=k

Iz

dx 2

a 2dx

/Jf

V

)

C4 V'Y

J

C

l (c2

J

dx

J

I

axi

iuiz dx + 3 f 1Du1z dx)

where u is computed at the point (-y (xl - vt) , x2i x3) . Making a change of variable as we did for the energy, we get f Iu12 dx + 3 Pi ('P") = CZ (w2 C2

=

J

IVu12 dx

Op") c2

=7E(00)v where u is computed at the point (x1i x2, X3). Finally, using the vector notation, we can write E (V)v) P(7/, ' )=

C2

V. v=ry E (PO) C2

By (39) and (41), it follows that the 4-uple (P

(41) E

is a 4-vector. In

fact

E (/v)

_

E

[ P (i& ) 1 and it is well known that the expression

'Yc2

c2

(42)

Yv

-YV

is a 4-vector in Minkowsky space, called 4-velocity (see, e.g., [10]).

2.5. The mass of solitons In this section, we will define the mass of a soliton within the framework of this theory. First of all notice that the concept of momentum comes before the notion

of mass; in fact the momentum is defined as the integral of motion due to the homogeneity of space. Then mass can be defined as the ratio between momentum and velocity:

M=-P V

Notice that the mass is a scalar and not a tensor since, by Eq. (41), P and v are parallel.

154

Vieri Benci and Donato Fortunato

If we take this definition of mass, by Eq. (41), we get that

m( v) _

=7.6 (110) CZ

c2

(43)

Then we get the second remarkable fact of theory of relativity: Theorem 7. The mass increases with velocity by the factor 1

1'=

3

and

Let us finally remind that the above properties can be proved also for a class of topological solitary waves (see ([2], [1])).

3. Some remarks on the Planck constant This model is based on very general assumptions and it does not have anything to do with Quantum Mechanics. Nevertheless, it presents some phenomena which are considered typical of Quantum Mechanics. This fact contradicts the idea, shared by most people, that no phenomena of Quantum Mechanics can be described in a world where the observer does not play any role. Actually, the model which we have presented so far presents some of these phenomena such as a direct proportionality between frequency and energy.

3.1. The De Broglie relations In our picture, a solution of type (18) represents a particle, but it has the structure of a standing wave: it reminds us of the big discussion in the beginning of this century about the dualism between waves and particles, one of the starting points of Quantum Mechanics (see, e.g., [6]). In this section, we will spend few words to see what our model can say about this. We recall that a model involving topological solitary waves has been discussed in [4]. If we look at the solution (24) with an apparatus at rest with dimension of the order of a wavelength, it is not adequate to consider only what happens at the barycenter q(t). We need to consider the space vibrations. Our solution will appear as a wave packet having the following form: 11v (t, x) = 1' (t, x1, x2, x3) = u ('Y (x1 - Vt) , x2, x3)

(44)

with k = ry-'wo and w = rywo. These relations can be written in the quadrivector notation: 4,

I

k

Wo

J=C 1

-rV

Comparing the above equation with (42), we get

t'('Pv)_hw

(45)

Some Remarks on the Semilinear Wave Equation

155

P(& )=hk

(46)

and

with (47)

where h has the dimension of action and is reminiscent of the Planck constant. More or less this is the argument which De Broglie used to deduce his celebrated relations (45) and (46). It is interesting to note that also (45) and (46) are a general consequence of A-1, A-2 and A-3.

For a given stationary wave O(t, x) = u(x)e-''t, h depends on £ (0) and w; however, even if Eq. (1) has continuous family of orbitally stable stationary waves,

we may assume that only one, namely Oo, has a strong stability and the others will evolve toward tpo. Thus, if a field has such property, all the standing waves are the same up to space translations and phase shifts. The point is that different field equations of type (1) might have different solutions of type (18) which give different values of h. In the next section, we will show that it is possible to build a model in which all the solutions of Eq. of type (1) satisfy the relation (47) where h is independent of the field equation. 3.2. The universality of the Planck constant

In order to continue our discussion we need to introduce the notion of maximally stable solitary wave. Let i(t, x) be an orbitally stable solitary wave. We know that in many cases it is characterized by a unique parameter, for example by its energy. Under small perturbations, it preserves its energy and the other quantities determined by it such as its frequency and its norm, both in L2 and in H I. For any orbitally stable solitary wave we can define its index of stability as follows: K = K (') is the index of stability of tk if any perturbation w(t, x) with energy less than K does not destroy the stability of 0. Definition 8. An orbitally stable solitary wave Oo is called maximally stable if K (0o) = max {K (0) 1

is an orbitally stable solitary wave).

(48)

In general we do not know if a given equation has a maximally stable solitary wave. For example, the field equation

CIO +'-11010=o has a family of orbitally stable solitary waves with rest frequency w E (1/f, 1) . Since this interval is open, we do not know if the maximum defined by (48) is achieved. However, if a field equation has maximally stable solitary waves, it is clear that they will "prevail" on the others. Having the notion of maximally stable solitary wave, we can continue our discussion on the Planck constant. We assume to have many field equations of type (1):

DO +W,(0) = 0, j = 1,...,N.

(49)

Vieri Benci and Donato Fortunato

156

We make the following assumptions:

the equation 1 (92j

c 8t2 -AT +W'(W)=0 has a unique maximally stable standing wave T(t. X) = U(x)e-tO2t

(50)

Wj has the following form:

Wj (u) =

E

W (Eju).

(51)

The function U appearing in (50) satisfies the following equation 02

-DU- 2U+W'(U)=0.

Theorem 9. If Oj(t,x) =

u(t,x)e-"j'

(52)

is the maximally stable solution of (49),

then

C(0j)=h'wj1 where

Ju2dx+1

b.=

JIvuI2dx

is independent of j.

Proof. A standing wave of Eq. (49) has the following form: =uj(x)e-iwjt

-Oj(t,x)

(53)

where the uj's satisfy the following equation w2

1 -Duj- 2u+ :;W'(cjuj)=0. Ej

(54)

C

This standing wave is maximally stable provided that it corresponds to the solution (50) resealed in a suitable way. If we set

uj=MyU Ej(X )

(55)

replacing (55), in Eq. (54), we get s

z

-DU-E''U+ EjMj W'(EjMjU)=0. C !2

1

Then, comparing (56) and (52) we get

Mj=ej

11 wj =:E . Ej

(56)

Some Remarks on the Semilinear Wave Equation

157

Then, by (40) and (55) we get

IC2If J

EIwjI)

=

//

u'+3IIwjl

.!'Vujl2

IIM;e;f U2+3I1 IM2ej f IVU12

I

Ju2+fIvuI2. Setting

h=

Ju2dx+flvuI2dx

we get the conclusion.

The fields of type (49) present another interesting feature: Theorem 10. If a/ij (t, x) = u(t, x)e-'"'j' is a maximally stable solution of (49), then S2

f U 2dx

namely, the charge of a maximally stable standing wave is independent of j.

Proof. By (17), we have that

C (b,) = cj f u2dx

_ j M, e f U2dx =

E

()2fu2ds=±fu2. ) Ci

C1

Remark 11. If we identify the solitary wave rlij with a charged particle, C (0j) is proportional to the electric charge of Oj (see (87)). Then, in this scheme, the universality of the Planck constant is strictly related to the fact that all the solitary waves tpj have the same charge (in absolute value). Thus, in this model, two important constants of physics derive by the same assumptions on the structure of the fields, namely from (51).

4. Interaction of solitons with the electromagnetic field 4.1. The Maxwell equations

In Section 1 we introduced the Lagrangian G (see (7)) which gives rise to the simplest Poincare invariant equation (5)) for a scalar field 0. If we regard rb as a zero form, (7) can be interpreted as the 4-form in the space-time R4, given by (*di) A dpi = Gdxl n dx2 A dx3 A dt

where * denotes the Hodge star operator with respect to the Minkowski metric.

Vieri Benci and Donato Fortunato

158

Now let A) = (ca,A1, A2, A3)

be a R4-valued vector field defined on the Minkowski space-time R°. As before we can construct for (cp., A) the simplest Poincar6 invariant Lagrangian L 1, by regarding this vectorial field as the 1-form on RI A = Al dx j + A2dx2 + A3dx3 + ,pdt

and considering the 4-form in R'' given by (*dA) A dA = Lldxt A dx2 A dx3 A dt where 1 OA

L,

=2

C at

z

+V

-2IVxAle

.

(57)

The Euler-Lagrange equations are:

1a

- car

l-A+Vp =Vx(VxA) c at

(58)

V - OW + VIP f = 0.

(59)

H=VxA

(60)

If we set

and

E=-cBA

-Vcp

(61)

(58) and (59) become respectively

IaE=VxH

(62)

c at

(63)

V E = 0.

Moreover clearly (60) and (61) imply that

V.H=0 xE_

- c 8t

(64)

.

(65)

(62), (63), (64) and (65) are the Maxwell equations for the electromagnetic field (E ,H).

Some Remarks on the Semilinear Wave Equation

159

4.2. Deduction of the equations The Lagrangian density relative to (1) is given by

0:-

[2_IvI2] - W (i) .

1[1

(66)

We want to couple the complex field z' with the vector field represented by the one-form A. We can write 'CO = 2 (*di)) A dzb - W (0) dxl A dx2 A dx3 A dt

(67)

where dV) is the 1-form

xo=ct

dpi=

Here and in the following we shall tacitly sum on the repeated indices. Now also A is a 1-form. Then we consider the (Weyl) covariant external differential defined by dAlb = di/b

+, e he

=

aa/i

axe

dx,

+,e 0A = D,i,bdxj

(68)

where i is the imaginary unit, n a coupling constant and Do, Dj are the so-called (Weyl) covariant derivatives

ieAi, i = 1, 2,3.

Do = 1 a +ie p, Di = fix Also we will use the notation

D=V - i

A.

at with Do and V with D, we have the following Lagrangian density for the interaction between 0 and A

Then, if we replace in (67) dip with dAm, which amounts to replace in (66)

2

Gw=21 1I-

-IVVi-i

Affil2

-W(bi).

(69)

then the total action is given by

rr

S=J J (1w+L1)dxdt

(70)

where L, is the Lagrangian (57) for the vector field (
S

2 f (*dAb)A dA1b +

f(*dA) A dA - Jwdxdt 2

(71)

Vieri Benci and Donato Fortunato

160 or

S(7G. p, A) =

2

t Jf [IDo'GI2 - IDTGI2] dadfJWdxdt

1

+2 ff [2

I

2 ID x

r_

a 2

= 2

Jf I

+2 Jf [I

c 8t

+ 1-h

+VwP

2

C

I

AI2] dxdt VIP

-

-iA

- I© x AIaJ dxdt

a

dxdt

l

-

(72)

JJWdxdt.

We point out that, when the vector field (cp, A) is assigned, we can take only the variation of S with respect to 0. Then we get an equation whose only unknown is the complex field 0

D21p - D20 + W'(*) = 0.

(73)

Now it is useful to write 0 in polar form O(x, t) = u(x, t) eiS(r,t)/h, u, S real functions. So (72) takes the following form:

()2 -Ivu12I -W(u) I dxdt

(2 (ca

J

/

r r Yh27 J J

+2

1

1

S.(u,S,W,A)=J

[ivs - eAla - 1 +O`,12

f

I as

-

auadxdt + ec,)

(74) - IV x AI2 dxdt. (I c 81t Making the variation of S. with respect to 8u, 6S, 60 and 6A respectively,

l

we get /

O u + W'(u) +

[IVS T2

a 5i

"S +

U21

-C -5i

(c 8t + V x (V x A) +

- eAI2 - I _ 8i +

ecp)2]

u=0

+ V - [(-VS + eA) U2] = 0 W

c it (c 8t +

(c 8t

o)

+ eW) u2 (OS - eA) ua.

(75) (76) (77) (78)

Now set

E

(c 8t + v',

(79)

Some Remarks on the Semilinear Wave Equation

161

H=VxA

(80)

e

lOS

P=h

cat

+ ecp u2

(81)

e

j = h (-VS + eA) u2. So the equation (76) is a continuity equation a 5jp+V.j=0

(82)

and (77) (78) are the second couple of the Maxwell equations with respect to a matter distribution whose charge and current density are respectively p and j

-V.E=p

(83)

1 OE

cat - VxH=j

(84)

Finally (79) and (80) give rise, as usual, to the first couple of the Maxwell equation

VxE+-- =0 V H =0.

4.3. An existence theorem Now we are interested in scalar fields r' = u(x)e-'gym (standing waves) in the electrostatic framework. Then we look for solutions of (75),...,(78) of the type

u = u(x), S = -wt, A = 0, W = cp(x). With this ansatz, the equations (76) and (78) are identically satisfied, while (75) and (77) become

-Au -

2

(-w +

Ocp =

(-

We observe that in this case (81) becomes` P=

u + W'(u) = 0 + e
+ ecp I u2 C

- he u2.

(85) (86)

(87)

The following result can be proved (see [3])

Theorem 12. Assume that the nonlinear term has the form W'(u) = wou - IuI,-2 u with I awl < Iwol and 6 > p > 4

then (85), (86) possess infinitely many solutions (u,cp) with u E Hl(R3) and fR3

dx < oo.

Vieri Benci and Donato Fortunato

162

References [1) V. BENCI, D. FORTUNATO, Solitons and relativistic dynamics, in Calculus of Varia-

tions and Partial differential equations, G. Buttazzo, A. Marino and M.K.V. Murty editors, Springer (1999), 285-326. [2] V. BENCI, D. FORTUNATO, L. PISANI, Soliton like solution of a Lorentz invariant equation in dimension 3, Reviews in Mathematical Physics, 3 (1998), 315-344. [3] V. BENCI, D. FoRTUNATO, Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equation, Reviews in Mathematical Physics, 14, No. 4 (2002), 409-420. [4] V. BENCI, Quantum phenomena in a classical model, Foundations of Physics, 29, 1-29 (1999). [5] H. BERESTYCKI, P.L. LIONS. Nonlinear Scalar Field Equations, I - Existence of a Ground State, Arch. Rat. Mech. Anal., 82 (4) (1983), 313-345. [6] L. DE BROGLIE, Un tentative d'interpretation causale et nonlineaire de la Mecanique ondulatoire: la theorie de la double solution, Gauthier-Villars, Paris, 1958. English traslation: Non-linear wave mechanics, a causal interpretation, Elsevier, Amsterdam, 1960.

[7] K. DODD. J.C. EILBECK, J.D. GIBBON, H.C. MORRIS, Solitons and Nonlinear Wave Equations, Academic Press, London, New York, 1982. [8] B. FELSAGER., Geometry, Particle and fields, Odense University press (1981). [9] I.M. GELFAND. S.V. FOMIN, Calculus of Variations, Prentice-Hall, Englewood Cliffs, N.J. 1963. [10] LANDAU L.,LIFCHITZ E., Thdiorie du Champ, Editions Mir, Moscow, 1966. [11] R. RAJARAMAN, Solitons and instantons, North Holland, Amsterdam, Oxford, New York, Tokio, 1988.

[12] J. SHATAH, Stable Standing waves of Nonlinear Klein-Gordon Equations, Comm. Math. Phys., 91, (1983), 313-327. [13] G.B. WITHAM, Linear and nonlinear waves, John Wiley and Sons, New York, 1974.

Vieri Benci Centro Interdisciplinare per lo Studio dei Sistemi Complessi (CISSC) and Dipartimento di Matematica Applicata "U. Dini" University di Pisa

Via Bonanno 25/b 1-56126 Pisa Italy E-mail address: benci®dma.unipi.it Donato Fortunato Dipartimento Interuniversitario di Matematica. Universit& di Bari

Via Orabona 4 1-70125 Bari Italy

E-mail address: fortunatmdm.uniba. it

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 163-179 © 2003 Birkhauser Verlag Basel/Switzerland

Unique Continuation Principles for Some Equations of Benjamin-Ono Type Rafael Jose Iorio, Jr.

1. Introduction By equations of Benjamin-Ono type we mean either

8eu (t) + aLu (t) + F (u (t)) = 0,

(1)

or

8t (u (t) + aLu (t)) + F (u (t)) = 0, (2) where L is a linear (possibly unbounded) operator, F is (in general) a nonlinear function of its argument and a denotes the Hilbert transform (af) (x) = PV 1

fl)

J Y-x dy.

(3)

In what follows we will be mainly interested in determining conditions under which

the solutions of the Cauchy problems associated to (1) and (2) are identically zero. Moreover we will concentrate on the case of the generalized Benjamin-Ono equation (GBO), namely (1) with L = 8= and F (v) = 8, (fi (v)) where 4' maps an appropriate function space into R. To explain what we have in mind, consider the following problem

1 8tu (t) + au (t) = 0 l u (0) = 4 E L2 (R).

(4)

Recall that if f E L2 (lR) then (af )^ where

isg

f (e) , i; a.e.,

(5)

-1 if l;<0 sgn(i;)=

0 if C=0 1 if C>0

(6)

and g denotes the Fourier transform of the tempered distribution g. In case g E L' (1R), our definition is

3W =

()4f9(x)exP(_ix)dx. 21r

(7)

Rafael Josl Iorio, Jr.

164

In particular o is a unitary operator in L2 (J) and

a2 = -1.

(8)

For a proof of these facts see ([3]) or ([4]). It is easy to verify that the unique solution of (4) is given by the following unitary group

u (t) = exp (-at) ¢ = (cx s t - a sin t) 0, t E R.

(9)

Assume that there exists R > 0 and r # 0 such that, sin r 0 0, and supp (0) , supp (u (r)) are contained in [-R, R]. We claim that under these conditions we have u - 0. Indeed, combining these assumptions with (9) we conclude that supp (a4)) C [-R, R] .

(10)

supp (P±0) C [-R, R]

(11)

Pf = 2 (1 T ia)

(12)

Therefore, where

are the projections onto the Hardy spaces 5)

(R) = {gyp E L2 (R) IX.T

(E) = 0a.e. }

(13)

and Y:F denote the characteristic functions of (-oo, 0) and [0, oo) respectively. But then

r

R

`

R

(P54)" (z) = (2 f (Pf4)) (x)exp(-izx)dx, z E C

(14)

are entire functions. In particular they are continuous. In view of (12) and (13) we have

(P+4))A(i) = 0, d1; 0.

(15)

Hence (P±Q)n = 0 and it follows that 0 = 0. This proves our claim. Remark 1. Note that the preceding arguments show that if 0 E L2 (R) has compact support and is such that arb also has compact support, then 0 = 0. Remark 2. In Section 3 we will present a more fruitful way of obtaining this result.

These ideas can be extended to nonlinear equations to a certain extent. Let L2 (R) be such that F (0) = 0 and

F : L2 (IR)

11F (v) - F(w)po < a (IIvIIo, Ilwllo) llv - wllo

(16)

where a : (0, oo) x [0, oo) -* [0, oo) is a continuous function, nondecreasing with respect to each of its arguments and 11- 110 denotes the L2 norm. Combining Banach's

fixed point theorem and Gronwall's inequality, it is possible to show that the

Unique Continuation Principles

165

problem

1 Btu (t) + au (t) + F (u (t)) = 0

(17)

u(0)_0

1

is globally well posed.'

Theorem 3. There exists a unique u E C (R,L2 (R)) such that u (0) and the equation in (17) is satisfied with the time derivative computed in the L2 norm, that is,

hl lim -.0

u(t+ h)

hh

(t)

+ou(t)+F(u(t))

= 0.

(18)

0

Now we are in position to prove a unique continuation principle for (17).

Theorem 4. Let u be as in Theorem 3. Assume that (a) there exists a to E IR and a positive R such that supp (u (t)) C [-R, R]

(19)

for all t in some neighborhood of to. (b) F is local, that is, if v E L2 (IR) has compact support then so does F (v). Then u =- 0.

Proof. Our assumptions imply that supp(u(t°+hh-u(to))

C [-R, R]

(20)

for all sufficiently small h. In view of (18) there exists a sequence (hn) such that

u (to + hn) - u (to) hn

- (au (to) + F (u (to)))

(21)

almost everywhere. Thus au (to) + F (u (to)) must have compact support. Since F is local and u (to) has compact support, it follows that a (u (to)) also has compact support. Remark (1) implies that u (to) = 0. Uniqueness implies the theorem. 0 As can be seen from the preceding remarks, there is a huge gap between the linear and the nonlinear results. At the moment it seems that this gap cannot be bridged unless we introduce more detailed structure than that allowed in Theorem 4. In what follows we will present some results in this direction. This article is organized as follows. In Section 2 we sketch the well-posedness theory associated to the generalized Benjamin-Ono equation. Section 3 presents two closely related unique continuation principles for GBO. Finally, in Section 4, we make a few comments on the existing literature, and indicate some open problems. ' In order to keep this paper reasonably short, here, and in what follows, we shall not dwell on the details of such applications. They are rather standard. See for instance ([5)), Sections 5.1 and 6.1 of ([8)) or Chapter 14 of ([16)).

Rafael Jose Iorio, Jr.

166

Before proceeding it is convenient to establish notations that will be used throughout this article. Let s E R and denote by H8 (R) the L2 type Sobolev spaces, that is, the set of all tempered distributions f such that 11f112

= f (1 +C2)8

(C)I2 If(C) 12

< oo.

(22)

As is well known, they are Hilbert spaces with respect to the inner product

(f19)8= 1(1 + 2)8f(e)(23) Define the usual weighted L2 spaces by the formula (24)

L2 (R) = (Hr (R) )A , r E R.

The norm and inner product in these spaces are given, respectively, by 1If111;:=

(25)

/ (1+x2)rlf(x)12dx

and

(f 19)1; = fR (1+x2)r f (x)9(x)dx.

(26)

We will also have the occasion to use the following weighted Sobolev spaces

&r = ,3s,r (R) = H8 (R) fl L2. (R). They are all Hilbert spaces when provided with the inner product

(27)

(28) (f i9)A,r = (f I9)A + (f i9)1.r . The corresponding norm will be denoted by II'Ile,r Since in what follows we will be interested only in real-valued solutions, it is convenient to introduce

he (R) = Re H8 (R).

(29)

fA,r = h8 (R) n 9A,r

(30)

and

Constants whose precise values are of no consequence to our arguments will be denoted simply by C. If X and Y are Banach spaces, we denote the set of all bounded linear operators from Y into X by B (Y, X) If Y = X we write simply B (Y). Finally, [A, B] denotes the commutator of the linear operators A and B.

2. The Generalized Benjamin-Ono Equation: Local Well-Posedness In this section we will sketch the well-posedness theory associated to the real-valued solutions of the Cauchy problem associated to the generalized Burgers-BenjaminOno equation (G13130). namely 2

fl atu(t)+Qalu(t)+O 4 (U (t)) _ u(0)=0Eh8(R)

u(t)

2To simplify the notation, from now on we will write O (4 (v)) = O 4' (v).

(31)

Unique Continuation Principles

167

where s > 2 is fixed, p > 0, and 4): h8 (R) -' h8 (R). If µ = 0 we refer to the pde in (31) simply as the generalized Benjamin-Ono equation (GBO). Examples of such 4i 's are wn+1

$ (w) - P+11

(32)

4i (w) = cos (w) .

(33)

where p > 1 is an integer and

We begin with the linear equation. If r E R and 0 E H' (R) , a straightforward application of the Fourier transform shows that the unique solution to

f atv (t) + aaiv (t) = u8 v (t) 1

(34)

u (0) _ 0 E Hr (R)

is given by

v,, (t) = E (t) 4 = exp

[Fe. (t, )

v

,

(35)

where

-Q,, = µa? - 2x02,

(36)

F,, (t, ) = exp [(-µ + 2ih (f )) 12t] .

(37)

and

Combining the Fourier transform with the parabolic character of the pde in (34) we obtain the following Sobolev space version of the smoothing properties of the heat equation. Theorem 5. Let p E (0, oo), A > 0 and 0 E H8 (la). Then there exists a constant K,, > 0, depending only on A, such that [1+(ZµtyiA,II0II8.

IIE,.(t)0Ilr
(38)

Moreover the map t E (0, oc) ' --4 E,, (t) 0 is continuous in the topology of Hr+a (R).

Proof. The proof is contained in any of the following references: ([5]), ([7]), and 0 Theorem 4.17 and Exercise 7.96 of ([8]). It is now easy to show that (31) is equivalent to the integral equation

u (t) = E,, (t) , - J Eµ t (t - t') 0x41 (u (t')) dt'

(39)

and a standard application of Banach's fixed point theorem and Gronwall's inequality yield the next theorem. For details see ([5]), ([7]) and Theorems 8.1 and 8.2 of ([8]).

Rafael Jose Iorio, Jr.

168

Theorem 6. Let p > 0, t¢ E W (IR), s > z. Assume; that 8x4' is Lipschitz in the following sense: there exists a continuous function L : (0, oo) x [0, oo) -* [0, oo), nondecreasing with respect to each of its arguments, such that

-

(40) 8,'' (w)II.-i < L (I1vll , IIwIIB) Ilv - w118 . Then there exists a unique u = u, E C ([0,T] ; he (R)) such that u, (0) = 9S and the pde in (31) is satisfied with the time derivative computed with respect to the norm of he-2 (R). Moreover, 11O .T.0 (v)

u,, E C((0,T];h°° (R))

(41)

for all u > 0, where h°O (R)

hr

rn

(42)

(R) ,

provided with its usual Frechet space structure.

In general, the existence time determined in the preceding theorem tends to zero asp tends to zero so that to establish existence of solutions with p = 0, we must be able to extend all the u,, 's to an interval which is independent of p. Lemma 7. Let ¢ E h" (IR), as before and assume that I(w 18A (w)).1 :5 91

(IIwlIA)

(43)

,

where g, : [0, oo) -i [0, oo) is continuous. Let p denote the maximal solution of

J 8ep (t) = 2gi (p (t))

(44)

p (0) = 110112

Then

Ilu, ll8

- p (t)

(45)

for all p > 0, whenever both sides are defined. Since p does not depend on p it follows that all the uM 's can be extended to any closed interval [0, T] contained in the maximal interval of existence of p. Proof. In view of (41) we have et Iluµ (t)IIA = 2 (u, (t)18tu, (t) )R = 2 (u,+ (t) IpOsu,, (t) - a8zu,, (t) - 8x41 (u,, (t))),

(46)

_ -2p IlOxuu (t) 112 - 2 (u,, (t) la8xu,, (t)), - 2 (u,, (t)18x4; (u,, (t)) )A .

Since the first term on the last member of (46) is nonpositive and the second is zero because a8-2 is antisymmetric and we are dealing with real-valued solutions, we obtain (t)112I

8t IIuN

5

l(u,, (t) 1,9.,,p (u,, (t)))gl < 9i

and the lemma follows at once. {Note that Br0 (0) = 0.

(huts

(t)IIs)

(47)

0

Unique Continuation Principles

169

Remark 8. Sufficient conditions for the validity of (43) can be found in Appendix A of [11]

A long, but by now standard argument, implies existence and uniqueness for p = 0. See ([5]), ([7]), Section 8.1 of ([8]) and ([12]).) Theorem 9. Assume that 4i satisfies the conditions of Theorem 6 and of Lemma 7. Let 0 E h' (R) and [0, T] , T > 0, be any closed interval contained in the maximal interval of existence of (44). Then there exists a unique u = uo E C ([0,T] , h' (R)) such that uc (0) _ 4) and 8tui (t) + era2u0 (t) + ay4p (uo (t)) = 0, with the time derivative computed with respect to the norm of h8-2 (R).

(48)

Remark 10. It is possible to show that the solution of GBBO depends continuously on the initial data in the following sense. Let p > 0 be fixed, 0,, E h' (R), n = 1, 2, ... , oo be such that 0,, -L 0,,.. Let u E C ([0, T J ; h' (R)) denote the corresponding solutions. Then if T' E (0,T.), the solutions u,,, n < oo can be extended to [0, T'] for all n sufficiently large and

sup Dun (t) - u.

n limou [0,1,,l

o-

If µ > 0 this statement follows directly from the analysis of the integral equation (39). In the case of GBO, one must resort to the Bona-Smith approximation procedure. (See ([1]) and Chapters 6 and 8 of ([8]).) Alternatively one can use Kato's theory of quasilinear equations to get continuous dependence for each fixed p > 0. (See ([13]).)

Now we turn to the 132,2 theory. Assume that µ > 0 for the moment. Multiplying (39) by x2, noting that 8.,4t (u,,) E C ((0, T) ; h'-i (R)) and using the exponential decay of F. (t, l;) it follows that x2u', E C ((0, T] ; L2 (R)) ,

(49)

for fixed p > 04. Let

w(x)=1+a2

(50)

and note that IIuu (t)112,

at 2 = 2 (wuµ (t) I w8tuN (WO = 2 (WUF, (t) Iw (Aaiu (t) - Oaru,. (t) - 8=,p (u (t))) ) .

(51)

The first two inner products in the last member of (51) can be rewritten as follows (WUµ (t) Iwp zu (t) )1

(52)

p (Wu' (t) I P, ax] it (t) ) 0 2

4For details see (6) where the case 4' (v) = 2 is considered.

Rafael Jost Iorio, Jr.

170 and

(wuµ (t) 1w,7eiuN (WO = (wu, (t) I [w, oo] U, (t) )o ,

(53)

because a88 is antisymmetric and we are dealing with real-valued functions so that, (wu1, (t) I o,wu,, (t)),) = 0. But [w a8=] +l _ [w, QI

V, + o [w, 8.1] Ip

(54)

and

1a x2 _ y2 8

[w, v] a8 ii (x') = pvf y-x 7r

=-1n

(y) dJ

(55)

(x+y)d (Y)dy=0

where the last equality was obtained integrating by parts''. Therefore (53) becomes (wu,, (t) I [w, o'dx] u, (t) ) 0 = (wuµ (t) 10, [w, e=] U. (t) ) 0 .

(56)

[w, 8=]' = -20 - 2x8?t(,.

(57)

Now,

Integration by parts shows that if O E f2.2 then xO IIxa. 1'II

E L2 (R)6 and

C 111p,12

<-

(58)

.2

Combining these results we obtain 8(IIuN

(t)112

(t)112

0.2 -

IIuµ

.2 + 2 I (wu,. (t) Iw8=4i (u,, (t)) )oI

(59)

Lemma 11. Let 4I+ satisfy the conditions of Theorem 9 with s = 2 and let ¢ E f2,2. Assume that (a) 8 4' (.) maps f2,2 into L2 (R) and

(b) there exists a continuous function g2 : [0, oo) , [0, oc) such that I (W10 I w8t4' (V') )()l <_ 92

(II0II2,2) .

(60)

Let p be the maximal solution of

J 8(P(t)=CP(t)+2(91(P(t))+92i(P(t)))

l

P (0) = ll011z,2

(61)

Then 1lun

(t)112.2 < P(t)

(62)

whenever both sides are defined.

Remark 12. Assumptions (a) and (b) of Lemma 11 are satisfied by the maps defined in (32) and (33). "Strictly speaking, the computation in (55) holds only if +' is sufficiently smooth and falls off sufficiently fast. However, an easy limiting argument settles the account for 0 E 12.2 6This is a special case of Theorem 16.

Unique Continuation Principles

171

Once again, the standard limiting argument implies existence in the case p = 0. Uniqueness is trivial in view of the h8 theory. For details see ([5]), ([6]) and Chapter 8 of ([8]).

Theorem 13. Let 0 E f2,2. Then there exist a T > 0 and a unique u = uo E C ([0, T] ; f2,2) such that u (0) = 0 and

atuo (t) + ca=u0 (t) + 8x' (uo (t)) = 0 with the time derivative computed with respect to the norm of h8 (R).

(63)

Remark 14. Note that the previous theorem says that the solution remains in f2.2 while it exists. This is also true for uE µ > 0, as we have noted in (49).

3. Unique Continuation for GBBO, it > 0 Consider (4) once again. Suppose that x0 E L2 (R) and that there exists a r such that sin r 34 0 and xu (r) also belongs to L2 (R). Then (9) implies xa¢ E L2 (R) . But then y0 (64) ( y ) dy + ( a ( x 4')) (x ) x (a0) (x) = pv 1 f x

rr Ry-x _-f (y)dy+a(x4')(x). R

Since a (x4') E L2 (R) we conclude that

(2a)' 3 (0) =

f

0 (y) dy = 0.

(65)

Conversely, if (65) holds then (64) implies that xa4) E L2 (R). A similar computation, using the formula k-1

xk _ yk

Exiyk-1-)

(66)

yields the following result.

Proposition 15. Let k be a positive integer and assume that 0 E Lk (R). Then a¢ E Lk (R) if and only if

(0) =

a4k -1;(0) = 0.

(67)

In particular if the above assumption is satisfied for all k then 0 has a zero of infinite order at t; = 0. Now, if 0 and u (r) have compact support where r is such that sin r 0, then the assumptions of the previous proposition are satisfied for all positive integers k. This implies 0 = 0 so u (t) = exp (-at) 0 = 0 for all values of t. We will now apply these ideas to GBO. Before proceeding, it is worthwhile mentioning a technical result that will be used several times in the sequel. For a proof see Appendix A of ([5]) or H. Triebel's book ([17]).

Rafael Jose Iorio, Jr.

172

Theorem 16. Let r E N and 0 E .Fr.r. Then x'ai,O E L2 (R) for all j, k E N such that 0 < j + k < r and there exists a constant C depending only on r such that IIxJaX0II()

CIIOIIr,r -

(68)

First we consider the linear equation. Combining the Fourier transform, Leibniz's rule, Theorem 16 and the formulas

a F,, (t, ) = (-2q) (p - 2ih (ty)) F,, (t,

(69)

0 F,, (t, ) = -2t (µ - 2ih (a)) F,, (t, ) + (-2t;)2 (µ - 2ih (S))2 F. (t, f) b1{ F,, (t, f) = 8itb + 3 (-2t)2 t:2 (p - 2ih

+ (-2t)3 s (i1- 2ih (t)); F, (t, C) F,,

8itb(.i)

aktAe.n(k)

+

F (t, )

( / L - 2ih( ))k F(t, t;) , j > 4,

(70) (71)

(72)

k=1

we obtain the behavior of the solutions to the linear equation in the weighted L2 spaces under consideration. Theorem 17. Let r E N and 0 E f,.,. (R). Then for each fixed p > 0 we have (a) v,, E C ([0, oc) ; fr,r) if r = 0, 1, 2 and satisfies (73)

II 1',, (t)Ilr.r < eµ (t)114IIr,r

where 8,, (t) is a polynomial of degree r with positive coefficients depending

only on r and p. (b) if r > 3, the following statements are equivalent q,, E C([0+00);fr.r), 3 t ( < t2 such that vµ (ti) E f r,r,

(74) 1, 2,

(75) (76)

i ii,,(t.0)=0,j=0,1,...,r-3,VtE(0,oc).

(77)

In this case an estimate of the form (73) also holds.

Proof. See ([51), ([71) and ([91). We just note that if r > 3 we obtain r-3

(x'*v,, (t))A (t) _ (xrE, (t) 0) A

cjb0)O(i) + (R (t))A (a) , r > 3 (78)

8it i=o

where b denotes Dirac's delta function, ak E R and m (k) E { 1, 2, ... , k}, and

REC([0,oo);L2(R)).

0

The next corollary is an immediate consequence of Theorem 17.

Corollary 18. If 0 and v,, (r) have compact support for some r > 0 then v1 - 0. At this point it is convenient to introduce the following (non-standard) definition.

Unique Continuation Principles

173

R+ = {B = (91i92,...,0")I9e > O,e = 1,2,...,n }.

(79)

Definition 19. Let

If n = 1 we write IRA. = R+. A function R : llt+ -+ S' (R) is called a regular map if there exists a finite number of (Banach) subspaces X j C S' (R) , j = 1, 2, ... , N and maps Rj E C (1R+, Xj) such that N

R=ER,

(80)

j=1

and R (9) = (Rj (0))A is a measurable function f o r all j = 1, 2, ... , N. .,

In particular, k (9) = (R (0))A is a measurable function. We will write R E reg (N, X1, X2, ... , XN.)

Remark 20. In the sequel, the letter R, and symbols like R, R#, will be reserved to denote regular maps, whose precise form is not relevant to our arguments. Thus,

in different formulas, R, R, R# ... may represent different regular maps, but in all instances they have the properties described in Definition 19. Assume that? 0 E f3,3 ti f2,2, and let u = u,, E C ([0, TJ ; f22), A ? 0 fixed, be the solution constructed in Theorem 13. Multiplying the integral equation (39)8 by x3 and using (78) with r = 3 we obtain

8it (0) 6 -

(a3u (t))"

It (x3E,, (t - t') 8x (u (t')))' dt'. t

J

(81)

To proceed we must examine the behavior of the nonlinear term in (81). Let T > 0 be fixed and E f2,2. A formal computation combining (69)-(71) and Leibniz's formula implies _i&43 (F (r,C)(8x4i(V,))"(C))

=-i [8iTo(aX))A (0) + Qi

2

Q1 (T,

t)

t

t

j=0

j=3

where

(82)

= (aO F,, (T,

,3

(a

t) - 8ir6) (ax b (V,)) A (S)

(83)

Since

(axe (0))^ (0) _2()

JR

(ax4, (V,)) (x) dx = 0

X means that the inclusionY C X is continuous and dense. sif µ = 0 we regard (39) as an identity in Ha-1 (R) 7The symbol Y

(84)

Rafael Jose Iorio, Jr.

174

we conclude that (X3 E,

cc C (T)O 4 (W})^ (f) _ -Z8 (F.. (T, S) (8x+(' ))^ (b))

(85)

z

_ -i [Q1T)+CJ

(8{F (T, )) (0,3-j (ash ())^

Theorem 21. Let 0 E 1:i,:1 and u be as above. Assume that maps 12.2 into Li (R), (b) (t, t') E R+ «-, X3 E,, (t - t') 8x$ (u (t')) is a regular map, (c) there exists a t 1 > 0 such that x3u (t1) E L2 (Ill). (a) O r4'

Then 0 (0) = 0.

Proof. Combining (81), (85) with our assumptions we obtain (xsu (t 1))

^

8it1 ¢b (0) 5

(R (t)) ^ v),

(86)

where R is a regular map. Hence the left-hand side of (86) belongs to L2 (Ill) if and only if 0 (0) = 0. Remark 22. Assumptions (a) and (b) in Theorem 21 are rather implicit. They are satisfied by the 0 's defined in (32). (See ([9]) where the case p = 1 is considered. The case p > 1 is easier and can be dealt with much in the same way.)

Remark 23. Note that if 0(0) = 0 it follows that (u (t))^ (0) = 0 for all values t in the interval of existence. Indeed, taking the Fourier transform of (39) (which makes sense even for it = 0, because of assumption (a) in Theorem 21) we obtain

(u, (t))^ = F,, (t, )

.) -JF t (t - t', (824' (U'. (t')))^ (') dt'. 0

Since the Fourier transform of the nonlinearity is automatically equal to zero at t; = 0, we are done. In view of the preceding remark it is natural to seek solutions to GBO in f:1.:; _ {

E N3 I (0) = 0}

.

(87)

Once again, Banach's fixed point theorem, Gronwall's inequality and the standard limiting process can be applied to obtain existence and uniqueness. (See ([5]), ([7]) z

and Chapter 8 of ([8]), where 4 (v) = 2 is considered. The proof in the general case is the same.)

Theorem 24. Let 0 E 11.3 and

be as in Theorem 21. Then the solution u,,

constructed in Theorem 13, A > 0, belongs to C ([0,T]; f3,3).

Unique Continuation Principles

175

Our next task is to establish the unique continuation principle for GBBO with Eu>0.9

Theorem 25. . Let 0 E NA - f2,2 and u,, E C ([0, T] ; f2,2) be the solution constructed in Theorem 13, p > 0. Assume that 4> satisfies the conditions of Theorem 21 and (a) axe () maps f3,3 into L2 (R); (b) 4> (v) > 0 for all v E L2 (R) and if 4' (v) = 0 then v = 0;

(c) (t, t') -i (x4Eµ (t - t')

ax -t (u

(t')))'

- 8i (t - t') (1 + c3) bat (ax4> (u (t'))") (0), where c3 is defined in (90), is a regular map. Suppose also that one of the following conditions hold (d) There exists 0 < t1 < t2 such that u, (t3) E f4,4, j = 1, 2; (e) There exists t j > 0 such that u, (t 1) E f4,4 and (88)

Then u (t) = 0 for all t E [0,T]. Proof. Write u = u,, for simplicity's sake. Since f4,4 -+ f3,3 Theorem 21 implies that ¢ (0) = 0, so that u, E C ([0, T] ; f3,3) in view of Theorem 24. Multiply the integral equation (39) by x4 and take the Fourier transform to get (x4u (t)) A (S)

_ (x4E (t)

f

0)A

0

e

(X

4E, (t - t') ax$ (u (t'))) dt'.

(89)

Combining Leibniz's rule with formulas (69)-(72) and Theorem 16 we obtain (x4Eµ (t) 0) A (S) =

oor (Fµ

(t,.) (S))

(90)

4

_ 1: cjaf F, (t, t;) of -'¢ (C) = 8itb' + c38itb4' + R (t, C), t=o

where R E C ([0,T] ; L2 (R)). For all appropriate test functions V we have

('fib', wv) = l6'o&) = - (6, (ow)) (011 + IPw) (0) = -0" (0) w (0) - ' (0) (0"(0) 6, (G) - (I p (0) 5, V) = - (V (0) 6,

(0)

+ (' (0) 6', v')

(91)

Therefore,

06' = -0'(0)6+I'(0)6'.

(92)

Since ¢ (0) = 0 we get (x4E, (t) .0) A

8itb (1 + c3)

'Conclusion (d), with 0 (v) = 2 , was obtained in [9].

(0) + R (t)

(93)

Rafael Jose Iorio, Jr.

176

Next we turn to the nonlinear term in (89). Since (a=4' (u (t')))" (0) = 0 for all t', (92), (93) (with 0 replaced by a=4 (u (t'))), and assumption (b), imply that (x4 E,, (t - t')

= 8i (t - t') 6'

(u (t')))" (ar.p

.944 (F,

(u (t'))")

(a=0 (u (t')))" ( ))

c38i (t - t') jot (a=4' (u (t')))" (t) + R# (t, (94)

= 8i (t - t') (1 + c3) 604 (ax' (u (t'))") (0) + R# (t, t',

where R# is a regular map. Therefore

(xau(t))"(S) =815(1 +c:,) (ti' (0) - f` (t - t') a{ (ast

(u(t'))") (0)dt') +R(t,t;`)

0

(95)

where R is a regular map. Note that

r atJ xu(t,x)dx

(96)

R

= f x (j. O u (t) - o a2u (t) - a,,-k (u (t))) dx = j 0 (u (t)) dx, and

(a=4' (u (t))") (0) = i (21r)J xay0 (u (t)) dx = -i (2a)

j4(u(t))dx.

R

(j(t

(97)

These formulas imply that (95) can be rewritten as (x4u (t))"

= K6 where

(98)

xu (t'x) dx) dt') +

- t') ae

(t,

UR

K = -8 (2ir)- 4 (1 + c,3). Integration by parts shows that

t j xO(x)dx+J(t-e)(fR xu(t',x)dx)dt'=

J( jxu(t',x)dx)0

0

R

(100)

Thus

(x4u (t))"

KJ uOt

(r xu (t, x) dx) dt') + R (t, t;)

(101)

R

Since x4u (t1) E L2 (R), we must have fr.,

Jo

(jxu(t'x)dx) dt' = 0,

(102)

Unique Continuation Principles

177

, dt' (fxu(t'x)dx)

(103)

so that the function

F (t) =

J

satisfies F (0) = F (tl) = 0. Thus, there exists a Ti E (0, t1) such that

F'(T1)= fR xu (ri, x) dx = 0.

(104)

We are now in the position to prove the conclusions of the theorem. Proof of (d). Since x4u (tj) E L2 (1R), we can repeat the above arguments with 0 replaced by u (t 1) and u (t1) replaced by it (12) to conclude that there exists a r2 E (0, Tl ), such that F' (T2) =

JR

xu (T2, x) dx = 0.

(105)

Equation (96) implies that for t, T E [0, T] we have

JR xu (t, x) dx = J xu (,r, x) dx + it (fR 4 (u (t')) dx dt'.

(106)

Applying (106) with t = 7-2 and T = 7-1 we conclude that

fr2

4' (u (t')) dx dt' = 0.

(107)

f 4i(u(t'))dx = 0 `d t' E [T1,T2]

(108)

l

(IR

/

H ence so that 4' (u (t')) = 0, t' E [7-1,,r2]. Hence

u(t') =0dt'E [T1,-r2]

(109)

uniqueness implies (d).

Proof of (e). Taking T = 0 in (106) our assumptions imply that F(t) > 0 for all t E [0, T] and

F'(t)=J xu(t,x)dx>_0dtE(0,T).

(110)

R

Thus F is nondecreasing. Since F (0) = F (ti) = 0 we must have F (t) = 0 for all O t E [0, T]. The result then follows from (106) and our assumptions on 4).

4. Final Remarks First, it deserves remark that there are unique continuation principles for evolution

equations that are not of BO type. Saut and Scheurer ([15]) proved that if u is a sufficiently smooth solution of the Korteweg-de Vries equation (KdV)

aiu(t)+Ou(t)+u(t)8xu(t) =0

(111)

such that supp (u(t)) C (a, b) for all values of tin some interval (T1, T2) then u = 0. Their arguments employ a Carleman type estimate. Bourgain ([2]) reobtained this

178

Rafael Jost Iorio, Jr.

unique continuation principle using complex variable techniques. Stronger results were obtained in Zhang ([18]) and Kenig, Ponce and Vega in ([14]). Both works assume there are t1 < t2 such that u (tl), j = 1, 2, are supported in a half-line. Moreover that ([18]) deals with KdV using the inverse scattering transform while in ([14]) the authors consider the generalized KdV equation, make use of Carlemann type estimates to obtain their results. It should also be noted that ([2]) and ([14]) deal with equations which are much more general than KdV. The results of ([18]) and ([14] together with those obtained above for the linear GBBO suggest that Theorem 25 should be true assuming only that 0 and u (t I) belong to NA for some t 1 > 0. Although we believe this to be true, we have not been able to prove it so far. It is also natural to inquire about unique continuation principles for solutions supported in half-lines. In view of the methods used above this question leads naturally to the study of GBBO in asymmetric weighed L 2 spaces. These questions will be dealt with somewhere else. Another natural question, posed by Otared Kavian, is the following. Suppose that 0, b E f2,2 let u, v E C ([0,T] ; f2.2) be the corresponding solutions. Assume Q, - lp E f4,4 and that there are two times 0 < t1 < t2 satisfying u (tj) - v (tj) E f4,4. Is it true that

u=v?

Finally we wish to remark that Henrik Kalisch ([10]) obtained a result, similar to part (e) of Theorem 25, for the BBM-BO equation, namely

19tu+a=u-oa=atu+ua=u=0.

(112)

He assumes that u E C ([0, 71; h2 (ia) n L; (R)). However his proof can be modified to yield a unique continuation principle where 0 and u (t1) belong to h2 (R)f1L2 (la)

and the condition

f 0(x)dx>0

(113)

is satisfied.

Acknowledgement The author wishes to express his thanks to the local organizing committee, Daniela

Lupo, Carlo Pagani and Bernhard Ruf, for their hospitality and kindness during the Bergamo meeting.

References [1] J.L. Bona and R. Smith, The Initial Value Problem for the Korteweg-de Vries Equation, Phil. Tlans. Roy. Soc. London, Ser. A 278, (1975), 555-604. [2] J. Bourgain On the Compactness of the Support of Solutions of Dispersive Equations, Internat. Math. Res. Notices, 9 (1997), 437-447. [3] P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, Volume I. One Dimensional Theory, Birkhi user Verlag, (1971). [4] J. Duoandikoetxea Zuazo, Andlisis de Fourier, Addison-Wesley/Universidad Aut6noma de Madrid (1995).

Unique Continuation Principles

179

[5] R.J. Iorio, Jr., On the Cauchy Problem for the Benjamin-Ono Equation, Comm. PDE, 11 (1986), pp. 1031-1081. [6] R.J. Iorio, Jr., The Benjamin-Ono Equation in Weighted Sobolev Spaces, J. Math. Anal. Appl., Vol.157, No. 2, (1991), 577-590. [7] R.J. lorio, Jr., KdV, BO and Friends in Weighted Sobolev Spaces, in FunctionAnalytic Methods for Partial Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, vol. 1450 (1990) pp. 105-121. [8] R.J. Iorio, Jr. and V.B.M. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Avanced Mathematics, vol. 70, Cambridge University Press, (2001).

(9] R.J. Iorio, Jr., Unique Continuation Principles for the Benjamin-Ono Equation, Preprint IMPA, (2001). [10] H. Kalisch, Decay of Internal Waves in a Two Fluid System, Preprint (2002). [11] T. Kato, On the Cauchy Problem for the (generalized) KdV equation, Studies in Applied Mathematics, Advances in Mathematics Supplementary Studies, vol. 8, Academic Press (1983), 93-128. [12] T. Kato, Nonstationary flows of viscous and ideal fluids, J. Func. Anal. Vol. 9, No. 3, (1972), 296-305. [13] T. Kato, Abstract evolution equations, linear and quasilinear, revisited, Lecture Notes in Mathematics, 1540, Springer-Verlag, (1992). [14] C. Kenig, G. Ponce and L. Vega, On the Support of Solutions to the Generalized KdV Equation. Preprint (2000). [15] J.-C. Saut and B. Scheurer, Unique Continuation for Some Evolution Equations, J. Difj. Eqs. 66, (1987). [16] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag (1983). [17] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, NorthHolland, (1978). [18] B.-Y. Zhang, Unique Continuation for the Korteweg-de Vries Equation, SIAM J. Math. Anal. 23, (1992), 55-71.

Rafael Jose Iorio, Jr. Instituto Nacional de Matematica Pura e Aplicada (IMPA). Estrada Dona Castorina 110 Jardim Botanico Rio de Janeiro, RJ, Brazil 22460-320

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 181-189 © 2003 Birkhauser Verlag Basel/Switzerland

Well-posedness Results for the Modified Zakharov-Kuznetsov Equation H.A. Biagioni and F. Linares Abstract. We establish local and global well-posedness for the modified Zakharov-Kuznetsov equation for initial data in H'(lt2). We use smoothing estimates for solutions of the linear problem plus a fixed point theorem to prove the local result.

1. Introduction In this note we consider the initial value problem associated to the modified Zakharov-Kuznetsov equation, U

(x, y) E IR2, t E R,

u(x, y, 0) = U0 (X, y),

(1.1)

where u is a real function. This model is a two-dimensional generalization of the Korteweg-de Vries equation. More precisely, in the context of plasma physics, Zakharov and Kuznetsov [15] showed that the propagation of nonlinear ion-acoustic waves in magnetized plasma is governed by the equation

ut + uxxx + uxy, + uux = 0. (1.2) On the other hand, Kakutani and Ono [4] established that the equation describing the propagation of Alfven waves at a critical angle to the undisturbed magnetic field was the modified Korteweg-de Vries equation. The two-dimensional equation in this physical situation is the modified Zakharov-Kuznetsov equation appearing in (1.1), see [11] for more details. Our purpose here is to establish local and global well-posedness for the NP (1.1).

In [3] Faminskii considered the NP associated to the equation (1.2). He showed local and global well-posedness for initial data in Ht(1R2), m > 1, integer. His method of proof was inspired by the one given by Kenig, Ponce and Vega [8] to show local well-posedness for the IVP associated to the Korteweg-de Vries in H'(R), s > 3/4. More precisely, he established a series of estimates for solutions of the linear problem and then he used these estimates in a regularized problem to establish local well-posedness. To prove the global results he made use of the This research was partially supported by CNPq-Brazil.

182

H.A. Biagioni and F. Linares

L2 and H' quantities conserved by the Zakharov-Kuznetsov flow (1.2). Here we show that the IVP (1.1) is locally well posed in H'(1R2) and using the conserved quantities associated to solutions of the equation in (1.1), that is,

I, (u(t)) = J u2(t) dxdy = R2

JudxdY, R2

I2(u(t)) = f (u= + uy - 4u4)(t) dxdy = R2

(1.3)

J(uo+ uoa - 4uo) dxdy, (1.4) Rz

we extend the local result to any time interval [0, T]. Our main results are: Theorem 1.1. For any uo E H'(1R2), there exist T = T(IIuoIIHI) > 0 and a unique solution of the IVP (1.1) defined in the interval [0,T] satisfying u E C([0,T] : H'(R2)), II d ull LTLvL;. + II azyull L-L2L2

(1.5) < 00,

(1.6)

<00,

(1.7)

and

(1.8)

IIUIIL=LO LT < 00.

Moreover, for T' E (0, T) there exists a neighborhood V of uo E H' such that the map uo u(t) from V into the class defined by (1.5)-(1.8) is Lipschitz. Theorem 1.2. Let uo E H'(R2). If IIuoIIL2 is small enough the solution u given by Theorem 1.1 can be extended to any interval of time [0, T]. To establish local well-posedness we use smoothing estimates for solutions of the linear problem and the contraction mapping principle, see [6]. The smoothing estimates employed in our argument of proof were shown in [3]. We notice that the method of proof given here can be used to obtain a similar result as Famisnskii's [3]. More precisely, it can be proved the following theorem.

Theorem 1.3. For any uo E H1 (R2), there exist T = T(IIuoIIHI) > 0 and a unique solution of the IVP associated to (1.2) with u(x,0) = uo(x) defined in the interval [0, T] satisfying

u E CQ0,T] : H'(R2)),

(1.9)

IlasuIIL.-L-LT + IlazaullL-LyLT

< 00,

(1.10)

Il8XUIIL-1L=L=

< 01),

(1.11)

and

IIUIIL=L-LT <00.

(1.12)

In addition, for T' E (0, T) there exists a neighborhood W of uo E H' such that the map T 0 -p 11(t) from W into the class defined by (1.9)-(1.12) is Lipschitz.

Zakharov-Kuznetsov Equation

183

We will not give the proof of this theorem since it follows closely the argument

explained in detail in the proof of Theorem 1.1. Since solutions of the NP (1.2) are conserved in L2 space it is natural to ask whether or not it is possible to have local solutions in that space or more generally what would be the largest Sobolev space where local well-posedness can be hold. Using the scaling argument we can have an insight in this question.

Observe that if u solves the initial value problem (1.2) with data uo then ua(x, y, t) = Au(Ax, Ay, A3t) also solves (1.2) with initial data ua(x, y, 0) = Auo(Ax,ay). Thus the highest derivative that leaves invariant the H'(11t2)-norm of ua is s = 0. In fact, (1.13)

IM-,0)IIE,. = A8II'uollH

where H'(R2) denotes the homogeneous Sobolev space of order s. From this we can deduce that local well-posedness for (1.2) could be realized in Sobolev spaces of indices greater or equal than 0. To establish better results than the one mentioned above for the KdV equation [6], Bourgain [1] introduced a new method to prove local and global wellposedness for data in L2. That method combined with the bilinear estimate method of Kenig, Ponce, Vega [5] allowed them to show local well-posedness in H', s > -3/4. These two methods have been proved useful to study another well-known generalization of the KdV equation, the Kadomtsev-Petviashvili (KP) equations. The difference between the treatment of KdV and KP equations is that one needs to use a sharp Strichartz estimate to handle the latter equations, see [2), [10], [12], [13] and references therein.

A final remark is regarding the global existence of solutions. The modified Zakharov- Kusnetsov equation has a similar critical character as the generalized Korteweg-de Vries equation,

ut + ux:x + uaux = 0.

(1.14)

Weinstein in [14] showed that for initial data in H'(R) with IIuoIIL2 < IIQIIL3 the NP for (1.14) has global solutions in HI(IR), where Q denotes the solitary wave solution of (1.14). Recently, Merle [9] has shown that for data in H(R) satisfying IIu00IL2 ? IIQIILz solutions of the NP for (1.14) blow-up in finite time. Thus it is

natural to ask whether the same results are shared by solutions of the modified Zakharov Kutnetsov equation. These questions and the ones mentioned above will be addressed somewhere else. This note is organized as follows. In the next section we list the linear estimates needed in the proof of Theorem 1.1. The proof of Theorem 1.1 and 1.2 will be given in Section 3. Before leaving this section we introduce some notation: The mixed space-time norm is denoted by 00

00

Tr

(f (J

IIfIIL:LyL;. _ f \-00 -00

0

If(x,y,t)It

dt)9/*dy) p1q dx)1/P

184

H.A. Biagioni and F. Linares

When t appears instead of T, the notation above means integration on the whole real axis with respect to t. The letter c denotes a constant that may change from line to line.

2. Preliminary results Consider the initial value problem Jui + u=s= + uxyy = 0, (x, y) E R2, t E R, lu(r, y, 0) = uo(x, Y) Solutions of (2.1) are described by the unitary group

(2.1)

-

u(t) = U(t)uo = f

that is,

ei(t(E'4+En)+xE+yn) uo(4, rt) d(drl.

(2.2)

Faminskii in [3] established the following estimates associated to solutions of (2.1). Lemma 2.1. Let 0 E L2(R2). Then II VU(t) OII

II U(t) 0II L LTL

<_

C IImlI L2,

(2.3)

01-0110-

(2.4)

In addition, if 0 E H"(R2). s > 3/4, then (2.5)

!5

We also have the dual version of (2.3), i.e.,

IloJ

SCIIfIIL-LdL;

(2.6)

0

The estimate (2.3) is known as a smoothing effect of Kato type, see [7]. The inequality (2.4) establishes a Lt' - LQ or Strichartz estimate for solutions of (2.1). Finally, we have the maximal function estimate (2.5) for the unitary group U(.). The proof of these estimates are contained in [3]. These estimates are the main ingredients in the proof of Theorems 1.1 and 1.3.

3. Proof of Theorems 1.1 and 1.2 Proof of Theorem 1.1. As usual we consider the integral equation form of IVP (1.1), that is,

u(t) = U(t)uo +

J0

t

U(t - t') (u28xu)(r) dr.

(3.1)

We define the operator

$(u(t)) = 4 ,,(u(t)) = U(t)uo + J tU(t - t') (u28xu)(r) dr, n

(3.2)

Zakharov-Kuznetsov Equation

185

and the metric space

Z. = {v E C([O,T] : H1(R2)) : lvi < a}

(3.3)

where

IVI

=

HI +IIVIILTL=°Lo +IIaXVIILTL2°LO

IIVIIL;.

LT

+ II axVII L-LYLT + II axyV II

We will show that there exist a and T such that

1. 4):ZT, ZT. 2. 4) is a contraction. We begin by estimating the H'-norm of D(u). Thus using Minkowski's inequality, group properties and Holder inequality we obtain

s C IIUOIIL2 +

II4)(u)IIL=v

f

I II (u2axu)(T)II L2v

0

< CIIIOIIHI +C

f 0T

dr

IIuIILW =v IlaXUIIL2 v dr

(3.4)

<_ CIlu0IIH1 +cT"/3 sup llaxu(t)IIL=,, IIUIIG;Lo. [0.T]

By using group properties, Minkowski's inequality and Holder's inequality we deduce II ax4)(u) II L2v

G CIIaXUOIIL2 +

f,

Ilax(u2axu)(r)IILsv dT

0

f1


IIu(aru)2IIL2dT


f

T IIuIIL- dr)112IIuaxuIIL2 L2

0

+ SUP IIu(t)IIL= [0.1']

f

T IlaxuIIL- d7-

0

CIIUOIIH1

IIUIIL=LO LT IlazuIIL-L2LT

+cT113 Sup Ilu(t)IIL=Y IlaxUIIL;.L=o, [0,T ]

(3.5)

H.A. Biagioni and F. Linares

186 and

I/T

IIu2O yUIIL2u dT + J

< C IlayuoIli, + c

yuOxUIIL2i, dr

1I

'J dIIuaxyUIIL2,dT+JT IlayulIL:yIIUIIL=,IIaxUIILxdT

IIuIIL


< c Iluollxi + c (J1 IIuII1,:, dr)'"2IIu8xyuII LLLT

(3.6)

0

+ Sup Ilayu(t)IILiy J` IIUIIL ,IIaXUIIL g dT [0,T]

0


T

+cT113 sup Ilu(t)IIH=w (0,T[

The estimate (2.4), Minkowski's and Holder's inequalities give II4'(u)II L4 L;'°LN C 1 1 U(t)u1v11 L ' m, m + 11 U(t) (J

U(-T)(u28xu)(T) dT) II Lr 11

V

1

(3.7)

< C IIuofI0 + J I II (u28ru)(T)42 dT O

cIluoll,,I +C j

lullIIxuIlLVdT

< C IluUII11. + cT'13 SUP Ilaxu(t)IILrn IIulI [0,TJ

Applying estimate (2.4), group properties, Minkowski's and Holder's inequalities and the argument in (3.5) it follows that

< C II U(t)axuoll L'TLoLy + IIU(t)( fo

U(-T)Ox(u28xu)(T) dr)

r1' ScIluolIHHI+CJO Ilu(a:u)2IIL L2+ J IIuIIL

<_ c Iluill,r + cT" 118xu1I i;.L e SUP

IIu(t)ll,,i

[oa'1

+ eT"/slluIlL .Lr,y Il ull

LT IlazullLi I,YL'q

IIL;.L-Ln

(3.8)

Zakharov-Kuznetsov Equation

187

Next we use the estimate (2.3), group properties and the argument in (3.6) to yield


(3.9)


clluolIW +cT'/3llaxullLaL. SUP Il(t)IIH T ° [0,T)

+ cT'/6IIuIIL; L° II UII L2LoLT Il a0UII L.°L2L ,, and

Ilasy4(u)II



(3.10)

0

G CIIuoIIHl +cT'3IIa=uIIL3

rL2

SUP

IIut)IIH

[0.7'1

+ cT'16IIuIIL;.Lc IIUIIL=Lv LT

Finally, the estimate (2.5), group properties and Sobolev spaces properties give

II'P(u)IIL2,Lv LT

< c(T, s) IluollH° + c(T, s)II (10 T U(-T)(u2axu)(T) dr) II O

< c(T, s) IIuOIIHl + c(T, s) II fTU(-T)(u2axu)(r) dTII Hl,, 0

Hence we can argue as in (3.6) to obtain the following estimate

<_c(T)IIuoIIHi+cT'/3Ila=III LsuPIIu(t)IIHi

(3.11)

[0.,11

+ c(T)T'/6II UII LILT.II UII

II axull L-L2L;

From the inequalities (3.4)-(3.11) and (3.4) it follows that 141(u)l < c(T)IIuoIIHi + c(T)(1 + T'16)T'/61u13

(3.12)

Choosing a = 2c(T)IIuoIIHI and T such that a2(1+T'/6)T'/6 < 1/2 it follows that 44 : ZT - ZT. To show that 45 is a contraction we follow the same arguments 0 as in (3.4)-(3.11) plus the choice of a and T.

H.A. Biagioni and F. Linares

188

Proof of Theorem 1.2. The claim in our result follows using the conserved quantities (1.3), (1.4) and the Gagliardo-Nirenberg inequality. Indeed, IIu(t)II' , = II u(t)III2 + 12(u(t)) + 1 Ilu(t)1114 = Iluoll%l + 12(uo) + 41Iu(t)1114

(3.13)

Iluoll2Ilou(t)ll 2.

Hence using the hypothesis we obtain IIu(t)IIHi < c(lluoIIHi) This a priori estimate and a standard argument yield the desired result.

References [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation, GAFA 3 (1993), 209-262. [2] J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, GAFA 3 (1993), 315-341. [3] A.V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations 31, No. 6, (1995), 1002-1012. [4] T. Kakutani and H. Ono, Weak nonlinear hydromagnetic waves in a cold collisionfree plasma, J. Phys. Soc. Japan 26 (1965), 1305-1318 [5] C.E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9, No. 2, (1996), 573-603.

[6] C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 48 (1993), 527-620. [7] C.E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. [8] C.E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347. [9] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578. [10] L. Molinet, J.-C. Saut and N. Tzetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, preprint 2001. [11] R. Sipcic and D.J. Benney, Lump interactions and collapse in the modified ZakharovKuznetsov equation, Studies in Appl. Math. 105, (2000) 385-403. [12] J.-C. Saut and N. Tzvetkov, The Cauchy problem for higher order KP equations, J. Differetial Equations 153 (1999), 196-222. [13] N. Tzvetkov, Global low regularity solutions for Kadomtsev-Petviashvili equation, Differential Integral Equations 13 (2000), 1289-1320.

Zakharov-Kuznetsov Equation

189

[14] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. [15] V.E. Zakharov and E.A. Kuznetsov, On three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286.

H.A. Biagioni IMECC-UNICAMP Campinas, 13081-970, Brazil E-mail address: hebemime. uni camp. br

F. Linares IMPA

Estrada Dona Castorina 110 Rio de Janeiro, 22460-320, Brazil E-mail address: linaresQimpa. br

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 191-195 © 2003 Birkhauser Verlag Basel/Switzerland

A Class of Isoinertial One Parameter Families of Selfadjoint Operators Orlando Lopes

1. Introduction If L is a (possibly unbounded) selfadjoint operator in a Hilbert space X, then the inertia in(L) of L is the pair (n, z) where n is the dimension of the negative subspace of L and z is the dimension of the null space of L. A one parameter family L(t) of selfadjoint operators is isoinertial if the inertia in(L(t)) of L(t) is independent of t. In this paper we show that certain families L(t) of selfadjoint operators depending on the real parameter t are isoinertial. Those families arise naturally when we study the orbital stability of double solitons (collision of two traveling waves) of integrable equations. The integrable equations have an infinite sequence of conserved quantities For a certain range of values of the real parameters a and Vj (u), V2(u), Q, a double soliton u(t) is a solution of the evolution equation that lives in the set of the critical points of the conserved quantity V(u) = V3 (U) + ctV2(u) + 3V1(u).

Using some techniques developed in [4] for finding local minimizers of constrained variational problems, the proof of the orbital stability of the double soliton u(t) is reduced showing that the inertia of the selfadjoint operator L(t) associated to the second derivative V"(u(t)) calculated along the soliton is constant and has a certain value.

In the case of the KdV equation, the stability of double solitons has been proved in [4] and it follows from the fact that the corresponding family of selfadjoint

operators L(t) is isoinertial with inertia equal to (1, 2) (that is, for any t, L(t) has exactly one negative eigenvalue and zero as a double eigenvalue). In order to show that such a family is isoinertial the authors use very strongly the fact that, in the case of the KdV, L(t) is an ordinary differential operator. This is so because the conserved quantities for the KdV are local functionals in one space variable. Since double solitons arise also in the case of integrable equations with more complicated conserved quantities (for instance, BO, ILW and KP equations), it is desirable to develop new methods to cover those cases.

Orlando Lopes

192

In this paper we prove an abstract result that shows that the families of selfadjoint operators that arise in the context of the stability of double solitons are isoinertial. The proof is independent of the structure of the conserved quantities. As a consequence of this result and with some extra effort, we can show that the calculation of the inertia of such families can be reduced to the calculation of the inertia of two stationary "limit" operators L, and L2 (which can still be a hard task). The operators L1 and L2 are obtained by replacing in V"(u(t)) the double soliton u(t) by the simple solitons (traveling waves) ul and u2i respectively.

2. Statement and proof of the main result We begin by recalling the definition of inertia of a real symmetric matrix.

Definition. If A is a real N x N symmetric matrix then the inertia in(A) of A is a triplet (n, z, p) of nonnegative integers where n, z and p are the number of negative, zero and positive elements (counted according to their multiplicity) of the spectrum of A . The next result is known as Sylvester Law of Inertia (see [1]).

Theorem 2.1. If A is a real symmetric N x N matrix and M is a real nonsingular (not necessarily orthogonal) N x N matrix then in(MAM*) = in(A). The unbounded selfadjoint operators L we will be dealing with in this paper satisfy the following property: there is a ry > 0 such that the spectrum of L to the left of 'y consists of a finite number of eigenvalues and the corresponding spectral projections have finite-dimensional range. For that class of selfadjoint operators we give the following:

Definition. The inertia in(L) of a selfadjoint operator as above is the pair (n, z) of nonnegative integers where, as in the case of matrices, n is the dimension of the negative subspace of L and z is the dimension of the null space of L. Now we can state the following Generalized Sylvester Law of Inertia:

Theorem 2.2. If L with domain D(L) is a selfadjoint operator as above and M is an invertible bounded operator then in(MLM*) = in(L), where MLM' is the selfadjoint operator with domain (M*) -I (D(L)). The proof of Theorem 2.2 follows exactly as in the matrix case ([1]) because it uses the variational characterization of the eigenvalues of selfadjoint operators. A very famous device for finding isospectral families of selfadjoint operators (families with constant spectrum) is the so-called Lax pair ([2]). The motivation is the following: let L(t) be a family of selfadjoint operators and let us impose that

L(t) = M(t)L(0)M'(t)

(1)

for any t, where M(t) is orthogonal and satisfies a linear evolution equation

M(t) = B(t)M(t) M(0) = I

(2)

Isoinertial One Parameter Families of Selfadjoint Operators

193

where B(t) is skew-adjoint. Differentiating (1) with respect to t and using (2) we get

L(t) = B(t)L(t) - L(t)B(t).

(3)

Conversely, if (3) holds with B(t) skew-adjoint then L(t) = M(t)L(0)M'(t) where M(t) is the (orthogonal operator) solution of (3) and this implies that the spectrum of L(t) is constant. Now suppose that instead of constructing isospectral families, we want to construct isoinertial ones. Then in view of Theorem 2.2 we impose

L(t) = M(t)L(0)M'(t)

(4)

and we assume that M(t) evolves in time satisfying a linear equation

M(t) = B(t)M(t) M(O) = I.

(5)

Differentiating (4) with respect to t and using (5) we get

L(t) = B(t)L(t) + L(t)B*(t).

(6)

Conversely, if (6) is satisfied then (4) holds and if M(t) is invertible then, according

to Theorem 2.2, the inertia in(L(t)) of L(t) is constant. So, (6) governs isoinertial flows. For further reference we set our framework. Let X be a real Hilbert space with scalar product (,), L(t) and B(t) linear operators depending on the real variable t E R satisfying the following hypotheses: H.1 for any t E R, L(t) is a selfadjoint operator with domain D(L(t)) independent of t;

H.2 for any t E R, B(t) and B'(t) are closed operators with common domain independent of t such that the Cauchy problem

u = B(t)u

(7)

is well posed for both positive and negatives times (that is, (7) defines a propagator M(t, s) for t, s E R); H.3 for a dense set of elements h E X we have

d L(t)h = B(t)L(t)h + L(t)B*(t)h, the derivative being taken in X.

Theorem 2.3. Under assumptions H.1 to H.3, the inertia in(L(t)) of L(t) is independent of t. Next we show how Theorem 2.3 can be used to construct isoinertial families of selfadjoint operators related to special solutions of some nonlinear evolution equations (the double solitons of integrable equations fit in that class). Let us consider an abstract evolution equation

u = f(u)

(8)

Orlando Lopes

194

in a Hilbert space X and we suppose that it has a first integral V(u). In order to be precise, we consider three Hilbert spaces X, X1 and X2 and we make the following hypotheses:

HH.1 X2 C X 1 C X with continuous embedding; the embedding from X2 into X 1 will be denoted by i; HH.2 V : X, R is a C3 functional; HH.3 f :.X2 X, is a C2 function; HH.4 for any u E X2 we have

V'(i(u))f(u) = 0.

(9)

Let us denote by (,) the scalar product of X and suppose that L(t) : D(L) C X -p X is a selfadjoint operator with constant domain D(L) such that (L(t)h, k) = V"(u)(h, k) for h and k in a subspace Z C D(L) fl X2 which is dense in X. We

consider also an operator B(t) : D(B) C X for h E Z.

X such that B(t)h = -f'(u(t))h

Our final assumption is: HH.5 the closed operators B(t) and B` (t) have a common domain independent of t and the Cauchy problems

u = B(t)u

is = B"(t)u are well posed for both positive and negatives times in the space X.

Theorem 2.4 Suppose assumptions HHl to HH5 are satisfied and let u(t) be a strong solution of (8). Then: (i) (P. Lax [3]) if for some to we have V'(u(to)) = 0 then V'(u(t)) = 0 for any t; in other words, the set of the critical points of V(u) is invariant under (8);

(ii) if u(t) is as in the first part (that is, u(t) is a solution of (8) satisfying V'(u(t)) = 0 for all t E R), then the inertia in(L(t)) of the selfadjoint operator L(t) associated to V"(u(t)) as above is independent of t. Proof. Differentiating once and twice (9) with respect to u we see that for u, h, k E X2 we have V"(u)(f (u), h) + V'(u)(f'(u)h) = 0

(10)

and

V"(u)(f (u), h, k)

+

V"(u)(h, f'(u)k)

+

V"(u)(k,f'(u)h) + V'(u)(f"(u)(h,k)) = 0.

(11)

Moreover, if u : R ---- X2 is a strong solution of (8) then

dV'(u(t)) dt and

h = V"(u(t))(f(u(t)), h)

dV"(u(t)) (h. k) dt

= V"'(u)(f (u), h, k).

(12)

(13)

Isoinertial One Parameter Families of Selfadjoint Operators

195

Now, if u(t) is a solution of (8) then, for any h E X2, (10) and (12) imply

dV'(u(t))h dt

= -V'(u(t))(f'(u(t)h).

(14)

If we set v(t) = V'(u(t)), from (14) we see that dt = B'(t)v(t). So, if for some to we have v(to) = 0, from assumption HH5 we conclude that v(t) = 0 for any t and this proves the first part of the theorem. In order to prove the second part, let u(t) be a solution of (8) such that V'(u(t)) = 0 for any t (according to the first part, it is sufficient that V'(u(to)) = 0 for some to). Using (11), (13) and that V'(u(t)) = 0 we get dV"(u(t))

(h, k) = -V"(u(t))(h, f'(u(t))k) - V"(u(t))(k, f'(u(t))h)

dt for any h, k E X2 and then

dt L(t)h = B(t)L(t)h + L(t)B'(t)h

for h in a dense set of X; so, in view of Theorem 2.3, Theorem 2.4 is proved.

References [1] Golub, G. and van Loan, C., Matrix Computations, The John Hopkins University Press, Baltimore and London, Second Edition, 1989. [2] Lax, P., Integrals of nonlinear equations of evolution and solitary waves, Comm.Pure Appl. Math., 21, 1968, 467-490. [3] Lax, P., Periodic solutions of the KdV equation, Comm. Pure Appl. Math.,28, 1975, 141-188.

[4] Maddocks, J. and Sachs, R., On the stability of KdV multi-solitons, Comm. Pure. Appl. Math., 46, 1993, 867-902.

Orlando Lopes Departamento de Matematica IMECC-UNICAMP-C.P. 6065 Campinas, SP-Brasil CEP 13083-970 E-mail address: lopesmime. uni camp. br

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 197-215 © 2003 Birkhiiuser Verlag Basel/Switzerland

Traveling Waves in Nonlinearly Supported Beams and Plates J. Horak and P.J. McKenna

1. Introduction The observation of the traveling waves on the Golden Gate Bridge in 1938 described in [1] motivated the research of the traveling wave solutions of the nonlinear beam equation (1) utt + uxxxx + f (u) = 0 . In 1988, McKenna and Walter [9] proposed a model of a suspension bridge based on (1) with f (u) = max{u, 0} - 1. They proved existence of traveling wave solutions

u(x, t) = z(x - ct) + 1 by explicitly solving two ordinary differential equations obtained for each of the linear parts of the piecewise linear function f. Later, Chen and McKenna [6] found a more general existence proof that does not rely on the explicit solution of the two ordinary differential equations. The existence proof of traveling wave solutions of (1) is based on a variational formulation and employs the Mountain Pass Theorem. The nonlinearity f belongs to a certain class of functions that are "similar" to u+ - 1 in a qualitative way. The authors also found several traveling wave solutions numerically using the Mountain Pass Algorithm. With these numerical solutions they performed a series of numerical experiments [5] to study their stability and interaction properties.

The main objective of the current work is to study equation (1) in higher dimensions. The nonlinear plate equation, that will be studied, has the form Utt + 02u + f (u) = 0

,

(2)

where N E N, u = u(x1, x2, ... , XN, t) is an unknown real function, f (u) is a given real function. Traveling waves are solutions u of (2) of the form

u(xlex2>...7XN,0 = Z(xl -Ct,x2,...,xN)+1 , (3) where c E IR is a given constant - the speed of the wave in the x1-direction, z(xl, x2, ... , XN) with x = (x1, x2i ... , XN) E RN is a new unknown function such

that z(x)--+0as IxI-goo. In Section 2, a variational proof, via the Mountain Pass Theorem, of the existence of traveling waves for N E {1, 2, 3} and a wider class of nonlinearities f will be presented.

J. Horak and P.J. McKenna

198

The Mountain Pass Algorithm has proved to be a useful numerical tool in finding solutions of elliptic problems, whose existence is proved by the Mountain Pass Theorem. In Section 3, the main steps in applying this method to equation (2) are described. Section 4 will show the numerical solutions found for N = 2, as well as several numerical experiments showing stability and interaction properties of the traveling waves. A comparison with the results for N = 1 obtained in [5] and [4] will also be discussed.

It has been found that in jR2, as in R, there exist solutions that are very stable, since when two of these waves collide, they interact nonlinearly but emerge almost intact. At the same time, a fission of a wave into two simple stable waves is also shown in R2. The reason behind this behavior still remains an open question.

2. Existence of Traveling Waves The main ingredient of the proof is the Mountain Pass Theorem used to prove existence of critical points of a continuously differentiable functional on a Banach space [8, 11]:

Theorem 1. Let B be a real Banach space. Suppose I E C' (B, iR). Let there exist e 1, e2 E B and constants e, 6> 0 such that

IIaaQ(el) ? I(ei)+6 , e2 E B \ BQ(el) , I(e2) < I(el)

(4)

(5)

Then there exists a sequence {z,,.} C B such that s

and

0

as n --+ oo

(6)

where

s = inf max I(y(t)) ,

(7)

r={yEC([0,1],B)Iy(0)=el,y(1)=e2}.

(8)

?EI'tE]O,l]

Substituting (3) into (2), the plate equation becomes A2Z +

c2&Z

+ z + g(z) = 0

x E IR'S' ,

(9)

1

where g(z) = f (z + 1) - z. Our goal is to prove that this equation has nontrivial weak solutions under the following assumptions:

(I) N E {l.2.3}, (II) C'-' E (0, 2),

(III) g E C(!), g(z) = O(z2) as z -r 0, (IV) g(z) > (z + 1)- for z < 0 (where S- = max{-S,0}), (V) g is concave up on (-oc, 0), concave down on (0, +oo).

Traveling Waves in Nonlinearly Supported Beams and Plates

199

Assumption (I) implies that H2 = W2.2(RN) is embedded in L°°, so there exists a1 > 0 such that

V¢EH2,

(10)

where 11- 16 denotes the standard norm on H2 with the square given by 11x11112 =

IVz12 + z2] d. T.

f [(Oz)2 +

(11)

RN

Moreover, functions in H2 are continuous, and if { z } is a bounded sequence in H2, then there exists a subsequence that converges uniformly on any compact subset of RN. Equation (9) is the Euler-Lagrange equation corresponding to the functional

J RN

\2

/

f

I (Z) = 2

(Oz)2 - c2 (

z

\

I

+ z2 dx + f G(z) dx ,

///

(12)

RN

where G(z) = f

Under Assumptions (I) and (III) one has I E C' (HI, R), that is, I has a continuous Frechet derivative given by I'(z)O =

f [(z)() - c2a1 8x +z01 dx + f g(z)Odx

RN

0 E H2 . (13)

RN.

It will be useful to define another norm on H2: Lemma 1. Suppose that Assumption (II) holds. The map 11'11: H2 -> [0, oo) defined by IIZI12 =

J R

I (Oz)2 - c2

l

(-

+ z21 dx

(14)

is a norm on H2 equivalent to 11'11112, so there exists a constant a2 > 0 such that a211211112 :5 11211 <

Vz E H2 .

11x11112

(15)

Proof. The second inequality follows immediately from the definitions. To prove the first inequality, we note that for any t E RN the following inequality holds: N a-1 (N )

2

N

1> 2

3

2

N

fn + l

[(a=] fin) + a=I

.

(16)

J. Horhk and P.J. McKenna

200

For z E H2 let z denote the Fourier transform of z. Then IIzII2

=

>

Cons t.

J

SaIxI2

[(1)2_c2+11 N

2-c2 3 const.I

N

2

fQ

+

Ftp+1 Ixl2dt: a=1

a=1

RN

`

2 (17)

3 IIZIIjp Hence we can take a 1 = , / 2 T .

We will apply Theorem 1 to the functional I on B = H2. Lemma 2. Suppose that Assumptions (I)-(IV) hold. Then there exist e1, e2 E H2 and constants p, b > 0 such that (4), (5) are satisfied.

Proof. First, we will show that (4) is satisfied. Choose el = 0. Then I(el) = 0. The hypotheses of the lemma imply that Vk > 0 3Ck > 0 Vz E [-k, k] we have Ig(z)I CkIzI2. This also implies IG(z)I < 3 IzI3. Let p E (0,1), IIzIIH. = p (then I I z I I L °O < a 1), and k = a 1. We can make the following estimate:

I(z) =

f I:I3dx

2IIzII2+f G(z) dx > 2IIzII23k

RN

R^'

3 IIzIIL- IIZIIL2 ? 2a2IIZIIHz - 3ka1 IIZI13

>-

2a2IIZIIH

=

3kalp3 = 5 > 0, 1a2p22 2

(18)

where the last inequality holds for p small enough. Now, we have to show that (5) is satisfied, too. Let z E H2 \ {0} such that

z < 0, and Il = supp z be a compact set in RI. The hypothesis g(z) > (z + 1)implies that fRN G(z) dx < -2 fz<_1(z+ 1)2 dx. Then z

I(z) <

(Oz)2 - c2 8 J +z 2dx - 2 f (z + 1)2 dx

2f

z<-1

N

r

f I(Oz)2-C2

2 RN

IL

(( 1Oz

/2

] dx

(19)

\\

+2 f z2dx-- f (2z+1)dx. z>-1

z<-1

Traveling Waves in Nonlinearly Supported Beams and Plates

201

Let A > 0, then

I(Az)

<

2

[(Z)2_C2() jdx

2A2 r RN

+2

(Az)2dx J Az>-1

-

2Azdx

J Az<-1

(20)

[(Z)2_C2(\ ]dx++AIIzIL. \xl2

< AJ

RN

Thus I(Az) is estimated by a quadratic function of A. If we make sure that the coefficient of A2 is negative, then taking A large enough will cause I(Az) to be in the negative. Let A > 0 and denote i(t) = z(A). The coefficient of inequality (20) applied to z ZA2

2

(AEZ)2 - c2

I

atz

k_N2 - c2A2-N

dt

RN

(2

57X1

)

RN

_

\4-N IIAZI122

2

8z

- C2,\2-N

(21)

11L2

is, indeed, negative, provided we choose \ > 0 small enough. Hence we will take e2(x) = Az(Ax). We also need to choose A large so that IIe21IH2 > P. Then (5) is

0

satisfied.

We can conclude that under the hypotheses of Lemma 2 and according to the Mountain Pass Theorem 1 there exists a sequence

such that I(zn)-+s>0and I'(zn)-'0as n-goo.

{zn}CH2

(22)

Lemma 3. Suppose that Assumptions (1-III), (V) hold. Then a sequence {zn} C H2 such that I(zn) -p s and I'(zn) -> 0 as n oo is bounded in H2. Proof. Define a function

r(z) = 2G(z) - g(z)z ,

(23)

its properties are studied in the Appendix. Suppose { z } is not bounded, i.e., there exists a subsequence { zn, } such that 11 zn,11 W - oo as l - oo. To keep the notation simple, we will assume that {zn} is already such a subsequence. It is easy to verify the following: 0

=

0=

lim 00

21(zn) - 1'(zn)zn IIznII

1'(zn)zn

oo Ilzn 112

n

=

limoc J R^

= 1 + 1;1 fr RN

rizn)

dx

(24)

IIznII

g(zn)zn IIznI12

dx .

(25)

J. Horak and P.J. McKenna

202

By using (57) in Lemma 5 in the Appendix, where b1 will be determined later, (10), (15), and (24), respectively, we obtain 0

<

lim n-0o

r

I9(zn)zn1

dx <

IIznII2

IZnI>e

r(zn) dx

J

IIznII

/

J

b2r(zn)Iznl dx IIznII2

Iz..I>e

b2IIznIIL

lim noo

lim

n-oo

IIznII

<

b2 1-1-oo

a,

IIdx = 0. (26)

a2J

IIz n

RN

Iz, I>e

If we choose b1 in (56) small enough, then from (25) and (26) we get r

- lim

9(zn)zn

dx

f I9(zn)znI dx

lim

II Z. 2

J

Iznl<e b1Iz n I2

liminf Jr n-.oo

<

IIznII2

bl IIz,IIL2 dx < liminf

IZnl<<e

b2 a2

<

<1

IIznII2

n-°°

IIznII2

(27)

,

which is a contradiction.

Theorem 2. Suppose that Assumptions (I)-(V) hold. Then there exists a nontrivial weak solution of equation (9).

Proof. We are looking for a function v E H2 \ {0} such that I'(v) = 0. Let A > 0 be a constant such that IIznIIH2 <
VnEN.

(28)

Then there exists a subsequence {zn, } that has a weak limit z in H2. We need to ensure that the weak limit is a nonzero function. The hypotheses of the theorem imply that Vk > 0 3Ck > 0 Vz E [-k, k] we have Ig(z)I < CkIzl2. This also implies IzI3. Since IIznIILo < al IznIIH2 < a1A, we take k = a1A. Then, if we IG(z)I < denote Ck = 2 + Ck, we get

r(zn) = 2G(zn) - 9(zn)zn < CkIznI3 .

(29)

We can make the following estimates: 2s

= nlimo[21(zn) - I'(zn)zn] = limo f r(zn)dx <

liminf Ck n-»oo

f

RN

I zn 3 dx < limn-.oo inf Ck II zn II LO

II zn II

L2

RN

< CkA2liminf IIzfhILn-oo

(30)

Traveling Waves in Nonlinearly Supported Beams and Plates

203

This means that there exists a positive integer no such that IIZIIL- >

do > no

6kA2

(31)

hence for every n > n,) there exists xn E RN such that Izn(xn)I >

s

CkA2

>0.

(32)

Let us define a new sequence {vn} by vn(x) = zn(x + xn). Then the sequence

{vn} has the following properties:

oo,

IIvn11H2
Ivn(0)I > CkA2 > 0

(33)

There exists a weak limit v E H2 such that vn w, v in H2 as l -+ oo along a subsequence. For the sake of a simple notation, let N,} be already such a subsequence.

To prove I'(v) = 0, we only need to show

I'(vn)O-'I'(v)4asn-,oo

`q EH2.

(34)

Fix 0 E H2. The weak convergence of {vn} to v in H2 implies

fI

c2 av' ao + vncb] dx -,

ax, ax,

L

RN

f

(AV)(A0) - c2 av 190 + v¢] dx

ax, ate,

(35)

RN

as n -+ oc. Hence it remains to show that

fg(vn)dx_*fg(v)*dx

as n -* oo .

(36)

RN

RN

Both vn and v are bounded in H2 by A, so IIvn1IL IIvIIi.,, < a,A = k. Concavity of g on (-oo, 0) and (0, +oo) implies >

sup

I g(O - g(Q = B < oo .

4.(E(-k.k],t0(

4-(

(37)

The following estimate can be made

f Ig(vn) - g(v)II0I dx <

Of Ivn - vII0I dx ,

(38)

RN

RN

which means that it is enough to show

f Ivn - vII0I dx RN

0

as n

oo ,

(39)

J. Horak and P.J. McKenna

204

i.e., We > 0 3n bn > n fit, Iv,, - vI 101 dx < e. Fix e > 0. Then there exists a constant K > 0 large enough such that 2A IIbIIL2(izi>K) < 2. This means that

J

Iv, - vJj

dx < IIVn - vIIL- IImIILz(ixi>K) <

.

2

(40)

jzI>K

Sequence {v } has a subsequence that converges uniformly on compact subsets of RN, let be already such a subsequence. The weak limit v of in H2 is also its uniform limit. By choosing no large enough, the following inequality can be satisfied

f Iv,, - vII0I dx < IIvn - VII(.(jej
(41)

(x1
This proves (34).

Since v,, (0) - v(0) as n -, oo, (33) implies v(O) # 0, so v is a nonzero

0

function.

3. Mountain Pass Algorithm Since the existence proof in Section 2 was based on the Mountain Pass Theorem 1, it is reasonable to try to find some of the solutions of (9) using the Mountain Pass Algorithm. The algorithm is, in fact, a constructive form of the Deformation Lemma [11]. The method was described in [7] by Choi and McKenna for second-order semilinear elliptic problems. Chen and McKenna [6] successfully used

this method on a one-dimensional nonlinear beam equation problem. A shooting method of [3] is more efficient in the one-dimensional case than the Mountain Pass Algorithm. In higher dimensions, however, the Mountain Pass Algorithm is our first choice of a numerical method. This section will describe how to implement the method to solve equation (9). On a finite-dimensional approximating subspace of functions, we take a piecewise linear path from the local minimum el = 0 of I to a point e2 whose altitude 1(e2) is lower than that of e,. Then we find the maximum of I along the path. The point, at which the maximum occurs, is then pushed in the direction of the steepest descent thus lowering its altitude and deforming the path. The norm we choose to use is the norm defined by (14) on H2. This step is repeated until the critical point is reached, that is, until no further lowering of the local maximum along the path is possible. Since H2 with the inner product

(0, b)c= f [()(iW)

-t- c2 ±"ax, ax,

dx

0, VIE H2

(42)

RN

is a Hilbert space, the steepest descent direction v of I at u E H2 is opposite to the direction given by the gradient of I at u. Let -v E H2 be the gradient of I at

Traveling Waves in Nonlinearly Supported Beams and Plates

205

u, that is I '(u)cb _ -(v, 0),2 dO E H2 . By denoting w = u + v, equation (43)11 can be written as 8w 8 4 6 [(w)(&) - c2 8x, ax,

+ wcb] dx = -

(43)

VOEH2,

19(u)cbdx

(44)

RN

RN

hence a linear equation

a2 + w = -g(u)

02w + c2

(45)

1

needs to be solved for w with u given, or, equivalently, the minimum of the functional

f

J(w) =

J

(Ow)2 -c2

()2 +w2 dx + J g(z)wdx

2 RN

(46)

RN

needs to be found.

We would like to apply the algorithm for N = 2. Although the domain of solutions of (9) is the whole plane 1R2, for the numerical purposes we can work on a bounded rectangle R = [-KI, KI] x [-K2, K2] C R2 only, because our solutions

decay to zero as IxI - oc. The constants KI, K2 > 0 must be chosen sufficiently large.

A finite difference method with a uniform mesh with step sizes Ax,, = 2KQ/MQ (where M. E N, a E 11, 2}) is used. For a numerical representation w=(WI.1,W1,2,...,w1.nf2,W2.1,W2.2,...,w2.A121...,wnf,.M2)T

(47)

of a function w E H2 on the rectangle R we can approximate J(w) by

J(w)

Ox10x2 [WTBW + w1 g(z)J

(48)

,

where g(z) represents a vector of values of g evaluated at the mesh point values of z. B is an M1 x M2 matrix with the following structure:

B=

1

AZ'x'+

2

AX1X2+

1

AX212-

Ax'+E, (49)

(0x2)4 (0x1)2 (4x1)4 (&1)2(Ax2)2 where the matrices Ax- x", Ax' are block matrices with integer entries coming from the finite difference approximation of the derivatives with respect to x0,xp and x1, respectively, E is the identity matrix. To minimize (48), we need to solve a linear algebraic system

Bw = -g(z) .

(50)

A very efficient method to diagonalize matrix B is the Fast Fourier Transform [12].

We can further decrease the size of the computation by considering only symmetric solutions z such that

z(xI,x2) = z(-x1,x2) and z(x1,x2) = z(XI,-x2) d(x1,x2) E 1R2 .

(51)

206

J. Horak and P.J. McKenna

Hence it will be enough to find numerical solutions on a smaller rectangle, [-Ki, 0] x [-K2,0]. On the boundaries of this rectangle given by x, = 0 or x2 = 0 we will have to apply the Neumann boundary condition. On the rest of the bound-

ary, i.e., for xi = -K1 or x2 = -K2, we assume that the solution z has its values very close to zero. We can apply the Dirichlet boundary condition here or the Neumann boundary condition. To make the algorithm more simple, we will use the Neumann boundary condition on the whole boundary of [-K,,0] x [-K2,01.

4. Stability and Interaction Properties We will present solutions of (9) found for N = 2 using the Mountain Pass Algorithm. We choose the nonlinearity f in equation (2), and hence the function g in equation (9), to be

.f(u)=a°-g(z)=e2-1-z.

(52)

The same function was also used in [4, 5] for numerical experiments for N = 1.

Figure 1 presents a single-pulse wave with c = 1.00 and c = 1.40 (Axe _ 0x2 = 0.1). To test the stability of these solutions, we solve the initial-boundary value problem for the original equation (2) on the rectangle R with the mountain pass algorithm solutions used as initial data u(x,0) = z(x) + 1. A standard explicit central finite difference scheme is used with periodic boundary conditions. The computations were performed on dual-processor Pentium computers using the Message Passing Interface for the parallel C code [10]. The single-pulse waves in Figure 1 keep their shape over time, they are stable. Even with a small perturbation they maintain the same basic shape. In Figure 2, two single-pulse waves are sent towards each other. After the collision they emerge almost intact. The same behavior was shown for single-pulse waves in one space dimension in [5]. Figure 3 shows an interaction of two copies of a single-pulse wave with c = 1.35 traveling in perpendicular directions. There is more noise apparent after the interaction. Figure 4 presents a double-pulse wave with c = 1.00. In [4], a numerical evidence was given that the single and double-pulse waves in R bifurcate from zero amplitude at c2 = 2. A similar conclusion can be made based on our numerical experiments in R2. Figure 5 shows the minimum of the single and double-pulse waves showed in Figures 1 and 4 as a function of the wave speed c E (1, v r2). The stability test of the double-pulse wave is shown in Figure 6. The wave starts to fission into two single-pulse waves with different speeds - similar behavior as the one observed in one space dimension [4]. Because of the periodicity of the boundary conditions of the numerical scheme, after a long time the waves collide Figure 7. For a short time, a wave very similar to the original one is formed before breaking down.

Traveling Waves in Nonlinearly Supported Beams and Plates

(aI

(c) 0

207

A

-5 -10 -15 -20

-40

-20

-20

.10

0

-40

-20

0

-40

-20

20

40

10

20

20

40

20

40

(d) 0 -10 -15 -20

(h) 0.5

.. -0.5

FIGURE 1. Single-pulse wave for c = 1.00 and c = 1.40: (a,e) a view from below, (b,f) above, (c,g) a cross section at x2 = 0, (d,h) at xl = 0.

208

J. Horak and P.J. McKenna

-100

t.

0.0

t.

35.0

0..0

50

50

100

0.-0

30.0

41

-100

1

-50

t-

0

70.0

50

10

-0

0

-1

-100

50

so

100

FIGURE 2. Interaction of two single-pulse waves with c('') = 1.35, c(I) = -1.40.

Traveling Waves in Nonlinearly Supported Beams and Plates

209

( a)

ihi

1,a

FIGURE 3. Interaction of two single-pulse waves with c = 1.35 traveling in the positive x1-direction and the negative x2-direction.

J. Horak and P.J. McKenna

210

(c) 0

-5

. -10 -15 -20 -40

-20

-20

-10

0

20

40

10

20

(d)

-5

x -10 -15

-20

FIGURE 4. Double-pulse wave for c = 1.00: (a) a view from below, (b) above, (c)

a cross section at x2 = 0, (d) at xl = 4.5. 0

-5

-10 E

-15

-20

1

1.1

1.2

1.3

1.4

C

FIGURE 5. The minimum of the single- and double-pulse waves as a function of the wave speed c.

5. Open Questions Under the Assumptions (I)-(V), we have proved the existence of at least one solution, but there is enough numerical evidence of existence of many solutions. In [4], the authors made a first step at trying to classify various solutions of the beam equation in one dimension. Also, based on the numerical experiments, they

Traveling Waves in Nonlinearly Supported Beams and Plates

0:-0

0.0

5-

211

5 10 IS 0

-25 -40

-20

t-

0

20

75.0

40

-0

-5

10 15 20 25

-40

-20

0

20

40

(c) 5- 125.0

-40

-20

X,-0

a

20

40

0:-0

1- 200.0

tin

-40

-20

0

20

0

FIGURE 6. Fission of a double-pulse wave, c = 1.00.

The issue of noted that certain types of solutions do not exist for c close to multiplicity of solutions and the range of the wave speed c for which they exist needs to be investigated.

J. Horak and P.J. McKenna

212

-0

0. 1000.0

.10

-20

25 -20

0

20

40

(b) 0. 1005.0

x;.0

0 -5

10 15

20 25 40

-20

0

I 20

40

FIGURE 7. Collision of two single-pulse waves. (a) Fusion into a double-pulse wave. (b) Break-down of the wave.

In the one-dimensional case, various types of traveling wave solutions were obtained by the shooting algorithm [4]. In 1R only two types of waves have been found so far using the Mountain Pass Algorithm. Different choices of the endpoint e2 of the path may cause the algorithm to converge to different solutions. However, it is not clear whether more types of solutions can be found by this method. The existence question has not been satisfactorily answered yet. Although the

numerical evidence presented suggests that the traveling waves exist for f (u) _ e"-i - 1, this function does not satisfy Assumption (V). Another large area for further research are the stability properties of the waves. There are, so far, no theoretical results. Similar numerical results were obtained in RN for both N = 1 and N = 2. As experiments, like the one in Figure 2, indicate, for wave speeds c close to f the waves are very stable. For smaller c they appear to lose stability and more "noise" is present after the waves pass through each other. Another important phenomenon observed but left without an explanation is the fission of a double-pulse wave shown in Figure 6 for c = 1.00. More noise has been observed after a fission of the double-pulse wave with a larger wave speed.

Traveling Waves in Nonlinearly Supported Beams and Plates

213

Appendix Lemma 4. Suppose g satisfies (III), (V), let r(z) be defined by (23). Then r(z) > 0

for all z E R. If there exists zo E R such that g(zo) = 0 then g(z) = 0 for all z between 0 and zo.

Proof. Suppose z > 0 and zo > 0. By concavity of z on (0, +oo), we have 9(0 > 9(0) + (C - 0)

Then

9(zz

(E (0, z) .

(53)

d(- 9(z)z = 0 . J0 (g())

(54)

- 0(0) = (g(z)

x

r(z)=2 f 9(C) d(- 9(z)z > 2 0

The last inequality implies that if r(zo) = 0, then

f 9(()d( 0

f (9(oo)d(

f [g(() -(g(oo)]d(=0.

0

0

By concavity, the integrand is nonnegative, and by continuity of g, we get

9(() =

9(oo)

t1(E [0, zo] .

(55)

Since g(z) = o(z) as z -r 0, (55) implies 8S o 1 = 0, which in turn implies g(() = 0 for all C E [0, zo]. The proof is similar for z < 0 and zo < 0. 0

Lemma 5. Suppose g satisfies (III), (V). Let bl > 0 (small). Then there exist constants e, 62 > 0 such that I9(z)I < bj I z I I9(z)I < b2r(z)

for l zl < e ,

(56) (57)

for IzI > e .

Proof. Fix b> > 0. The existence of e such that (56) holds follows from the fact

that g(z) = o(z) as z-+0. We will prove (57) for z > e (the proof for the case z < -e is similar). Our hypotheses imply that g(z) < 0 for z > 0. We must find b2 > 0 such that b2r(z) + g(z) > 0 for all z > e. We will prove two inequalities:

b2r(z) + g(z) > b2r(e) + g(e) > 0 Vz > e

.

(58)

The second inequality is satisfied for b2 large enough because, by Lemma 4, r(e) > 0 and if r(e) = 0, then g(e) = 0. The rest of the proof will focus on the first inequality. This inequality is equivalent to 62

r(z) - r(e) > _ 9(z) - 9(e)

z-e

z- e

`dz > e ,

(59)

which, as we will show, holds true for 62 large. Using the definition of r and concavity of g, we can arrive at an estimate

r(z) - r(e) > g(e)(z - e) - e[g(z) - 9(e)]

(60)

J. Horak and P.J. McKenna

214

After dividing by z - e and multiplying by b2, we get b2 r(z)

- e(e) > b2

[g(e)_e9 (zz

-

9(e)]

(61)

If we compare this last inequality with (59), we find that to complete the proof, it is enough to show b2

[g(E)

eg(z) - g(E)] > _g(zz

-

9(e)

dz > e .

(62)

Both sides of this inequality contain - v ZZ-E e . It follows from the hypotheses that

this is an increasing nonnegative function of z. Denote L = lim._.e+ =z-e If we show that b2[g(e) - eL] > -L (63) for b2 large enough and if by choosing b2 > we ensure that the left-hand side of (62) will grow faster with z than its right-hand side, then (62) is satisfied. Hence, finally, the very last step of the proof is to show that (63) can be satisfied for some large b2. Concavity of g implies that the left-hand side of (62) is nonnegative, hence g(e) - eL > 0. If g(e) - eL = 0, then L = 2. Let C E (0,e). Concavity of g implies again th at

g(z) - g(e)

<

g(O - g(E)

z-e (-e If we take the limit as z - e+, we get g(O - g(e)

(- e

<

g(e) - g(0)

e-0

(64)

g(e) e

( 5)

which means that g is a linear function on (0, e). And if we take into account continuity of g and g(z) = o(z) as z - 0, we obtain g =_ 0 on [0, e]. Hence, if g(e) - eL = 0, then g(e) = L = 0, so (63) will be satisfied for any b2. If g(e) - eL > 0, then we can choose b2 large so that (63) holds true. 0

References [1] O.H. Ammann, T. von Karman, G.B. Woodruff: The Failure of the Tacoma Narrow Bridge. Federal Works Agency, Washington, DC, 1941.

[2] A.R. Champneys, P.J. McKenna: On solitary waves of a piecewise linear suspended beam model. Nonlinearity, 10 (1997) 1763-1782. [3] A.R. Champneys, A. Spence: Hunting for homoclinic orbits in reversible systems: a shooting technique. Adv. Comp. Math., 1 (1993) 81-108. [4] A.R. Champneys, P.J. McKenna, P.A. Zegeling: Solitary waves in nonlinear beam equations; stability, fission and fusion. Nonlinear Dynamics, 21 (2000), no. 1, 31-53. [5] Yue Chen, P.J. McKenna: Traveling waves in a nonlinearly suspended beam: some computational results and four open questions. Phil. Trans. Roy. Soc. Lond. A., 355 (1997) 2175-2184.

Traveling Waves in Nonlinearly Supported Beams and Plates

215

[6] Yue Chen, P.J. McKenna Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations. J. Dif. Eqns., 136 (1997) 325-355. [7] Y.S. Choi, P.J. McKenna: A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Analysis, Theory, Methods and Applications, Vol. 20, No. 4 (1993) 417-437. [8] J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences 74. Springer-Verlag, New York, 1989. 75-80. [9] P.J. McKenna, W. Walter: Traveling waves in a suspension bridge. SIAM J. Appl. Math., 50 (1990) 703-715. [10] P.S. Pacheco: A User's Guide to MPI. Department of Mathematics, University of San Francisco. [11] P.H. Rabinowitz: Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math., 65 (1986). [12] G. Strang: Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley, MA, 1986. 451-458.

J. Hor'ak Mathematisches Institut Universitat Basel Basel, Switzerland

E-mail address: [email protected] P.J. McKenna Dept. of Mathematics University of Connecticut USA

E-mail address: mckenna@math. uconn. edu

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 217-224 © 2003 Birkhauser Verlag Basel /Switzerland

Solitary Waves Solutions of a Nonlinear Schrodinger Equation A.M. Micheletti and D. Visetti Abstract. The aim of this note is to prove the existence of standing waves solutions of the following nonlinear Schrodinger equation

i!1;

= -v4'+ V(x)u +rN(+&),

where N(r,) is a nonlinear differential operator. In [81 and [91 Benci and the authors proved the existence of a finite number of solutions (Ic(e), u(s)) of the eigenvalue problem

(P,) -Au + V(x)u + eN(u) = µu where N(u) = -D,,u + W'(u). The number of solutions can be as large as one wants. Since W is singular in a point these solutions are characterized by a topological invariant, the topological charge. A min-max argument is used.

1. Introduction The first documented observation of a solitary wave was made in 1834 by the Scottish scientist and engineer John Scott Russell (see (171). The term solitary wave was coined by Scott Russell himself, mainly because this type of wave motion stands alone and apart from the other types of oscillatory wave motion. Using a very loose definition, solitary wave solutions are no more than waves which propagate without change of form and have some localized shape.

Strauss (see [18]) gives the following definition of a solitary wave:

"We define a solitary wave as a solution O(x, t) of a wave equation whose maximum amplitude at time t, sup, 10(x, t)I, does not tend to zero as t oo, but which tends to zero in some convenient sense as jxj -+ oo for each t. The convergence should have the property that physical quantities, such as the energy and charge, are finite. Particular types of solitary waves are (1) traveling waves 0 = u(x - ct) where c is a constant vector and (2) standing waves 0 = exp(iwt)u(x) where w is a real constant." In a paper of 1964 Derrick tackled the problem of giving a model for elementary particles. In that period Enz (see [14]) had considered the sine-Gordon

A.M. Micheletti and D. Visetti

218

equation and had shown that in the one-dimensional case it has time-independent solutions where the energy is localized about a point on the x-axis. Enz suggested that, since these one-dimensional solutions possess certain symmetry and topological properties, they might correspond in the three-dimensional case to discrete quantum numbers, such as charge or parity. In [13] Derrick wrote: "These suggestive results of Enz for the one-dimensional case then lead us to consider the following problem: Can (1) [the sine-Gordon equation] or some similar nonlinear equation have stable, time-independent, localized solutions in three dimensions? If such solutions exist then it

would be an attractive hypothesis that the allowed energies [to have stability] correspond to the rest energies of elementary particles. The answer given to the above question by this paper is no. The equation

z

A9- ca 5B =

2fl(O)

[... ] will be proved to have no stable, time-independent localized solutions for any f (0)." He presented some possible ways out of this problem and the first one was to consider higher powers for the derivatives in the Lagrangian function. Actually this conjecture has been considered by Benci, Fortunato and Pisani in a paper of 1998 ([7]): they proved that equation where ;o

:

1R:'

-A,p - Ono+W'(P) = 0, (1) -* IR` and W : R' \ -p JR is a singular function, admits

a nontrivial solution (a solitary wave) with the energy concentrated around the origin and with a particle-like behavior. The functions of the configuration space (and consequently the solutions of the problem) are characterized by a topological invariant ch(.), called topological charge, which takes integer values: hence, also from this point of view, there is an analogy with the case of the sine-Gordon equation. Therefore we can say that the existence of these concentrated solutions is guaranteed by topological constraints. The aim of this paper is to present some results of existence and multiplicity of standing waves solutions of the following nonlinear Schrodinger equation i

where ty : 1Z x JR , C and waves of equation (2)

= -A*+V(x)-O+eN(TI'),

(2)

is a nonlinear differential operator. The standing

i1(x,t) = u(x)e with u : 9

$t, are determined by the solutions of the following nonlinear eigenvalue problem

-Du + V(x)u + rN(u) = pu

(3)

N (u(x)e O') = e 'I'N (u(x))

(4)

provided that

Solitary Waves Solutions of a Nonlinear Schrodinger Equation

219

The nonlinear operator

N(u) = -Opu + W'(u)

(5)

can be extended to the complex functions in such a way to verify (4). The basic idea is to consider problem

-AU +V(x)u+E(-A u+W'(u))=/tu

(Ps)

as a perturbation of the linear problem (Po). In terms of the energy functional associated, one considers the non-symmetric functional

[vu2+ 2 V(x)Iu12 + EIVuIp +eW(u)J dx. (6) = as a perturbation of the symmetric functional Jo. Non-symmetric perturbations Jf(u)

f

of a symmetric problem, in order to preserve critical values, have been studied by several authors. We omit for the sake of brevity a complete bibliography and we only recall Bahri, Berestycki, Bolle, Ghoussoub, Tehrani, Rabinowitz, Struwe (see [3], [11], [12], [16] and [19]).

The existence result is then a result of preservation for the functional JE of some critical values of the functional Jo, constrained on the unitary sphere of L2(ft Rn+1). The hypotheses on the function V : f2 -+ R are the following: (V1)

1 bounded domain V E L" (1, lR)

(VII)

ft = Rn lim V (X) = +00

(V2)

ess inf V(x) > -oo

(V2)

V(x)e-1x1 E LP(Rn,R)

(V3)

ess inf V(x) > 0

xER"

IzI-+CO

xER"

We note that (1') is a technical hypothesis. We need it to prove the regularity of the eigenfunctions of the linear eigenvalue problem, but it may be weakened. The assumptions on the function W : Rn+I \ IR are the following: fl bounded domain

(Wi) W E C (iR' (W2)

\ {..},lR)

W(C)>_0V

SI = R

\

(Wi) W E C (R' (W2) 0

0 such that

(W3)

d E Rn+l 0 < I S 1 < Cl

3c1, c2 > 0 such that VV E Rn+1 0 <

Cl

Itl P-n

IfI P-"

and{-cl >0 (W4')

3c3, c4 > 0 such that


VVElRn+l c41E1

A.M. Micheletti and D. Visetti

220

In the case c = 0 the functional J is even and, using for example a category argument, one easily obtains the existence of infinitely many critical values of the functional JO constrained on the unitary sphere of L2(cl, R'+'). Here the energy functional J. is not even and we look for solutions either as minima or as min-

max critical points of the functional J constrained on the unitary sphere S of L2(9., R"+'). Since the topological charge divides the domain A of the energy functional J, into connected components A. with q E Z, these solutions have to be found in each component A4 fl S. First of all one has to verify that these components are

not empty and then in any A. fl S a suitable submanifold has to be constructed in such a way that the min-max technique can be applied. We can do that if the parameter ae is sufficiently small and the singular point C. is sufficiently far from the origin of IIt"+'. The number of solutions obtained is finite, but it can be as large as one wants, provided that e is suitably small and C. is suitably far from the origin. By some technical devices, it is possible to obtain the Palais-Smale condition

for the functional J constrained on the sphere S. The addition of the potential V breaks the translation invariance, so that the technical lemmas require some care. When we consider ) = 1R" the compactness is recovered by the coercivity of the potential V. The following results are established in [8] and [9]:

Given q E Z, for any C. = (CO, 0) (with Co > 0 and 0 E R") and for any s > 0, there exist µ1(e) and ui(e) respectively eigenvalue and eigenfunction of the problem (Pe), such that the topological charge of ui(e) is q. Moreover, given q E Z \ {0} and k E N, we consider C. = 0) with CO large enough and 0 E R". Let Aj be the eigenvalues of the linear problem (Pa). Then

for e sufficiently small and for any j < k with A,_i < a there exist uj (,-) and us(e) respectively eigenvalue and eigenfunction of the problem (PE), such that the topological charge of u, (s) is q.

2. Functional setting The functional space E is differently defined, depending on 0; more precisely it is the completion of Co (fl,1R"+') with respect to the following norm: IIDuIIt2 + IIVuIILr IIUIIE =

11V4u11L2

+ IIVUIIL2 + IIVuIIL,

if n is bounded if H is R"

where, if we write u = (ui,...,u,,.+1), rl+1

IIV#u

IlVuIIi _ i=1

IL2

V (x)Iu(x)I2dx)

(7)

Solitary Waves Solutions of a Nonlinear Schrodinger Equation

221

The energy functional J. associated to the problem (PE) is (6) and is defined on the open set

A = {u E Elf. ¢ u(1l)} .

(8)

The functional Jf is of class C' on A. The functions of the set A are characterized by a topological invariant called topological charge, which divides A into connected components

A= UAq, qEZ

with

Aq = {u E A / ch(u) = q}. We consider the constraint

S = {u E E/

IIUIIL2(0,Rnn+i) = 1}.

By means of some technical lemmas it is possible to recover the following compactness result: Lemma 2.1. The functional JE satisfies the Palais-Smale condition on Aq n S for

anygEZand0<e<1.

3. Critical values of the energy functional The first existence result is obtained by looking for the minimum of the functional JE on each component Aq n S, with q E Z, for any e > 0 and for any t;. = (0, t;):

Theorem 3.1. Given q E Z, for any t. = (0, 1;) with 0 E R' and t; > 0 and for any e > 0, there exists a minimum for the functional JE on the component Aq n S of An S. To construct the other critical values of the energy functional, it is necessary to recall now some classical notions about the linear eigenvalue problem (Po)

-Du + V (x)u = µu

in L2(11, IR"+1) There exists a non-decreasing sequence of eigenvalues {.Aj } jEN, counted with their multiplicity, with associated eigenfunctions {e, } jEN, normalized

in L2(fl,1R"+1). It can be proved that the eigenfunctions {ej}jEN have enough regularity to belong to the space E. We set for any j E N

Fj =span [e1,..,,e,], S(j) = Fj nS.

(9)

We introduce some suitable functions G£ of topological charge q and depending on e. For the sequel the choice of the G f's is crucial. These functions are constructed in such a way that we can build some suitable submanifolds of

A.M. Micheletti and D. Visetti

222

S, contained in A. n S, which are in some sense a copy of the finite-dimensional sphere S(j). With fixed k E N, these submanifolds are the following ones: /Vt E,J

-

CQ

/ u E S(j)

+ Pu

11G°

(10)

+pUIIL2(u,Rri}1)

where j < k, q E Z \ {0} and e, p > 0 are sufficiently small. Now we can consider the values c?i = inf sup JE(h(u)), hE't uEMs,i

where x-',, is the set of continuous transformations '{y

h : A. n S

A,, n S / h continuous, h Lt

, = idMq, }

.

At this point our aim is to prove that the real numbers ce,, provide some critical values for the functional JE. Indeed we get the following result either when Si is bounded or Si = R". Theorem 3.2. Given q E Z \ {0} and k E N, we consider (0, t;) E R"+1 with > 2mk, where mk = IIUIIL-(SI,R.-+1). Then there exists & E (0,1] such that for any e E (0, e] and for any j < k with \j -I < \j, we get that c9,j is a critical value for the functional JE restricted to the manifold Ay n S. Moreover cw,j_ 1 < c£ j and cE,j -i A for a -1, 0. To conclude we point out that it is possible to find symmetric solutions of the problem studied here (see [201). The symmetry considered is given by the following

action of the orthogonal group O(n) on the space E for Si = W' (see [4]): T:

O(n) x E -* (g, u)

E --> T9u

(12)

where, if we write u = (u, u"±1), with u E R", (13) T9u(x) = (g-,fi(gx),u,,+l(gx)) Requiring some suitable symmetry assumptions on the functions V and W, it is possible to obtain an existence and multiplicity result of symmetric solutions similar to Theorems 3.1 and 3.2.

Solitary Waves Solutions of a Nonlinear Sc.hrodinger Equation

223

References [1] A. ABBONDANDOLO, V. BENCI, Solitary waves and Bohrnian mechanics, to appear in Proceedings of the National Academy of Sciences of the United States of America. [2] M. BADIALE, V. BENCI, T. D'APRILE, Existence, multiplicity and concentration

of bound states for a quasilinear elliptic field equation, Calculus of Variations and Partial Differential Equations 12 3 (2001), 223-258. [3] A. BAHRI, H. BERESTYCKI, A perturbation method in critical point theory and applications, Transactions of the American Mathematical Society 267 1 (1981), 1-32. [4] V. BENCI, P. D'AVENIA, D. FORTUNATO, L. PISANI, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Archive for Rational Mechanics and Analysis 154 4 (2000), 297-324. [5] V. BENCI, D. FoRTUNATO, Discreteness Conditions of the Spectrum of Schrodinger Operators, Journal of Mathematical Analysis and Applications 64 3, 1978. [6] V. BENCI, D. FORTUNATO, A. MASIELLO, L. PISANI, Solitons and the electromagnetic field, Mathematische Zeitschrift 232 1 (1999), 73-102. [7] V. BENCI. D. FoRTUNATO, L. PISANI, Soliton-like solution of a Lorentz invariant equation in dimension 3, Reviews in Mathematical Physics 10 3 (1998), 315-344. [8] V. BENCI, A.M. MICHELETTI, D. VISETTI, An eigenvalue problem for a quasilinear elliptic field equation, to appear in Journal of Differential Equations. [9] V. BENCI, A.M. MICHELETTI, D. VISETTI, An eigenvalue problem for a quasilinear elliptic field equation on iR", Topological Methods in Nonlinear Analysis 17 2 (2001), 191-212.

[10] F.A. BEREZIN, M.A. SHUBIN, The Schrodinger Equation, Kluwer Academic PubUshers, 1991.

[11] P. BOLLE, On the Bolza problem, Journal of Differential Equations 152 2 (1999), 274-288.

[12] P. BOLLE, N. GHOUSSOUB, H. TEHRANi, The multiplicity of solutions in nonhomogeneous boundary value problems, Manuscripts Mathematica 101 3 (2000), 325350.

[13] C.H. DERRICK, Comments on Nonlinear Wave Equations as Model for Elementary Particles, Journal of Mathematical Physics 5 (1964), 1252-1254. [14] U. ENZ, Discrete Mass, Elementary Length, and a Topological Invariant as a Consequence of a Relativistic Invariant Variational Principle, Physical Review 131 (1963), 1392.

[15] A. MANES, A.M. MICHELETTI, Un'estensione delta teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. U.M.I. (4) (7), 1973, 285301.

[16] P.H. RABINOWITZ, Multiple critical points of perturbed symmetric functionals, Transactions of the American Mathematical Society 272 2 (1982), 753-769. [17] J. SCOTT RUSSELL, Report on waves, Rep. 14th Meeting of the British Association for the Advancement of Science, John Murray, London 1844. [18] W.A. STRAUSS, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics 55 2 (1977), 149-162.

224

A.M. Micheletti and D. Visetti

[19] M. STRUWE, Infinitely many critical points for junctionals which are not even and applications to superlinear boundary value problems, Manuscripts Mathematica 32 3-4 (1980), 335-364. [20] D. VISETTI, An eigenvalue problem for a quasilinear elliptic field equation, PhD Thesis, Department of Mathematics, University of Pisa, 15th June 2001.

A.M. Micheletti and D. Visetti Dipartimento di Matematica Applicata "U. Dini" University degli studi di Pisa via Bonanno Pisano 25/B 1-56126 Pisa Italy

E-mail address: a. michelettimdma. unipi. it E-mail address: visettiOdm.unipi.it

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 225-238 © 2003 Birkhauser Verlag Basel/Switzerland

Nontrivial Solutions of a Class of Quasilinear Elliptic Problems Involving Critical Exponents C.O. Alves, P.C. Carriao, and O.H. Miyagaki Abstract. In this paper we deal with the following class of quasilinear elliptic problems in radial form

J -(r' Iu'j'3u')' = \r6juj'u + r'IuI°-2u in (0, R)

(P)

l

u(R) = u'(0) = 0

where a, 0, 6, ry, q are given real numbers, \ > 0 is a parameter and 0 < R < 00.

Combining a version of generalized mountain pass theorem due to Rabinowitz and an argument used by Brezis and Nirenberg to overcome the lack of compactness due to the presence of critical Sobolev exponents we prove a result of the existence of a nontrivial solution of problem (P), for any A E (0, A ), where A > A1.

1. Introduction In this paper we deal with the following class of quasilinear elliptic problems in radial form

(P)

J -(r"Iu'13u')' = ar6IuIau+r'lul9-2u in (0,R)

l

u(R) = u'(0) = 0

acting (weakly) in absolutely continuous functions u : (0, R) - + R, A > 0 is a parameter, 0 < R < oo and a,,3, 6,'y, q are given real numbers. When considered acting in radially symmetric functions, say in a ball of R , let us point out that, for instance, the operators Laplacian (a = N - 1, Q = 0),

p-Laplacian (1 < p < N): (a = N - 1, 0 = p - 2) and k-Hessian (1 < k < N/2): (a = N - k, 13 = k - 1) are included. In 1983, among other results, Brezis and Nirenberg in [61 have proved that the problem

1 -Au=.1u+JuI2'-2u,

l

u=0

in

12

on

c312,

Research partially supported by CNPq - Brazil and PRONEX-MCT.

226

C.O. Alves, P.C. Carriao, and O.H. Miyagaki

has a positive solution if 0 < A < Al and N > 4. Here Al is the first eigenvalue of (-0, Ha (a)), !l is a smooth bounded domain and 2* := NN2 is the Sobolev critical

exponent. This result on the existence of a positive solution was extended by Guedda and Veron [12] for the p-Laplacian operator and by Clement, De Figueiredo and Mitidieri [8] for a class of quasilinear elliptic problems in radial form treated in (P), more precisely, they improved the earlier result by imposing the following

restrictions on the constants

,3 >-1 and a-Q-1>0, 9 = 4(-y) _

(1.1)

(7 + 1)(0 + 2)

a-'3-1

'

(d+1)(f3+1)-(a-(3-1)(Q+2) <0.

b>ry, 6>a-0-2and ry+l>a-Q-1, q>/3+2 and

0
(1.6)

where RR

Al = A,(R) = inf{ J r"lu'10+2dr :

R

JroIuI2dr = 1},

0

denotes the first eigenvalue of the eigenvalue problem (see [8, page 145])

(PL)

f -(r"ju'I0u')' = Ar6Iu]°u in (0, R) l u(R) = u'(0) = 0

and we can choose an eigenfunction 01 > 0 in (0, R). We recall that in view of [8, Theorem 3.1 and Theorem 4.1] problem (P) has no positive solution for any A > Al, while problem (P) has no nontrivial solution

when A<0. The main aim of this paper is to show that the techniques developed in Cappozi, Fortunato and Palmieri [7] as well as in Gazzola and Ruf [11] can be modified for the class of problems studied in [8] in order to show the existence of nontrivial solution for problem (P) when A E (0,,\*) (see below for the definition of A`). It is well known that there are many difficulties when we work with such

operators, because the space of functions used is no longer a Hilbert space, as well as, the operators involved are not linear any longer. In order to overcome the difficulties mentioned above, we gave a new proof of an abstract theorem contained in [4](see theorem 2.4), showing that the same still holds in Banach spaces, such a result can be found in Section 4 (see Theorem 3.1). In order to state our main result, we define the following numbers

n a-,(3-1 n rj:=d-m((3+2)+1, m:= 13+1

Quasilinear Elliptic Problems Involving Critical Exponents

227

and RR

A' := inf{ J r°1u'12dr : 0

fr5IuI2dr = 1 and

0} > Al. 0

0

Theorem 1.1. Assume that (1.1), (1.2), (1.3), (1.4) and (1.5) hold. Then problem (P) has a nontrivial solution, provided that one of the conditions below holds (a) 77 < 0, for all A E (0, A'), (b)

17 = 0,

for all A E (0, A'), A

A1.

Remark 1.1. This result is related to Theorem 0.1 in [7] and Corollary 1 in [11]. Unlike the semilinear case where existence is proven for all A > 0, in the quasilinear

situation we are not able to solve for all A > A'. This is because we do not know the properties of the spectrum of our operator. But, we are working in order to get some information about this question.

2. Preliminary results We start recalling that the following properties were proved in [2] (see also [3]) for the case of the p-Laplacian operator.

Theorem 2.1. a) Al is isolated,

b) A' > Al Proof. a) Al is isolated Verification: Define

Z(u,v) :=< -Lu,

uR+2 _ vp+2 up+1

> - < -Lv,

where

u$+2 _ vR+2 vp+1

>,

R

Lw := (r°Iw'IFw')' and

< -Lw, z >:= f r°lw'1aw'z'dr. 0

Then

Z(u,v) > 0,

u,v > 0.

(2.1)

Next, we shall prove the following claim

Claim 2.1. If v is a eigenfunction associated to the eigenvalue A > 0, A 34 Al, then v changes sign, that is, of 34 0. In addition,

mes(SZ-) > (AC)_(f , C > 0, where 9- :_ {x E [0, r) : v(x) < 0}, some r < R. Verification: Let u and v be eigenfunctions with lull = Ilvil = 1, associated to the eigenvalue Al and A 54 A1, respectively.

C.O. Alves, P.C. Carriao, and O.H. Miyagaki

228

Suppose by contradiction that v does not change sign in [0, R). Thus

0 < Z(u, v) = (A, - A)(a, - -), which is impossible.

Since 5 > ry, taking v := v-, and by the Holder inequality we obtain I

IV-113+2 < ACIfl- I q-

+_, I IV-11;1+2

where C > 0. From this inequality we infer that IH- I > (AC) y-

+2

this finishes the proof of the claim.

Now, arguing by contradiction that there exists a sequence of eigenvalues A,, 0 A 1, which converges to A, . Let (un) be the corresponding sequence of eigenfunctions with IIu,,II = 1, dn. Then, we can assume that u -k u weakly in E, u u in L +2 and a.e. in (A , ),

[0, R).

But rR

< -Lun, a >= AJ r6I

A(u,,, v)

0

Thus, by the monotonicity of L, and keeping in mind that IA, ,l and I Iun I I are bounded we conclude that (un) is a Cauchy sequence in E. Hence u,, converges to an eigenfunction u associated to A1, which we can assume satisfies u > 0 in [0, R). From the Egorov theorem, we get a contradiction to Claim 2.1. This proves that A, is isolated.

b) A'>A,: Verification: By the definition we have Al < A*. Suppose by contradiction that At = A* := A. 'Take a minimizing sequence {un} C E to A,, that is

IIu,II'

A, IunI$+2,6 = 1

and

/r61hmundT

= 0.

So, passing to the subsequence if necessary, we have

u - u, in E, un.

u in L69+2 , a.e. in (0, R).

From this u = co,, for some c 0 0. On the other hand, since

un-.u in La, some rE(f3+2,q),

and

0

E L a«

Quasilinear Elliptic Problems Involving Critical Exponents

229

then R

0 = Jruadr

fr1udr = c, RR

0

0

which is a contradiction, proving the strict inequality desired and consequently the theorem. O We are looking for the solutions of (P), which are considered in the Banach

space E := XR of absolutely continuous functions u : (0, R)

R such that

u(R) = 0 and with the norm defined by 1/((+2)

/R

< cc.

Iull :=

(2.2)

0

It is well known that E =< 01 > ®E2, where RR

E2 :_ {u E E :

r6I0l I190,udr = 0).

J0

Also, let us denote by Ly(0, R) the Banach space of Lebesgue measurable functions u : (0, R) -* R such that R

Iul8,

:= IuILti := (f r7luledr)'/s < oo. 0

It is proved in [13] that, if -y + 1 > as°+2 - (/3+ 1)p}2 and /3+2 < q' < q(-y), then the embedding XR C Ly is continuous. It is also known that at q' = q('y) the embedding fails to be compact. As a consequence, since 6 + 1 > a - /3 - 1, we have continuous embeddings of XR in weighted Lebesgue spaces related to the exponents in problem (P): XR C L6+2(0, R)

and XR C L7(0, R).

Also the following embeddings hold

L7(0,R)CLa}2(0,R), if 6>ry, LP (0, R) C LT,(0, R),

-y +1>(2.3) if r < r

(2.4)

and we recall that (see [13]) (Lp(0, R))' = LP' 1_p,)(0, R), 1/p+ 1/p' = 1. Finally, we shall use the following number (see [8]) S := inf{IIuIIa+2 : u E E, Jul,,, = 1},

(2.5)

C.O. Alves, P.C. Carriao, and O.H. Miyagaki

230

the so-called the best constant of the Sobolev embedding E into L9 (0, R), which is independent of R and it is achieved when R = c by the functions

()

3'

8E G14

where

m-_ ti+Q+2-a a-(3-1 ' and

n

_ ry+Q+2-a

a-Q- 1

3+1

(0+1)(Q+2) T(74

s+t

Q+1 In the sequel, we will denote by S := Si

.+z

'.

.

3. Proof of Theorem 1.1 The proof is done by combining some of the arguments used in [6] as well as in [7] and [11] with a variational approach, more exactly, we shall apply a version of the Generalized Mountain Pass Theorem due to Ambrosetti and Rabinowitz (1] (see also Rabinowitz [14]) in Banach spaces, whose sketch of proof will be given in the next section and whose statement is the following

Theorem 3.1. Let E be a real Banach space and 1 E C'(E, l) be a functional satisfying the following conditions

(I1) I(u)=1(-u), 1(0)=0, foranyuEE (12) there exists d > 0 such that I satisfies the (PS), condition for all c E (0, d) (13) there are constants p, e > 0, e < d and a j-dimensional subspace Et C E with topologically complementary subspace E2 such that.

(i) 1(u) > 0 on (BP fl E2) \ {0} (ii) 1(u) > e on 3B,, fl E2 (14) there is a constant r > 0 and an in(> j) -dimensional subspace E3 C E such

that 1(u)
if e = 0, then I has at least m - j - 1 distinct pairs of nonzero critical points. Define a functional on XR by It

I(u) =

J

(0+2rnlu Ii3+2 _

Q+2ralul3+2 _ gr'lu]9)dr,

0

which is C' with derivative given by R

1'(U) I? =

f (r'lu'I 3u.'v' - Ar5 ul`3uv - r"1 lulq-2uv)dr. 0

Quasilinear Elliptic Problems Involving Critical Exponents

231

Using the embedding results we are going to show that I verifies the geometric conditions as well as other hypothesis given in the critical point theorem above. The following function will play an important role in our proofs. Define

uE(r) = O(r)luc(r), where 0 is cut-off function given by 1

0

in in

(0, Ro)

[2Ro, R),

with 0 O such that I(u) > e, IIuII = p, uEE2 Step 2. 1 satisfies the Palais-Smale condition (PS),, for all c E (0, (a++ - q )S), that is, for all c E (0, 0+2 )S), let {u. } be a sequence in E satisfying

-

a - c 1(u.)

and

1'(u.) - 0,

then {un} contains a subsequence convergent (strongly) in E. 1)S, where Step 3. sup{I(u) : u E W,} < (A+z -

Wf := {u E E : u:= u- + tu(, u- E < 41 >} Remark 3.1. Since we can choose e sufficiently small, actually we have a stronger inequality than that given in (ill), namely

sup{I(u): uEW,} <(a-1}-2 -9 1)S-e. Proof of Steps.

Step 1: Since u E E2, there are positive constants C1, C2, p and e < S such that ,3+1

2(1 -

1(u)

A )IIUIIA+2 _ C2IIUII°

> e>0, if IIuII=p, uEE2. Step 2: Firstly we will show that {un} is bounded in E. Combining the inequalities

c(1+IIunII)

I(un) -

a+21'(un)un =

+2

q)IunI9,.v

and

IlunllA+2 < C +aC(IuIq,-1) °_ +CIuIQ.7, C > 0,

we obtain

< C + AC(1 + IlunII) a' + C(1 + Ilunll), C > 0, then it follows that {un} is bounded in E. IIunJIA+2

C.O. Alves, P.C. Carriio, and O.H. M iyagaki

232

Hence, passing to a subsequence if necessary, we can assume u,, - u (weakly) in E. Arguing as in [6] (see also [10]), we infer that u is a weak solution of problem

(P).

Define w := u - U. Since I'(u,,)u,, =

and I'(u) = 0, using a result by Brezis and Lieb [5)

we have

+o(1) Suppose that IIu,,,II3+' -i P, then from this last inequality o,,(1) = IIw«IIF3+2 -

Iwn11,7 -

f as n - oo.

But

a+2I'(u)u-(1i+2

I(u)

I(u)

q)IuI°y.,?0,

thus passing to the limit in

I(u) we have


d-(13+2

y)2>(a+2-q)S>c-

which is a contradiction and we can conclude that i;' = 0, that is u,, -+ u (strongly)

in E. Step 3: We start by giving some estimates, which are obtained by arguing as in [6] and [8]. R

JrIuI

2d r

= S + Q($(3+2) ),

s (3+2)

:

=

+1

,

(3 . 1)

0

K

frIucIdr =

O(Es),

(3.2)

0

and R

f r6iu,I'+2 dr > Ce'

9+6+2-a

1

+t

C(E8(,i+2)+") ce8(%3+2) I log El

if rl < 0, if rf = 0.

3.3)

0

As in [6, 8], from (3.1) (3.2) and (3.3), if E is sufficiently small, we have l'3.}2

Iur I

- \Iu, 13+2.6 ,3+'2

-

S - aCEx03+2)+n + O(E'(4+2)) l S - ACER('r+2)I logEl + O(E«(d+2))

if q < 0

if 17=0.

(3.4)

Quasilinear Elliptic Problems Involving Critical Exponents

233

Now, since

su

I to

IIUII°+2 -

t > 0, u fixed}

,3+2

q

\lul0+2,a

IuIq.7

it suffices to prove that Claim 3.1. sup{IIuII1+2 - Alula+2,a : IuI972 = 1, u E WE} < S.

Cased > Al and q < 0: Let be u := u- + tuf, u- E < ¢1 >, t > 0, then t and Iu IA+2,6 are bounded. In fact, by the embedding of weighted spaces, we have lu-Iv+2.6

<-

CIulq,7+CItI°+2IucI4,..

Assuming that t is bounded and Jul,,, = 1, from (3.2) we conclude that u- is bounded in La+2(0, R). Next, we are going to prove that t is bounded. First of all, we shall prove the following inequality I Iu- + tuf Ig9.7

-

Iu-I

4.7 - I tuf I g4.7 l

< 1

Iu- Ig

4.7

Ctgf(a-;Z-)(q- 1)+7+1)(

+

)

C > 0.

Proof. It is standard to get Iu- + tu,1q,7 - Iu Iq,7 - Ituc Iq.7

< C1ltuclq-1,71u Ioo.7+C2ItucI1.7lu Iao.7 Since < 41 > has finite dimension and writing C2 := a.b we have +tuElq,7 - Iu Iq,7 - ItucI9.7

Iu

a1/9-1

<

C3tg-lE(a m)(9-1)+7+1)Iu IQ+2.d +

Iu-IQ

9/9 - 1

9.7

+

bgtgE(a--)(4-1)+7+1)4

9

Choosing a such that 4 4-1 °`/Q-` < 1/4, , we obtain lu +tuElq,7

- Iu Iq,7 - Itualq,7

< <

I0+2.6+ 4Iu Iq.7 C7tgf((8-m)(4-1)+7+1)(4/4-1) + 1Iu-Iq

2

+C4t9E((a-m)(q-1)+7+1)q

.7

This proves the inequality. Finally, from the inequality above (3.5) we get 1 = IUI4,7 >_ tgluElg4.

-C70f((5-m)(9-1)+7+1)(4/4-1)

If a is sufficiently small, from (3.2) we conclude that t is bounded. From an inequality in [15] it follows Ix + yI n - la + bl' < IxI + Iyll - IaI° - Ibin + CP(I xI p-' I yI + Ixllylp-1 + IaI°-' IbI +

IaIIbIp-1). (3.6)

C.O. Alves. P.C. Carriao, and O.H. Miyagaki

234

Take u E W with Jul q,., = 1. Using the inequality above (3.6), let us estimate the following IluIf3+2

<_

-

3+2

IuI3 +2,e

(al -'\)Iu

y+2 1,3+2.b +

Iltucll'i+2 - \Itu13+2.6

3+2 -j

Itu,1q+2

+C(t,u),

where C(t. U) C(Ilu-Its+llltu'll

=

IIIItu'II`3+' +

+ Ilu

Iu-l

3+2.aItuEI;3+2.a + Iu-13+2.6ltufl3+2,a)


Define

A(e.C,u) := (al -A)Iu I,3+2.e+Clu

A

I;3+2,SE((e_L)(9-1.j+7+])(7

71

then

A < 0 or A < Ce((+-

(3.7)

On the other hand, from (3.5) we infer that ltu,1,3+2 q,_ _< (1 +

3.8

Substituting (3.8) and (3.4) with q < 0 (the case :1 = 0 is similar) into I'y+2

I IUI

_ aI uI d+2,a

A+

I ItuEI IF3+2 _ \Itu I ;3+2,a 1tui13

;3+2

2

we have IIUII

+ <

13-«2

_ aluld+a.s

(S -

.1Ce"(.;+2)+n + O(EQ(3+2)))(1

+

Ce((A- m)(q-1)+-;+1)(

)WT)

S, for a sufficiently small.

0 The case A = \1 and , < 0: Let be ¢1 > 0 the first eigenvalue such that R

Jrii13+2dr = 1. 0

Denote by Pu the projection of u E E onto the space < 01 >, then R

Pu = (I r6I0iI30ludr)0l, 0

Quasilinear Elliptic Problems Involving Critical Exponents

235

and we will use the following notation i := u - Pu E E2. It is easy to see that IPU,Iv.7, IPuEIa+2.6 and IIPu,II are O(E(y-*)+7+1)

Then for i < 0, the equation (3.4) still holds replacing u, with u,, where

u, := of - Pu(. Let be u E W, with Jul,., = 1, where Wf := {u E E : u = u- + tu u- E < 461 >, t E R}.

Then since u = u- + Pu- + tuG, using the inequality (3.6), we have IIuIIp+2 - \I IUIO+2.a

<

allu=la+2.a + IIPu

II3+2

_ a1IPu-Ip+2.6

+CEc(-"-a)(9-1)+7+1)(),

- a1ltufla+2.5 + thus the proof follows arguing as in the non-resonant case. IItaEII'3+2

0

4. Proof of the abstract Theorem 3.1 In this section, we will prove the abstract Theorem 3.1, whose proof is essentially that given in [4, Theorem 2.41 in the Hilbert framework. Instead of using the general index theory on Hilbert space we combine the genus of a set A in a Banach space E due to Krasnoselskii (see [1]), denoted by -y(A), with the pseudo-index theory (see [41), given by

i'(A) = min y(h(A) n S) hE* where

S E E = {A closed in E : A is symmetric with respect to the origin} and

H'={

4) : E

E : 4) is an odd bounded homeomorphism mapping in E

such that 4i(u) = u if u V I `(e, d). We say that y(A) = k for an A E E, if k is the smallest integer such that

there exists an odd continuous map 4i : A - IRk \ {0}. (See [1] for more details and their properties.) In our proof we take S = S, n E2 C I-1 [e, oc). We claim that there exists B = E3 such that B C I-1(-oo,r) and

i' (B) = min y(B n S) > dim E3 - dim El.

hEH Verification: Notice that E3 n h(S) is compact, then

y(B n S) = y(E3 n h(S, n E2)) = y(E3 n h(S. n E2,6)) where E2,6 is a 6-neighborhood of E2.

C.O. Alves. P.C. Carriao, and O.H. Miyagaki

236

On the other hand, let. R := E/E2,6 and P a projection of E on El along E>. Then (cf. [1, Lemma 1.2(1)])

y(R) 5 y(P(R)). From (see [1, Lemma 1.2(7)]) if y(P(R)) > dim El follows that

n P(R)

0,

which is impossible. So

y(R) < dim El Now, choosing e > 0 such that S, n El C R, we have dim El = y(S, n Et) < y(R),

y(R) = dim El.

dim E:; < y(Ei n h(S,, n E2.p)) + y(E: n h(S,, n R))

y(E:, n h(S, n R)) < y(h(S n R)) = y(Sp n R) < y(R) = dim E, Therefore

i'(E:t) = min y(E:3nh(SPnE2,p)) > dim E,3 - dimEt hEH

k

This proves the claim. Define the following numbers

c;:= inf sup l(u), k=1,2... . , k AEEk uEA

where

Ek:={AEE:i'(A)>k}. Thus it is standard to prove that

I(0) =0 < e < cl <...
k>1,k+p:5ktheny(KK)>p+1>2. From these claims above we complete the proof of Theorem 3.1

(4.3)

Quasilinear Elliptic Problems Involving Critical Exponents

237

References [1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349-381 (1973). [2) Anane, A.: Simplicite et isolation de la premiere valeur propre du p-Laplacien avec poids. C.R. Acad Paris. t. 305 Serie I, 725-728 (1987). [3) Anane, A., Tsouli, N.: On the second eigenvalue of the p-Laplacian. In: Nonlinear PDE 343 (Benkirane and Gossez, eds.), pp. 1-9. Pitman Math. Research 1996. [4] Bartolo, P., Benci, V., Fortunato, D.: Abstract critical theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Analysis TMA 7, 981-1012 (1983). [5] Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88, 486-490 (1983). [6] Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437-477 (1983). [7) Capozzi, A., Fortunato, D., Palmieri, G.: An existence result for nonlinear elliptic problems involving critical Sobolev exponents. Ann. Inst. H. Poincare 2, 463-470 (1985).

[8] Clement, P., de Figueiredo, D.C., Mitidieri, E.: Quasilinear elliptic equations with critical exponents. Top. Meth. Nonl. Anal. 7, 133-170 (1996). [9] Clement, P., Mandsevich, R., Mitidieri, E.: Some existence and non-existence results for a homogeneous quasilinear problem. Asymptotic Analysis 17, 13-29 (1998). [10] de Figueiredo, D.G., Goncalves, J.V., Miyagaki, O.H.: On a class of quasilinear elliptic problems involving critical exponents. Comm. Contemporary Math 2, 47-59 (2000).

[11] Gazzola, F., Ruf, B.: Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Diff. Eqns. 2, 555-572 (1997). [12] Guedda, M., Veron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Analysis TMA. 13, 879-902 (1989). [13] Kufner, A., Opic, B.: Hardy-type inequalities. Pitman Res. Notes in Math. vol. 219, Longman Scientific and Technical 1990. [14] Rabinowitz, P.H.: Some minimax theorem and applications to nonlinear partial differential equations. In: Nonlinear Analysis (Cesari, Kannan and Weinberger, eds.), pp. 161-177. Academic Press 1978. [15] Stroock, D.W.: A concise introduction to the theory of integration. Birkhauser 1994.

238

C.U. Alves, P.C. Carriao, and O.H. Miyagaki

C. O. Alves

Universidade Federal da Paraba Departamento de Mateinatica 58100-907 Campina Grande-PB, Brazil E-mail address: [email protected] P.C. Carriao Universidade Federal de Minas Gerais Departamento de Matematica 31270-010 Belo Horizonte-MG, Brazil E-mail address: [email protected] O.H. Miyagaki Universidade Federal de ViCosa

Departamento de Matematica 36571-000 Viccosa-MG, Brazil

E-mail address: [email protected]

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 239--244 2003 Birkhauser Verlag Basel/Switzerland

Solutions of Semilinear Problems in Symmetric Planar Domains ODE Behavior and Uniqueness of Branches Filomena Pacella and P.N. Srikanth Abstract. We prove an existence and uniqueness theorem for an "Initial Value Problem" in the plane, related to the semilinear elliptic equation

-Au = f (u) in the case f is a Cl-convex function. This result is applied to show the uniqueness of a global bifurcation branch for the problem -Au = up +Xu in S2 u > 0 in S2

u=0

on

852,

where 11 is a symmetric bounded domain in Il22.

1. Introduction and statement of results Let 52 be a smooth bounded domain in R2, containing the origin, convex in the direction xi, i = 1,2 and symmetric with respect to the hyperplanes {xi = 01, i = 1,2. The aim of this note is to point out that existence of solutions of the problem

-Au= .Xf (u) inn

U>0 U=0

ind on 852,

where A > 0. correspond to an existence and uniqueness theorem for an "Initial value problem" associated with (1.1). More precisely, denoting by

SZt={tx: we have,

xES2}, t>0

Filomena Pacella and P.N. Srikanth

240

Theorem 1.1. For each d > 0 there exists a unique t = t(d) > 0 and a unique solution u = u(d) in the space C3(52t) fl CI(?It.) of the problem

-Au = f(u)

=d u > 0 u=0

in 11t

u(0)

(1.2) in 12t

on 8S2t,

where f is a C2 -convex function, with f (0) = 0 and f'(0) > 0 (note f (s) > 0 for s > 0 follows from our assumptions on f). Moreover t and u depend continuously on d.

Remark 1.1. The meaning of the theorem is the following. If we fix a symmetric domain in R2 and consider all the homothetic domains It (i.e., if we fix a shape of a symmetric domain in R2 as we usually do considering a radial problem in a ball) then for every d > 0 the "initial value problem"

-Du = f (u) u(0) = d has one and only one positive solution which goes to zero in a "finite time" t, i.e., which takes the value zero on the boundary of some homothetic domain 12f. This result is analogous to that of Proposition 2.35 of [3] for radial solutions in a ball. We believe that this could be a first step in extending the well-known "Shooting Method" to semilinear elliptic problems in general symmetric domains. The main ingredient in the proof is a result from [1] which we shall recall. Definition 1.1. A positive function u E C' (N) will be called symmetric and mono-

tone if it is even in the variables xi(i = 1,2) and - > 0 in

S2i={xESl:x,<0}, i=1,2. Note that if u is a solution of (1.1) then by results in [2], u is symmetric and monotone.

Now we state without proof the following result from [1].

Theorem 1.2. Assume that u1 and u2 E C' (Q) nC'(S2) are symmetric and monotone functions satisfying

-Au=f(u)

in S2

(1.3)

with f convex. If u1(0) = u2(0) and u1 < u2 on 812 then u1 and u2 coincide. Note that for symmetric and monotone functions u, the value u(0) is precisely

maxu; hence Theorem 1.2 asserts that there cannot be two solutions of (1.3) 11

which have the same maximum as long as they are "comparable" on the boundary (comparable is used in the sense ul < u2 or u2 < ul on the boundary). In particular there cannot be two solutions of any Dirichlet problem associated to (1.3) having the same maximum.

Solutions of Semilinear Problems in Symmetric Planar Domains

241

Remark 1.2. If we now fix another domain A with the same property of Sl (i.e., symmetric and convex in the x1-directions) but with a different shape, then applying Theorem 1.1 to A and S2 with the same value d > 0 we get two solutions u and v, solving respectively (1.2) and

-Av = f (v) v

v(0)

>0 =d

v=0

in At. in At,

on 8At,

for some t'. Then by Theorem 1.2 we deduce that neither Sgt C At, nor At, C 12t,

otherwise u=-vandS2=A. Finally we apply Theorem 1.1 to show the uniqueness of a global positive solution branch for the following semilinear problem

-Du = up + Au

U>0

u=0

in S2

in11 on 8S2

where 0 is a symmetric domain as earlier. Here p > 1 and A > 0. Let us denote by Al the first eigenvalue of the Laplace operator in Ho (S2). The theorem we prove in the context of (1.4) is the following. Theorem 1.3. All solutions of (1.4), lie on the global bifurcation branch emanating from (,\1, 0) given by the Rabinowitz Theorem 14). Moreover this branch is a "simple continuous curve joining (,\,, 0) with (0, uo), uo being the unique positive solution

of (1.4) for J\ = 0 (see [1]). Remark I.S. As will be clear from the proof, similar results can be deduced for more general nonlinearities; the restriction to (1.4) is essentially due to the attention this particular nonlinearity has received over the years. We conclude by mentioning that Theorem 1.3, throws some light on the question of the uniqueness of the solution of (1.4). In fact we get the following result ([see [51) for the case of radial solutions in a ball). Corollary 1.1. If, for any A E (0, A1), the solution of (1.4) with Morse Index one is non-degenerate, then (1.4) has unique solution for any \ E (0, A1).

Remark 1.4. The final statement of Theorem 1.3 asserts that there are no "secondary bifurcation branches" and any degenerate solution can produce, at most, a turning point. This means in the figures below of the bifurcation branches only Fig. 2 is possible but not Fig. 1.

242

Filomena Pacella and P.N. Srikanth

X.

Fig 2

Fig 1

2. Proofs In this section we prove the results previously stated. Proof of Theorem 1.1. Under the assumptions on f , from the bifurcation result of Rabinowitz ([4), Theorem 2.12) we have the existence of a continuum r of solutions (A, ua) of the problem (1.1) which bifurcates from ( e1, 0) and goes to infinity

in the space X = IR x C',"(St). In fact under the assumptions we have on f, u.\(0) -, oo. with A bounded. Note that ua(0) 0 as A \I/f'(0). Hence along the branch of the Rabinowitz Theorem, we have that the values u,\(O) cover the entire interval (0,00). Hence given any d > 0 there exists (A, u,\) E t, A > 0 such that ua(0) = d and ua satisfies (1.1). Defining w(x) = u,\ (7.), we have that w is a positive solution of

-Aw = w(0)

f (w)

in S2\/-

= d

(2.1)

w=0

on i%1

Therefore setting t = f we have proved the existence statement of Theorem 1.1. Now, if for some t' > t there exists a solution w' of the same problem in Stt, with w'(0) = d then the functions w and w' would solve in fl the same equation and w' > w on t fl, . Thus by Theorem 1.2, w' = w, hence t = t'. The same argument applies if t' < t, proving the uniqueness of the number t for which (2.1) has a positive solution. It is now clear that Theorem 1.1 follows from the arguments

0

above.

Proof of Theorem 1.3. Again from the result of Rabinowitz and a priori bounds,

we know that there exists a continuous branch I' of solutions (A,ua) of (1.4), We shall restrict our bifurcating from (A1, 0) which exists for all A E attention only to the interval (0, Al ). Writing, h() x =

x u.%

1

Solutions of Semilinear Problems in Symmetric Planar Domains h satisfies,

-Ah = hP + h

h>0 h=0

in in

243

S2 f

fl f

(2.2)

on 8S2 f

and clearly h(O) 0 as A Al, while h(0) - oo as A 0, so that as (A, u,\) moves along F, the corresponding h(0) takes all values in the interval (0, oo). To prove the theorem we argue by contradiction and assume that for A E (0, A1) there exists a solution v of (1.4) such that (a, v) does not belong to the above branch F. Then, by scaling this function as we did earlier, we get a function k(x), solving -Au = uP + u in Ht u(0) = d > 0

U > 0

u=0

(2.3) in tl1

on i

l1.

4,

k(0) = v(0) But by Theorem 1.1, we know that in correspondence with d there exists one and only one number t such that (2.3) has only one solution in Qt. Since we showed before, that, for every d > 0 a solution of (2.3) already exists, obtained starting from a solution belonging to the branch I' (namely one of the functions h) we get Here

a contradiction. Hence all positive solutions of (1.4) lie on r. If F' C r is any simple continuous curve joining (A1,0) with (0, uo), we could respect for r' the same argument we used for r, implying that there are no solutions of (1.4) outside of F'. Thus r itself must be a simple continuous curve. Proof of Corollary 1.1. By the previous theorem, we know that for A > 0 all positive solutions of (1.4) lie on a simple continuous branch. Moreover solutions of Morse

Index one exist for all A > 0(0 < A < A1) and they are the only solutions near Al or near zero ([1]). Hence if nondegeneracy of these solutions is established then this is equivalent to proving that all solutions are nondegenerate for all A E (0, A1), which gives the uniqueness.

References [1] L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincare 16 (5) (1999).

[2] B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phis., 68 (1979), 209-243. [3] W.M. Ni and R.D. Nusshaum, Uniqueness and nonuniqueness for positive radial solutions of Du + f (u, z) = 0, Comm. Pure. Appl. Math. 38 (1985), 67-108. [4] P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journ. Funct. Anal. 7 (1971), 487-513. [5] P.N. Srikanth, Uniqueness of Solutions of nonlinear Dirichlet problems, Daft.. Int. Eq., 6 (1993), 663-670.

244

Filomena Pacella and P.N. Srikanth

Filomena Pacella Dipartimento di Matematica Universit'a di Roma, "La Sapienza" 1-00185 Roma, Italy

E-mail address: pacella®mat. uniromal. it P.N. Srikanth

TIFR Centre P.O.Box 1234 IISc Campus Bangalore 560012, India

E-mail address: srikanth(math.tifrbng.res. in

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 245-256 © 2003 Birkhi user Verlag Basel/Switzerland

Solutions of an Allen-Calm Model Equation Paul H. Rabinowitz and Ed Stredulinsky

1. Introduction The goal of this note is to describe some ongoing work of the authors on an AllenCahn type model equation. The equation serves as a model for phase transitions in a binary metallic alloy. Consider the partial differential equation

-Au + Fu(x, y, u) = 0, (x, y) E IR2. The function F is a double well potential with a pair of global minima, e.g., F(x, y, u) = a(x, y)u2(1 - u)2 with a(x, y) positive. The minima u = 0 and u = 1 of F are called pure states. Solutions of (PDE) with 0 < u(x, y) < 1 are called mixed states. It will be assumed here that F satisfies (Fl) F is C2 in its arguments and is 1-periodic in x and y, (F2) F(x, y, 0) = 0 = F(x, y,1) and for 0 < z < 1, F(x, y, z) > 0, (F3) F(x, y, z) > 0 for all x, y, z. Thus u =_ 0, u = 1 are solutions of (PDE). The main question of interest to us is to classify the mixed states which are asymptotic to the pure states. It will be shown that there are a large number of such states that can be obtained using elementary minimization arguments. Some earlier research in this direction was carried out by Alessio, Jeanjean, and Montecchiari [1]-[2]. Indeed their paper [1] was a primary stimulus for our study (PDE)

of this question. The paper [1] in turn was motivated by work of Alama, Bronsard,

and Gui [3]. The contributions of [1]-[2] will be indicated in §2. It also turns out that our research is related to the study of minimal laminations of a torus as studied by Moser [4] and Bangert [5]. The connections to [4] and [5] will be described at the end of §2. We divide the solutions of (PDE) into two classes. First there are relatively simple solutions that we call basic solutions. They will be described in §2 and the ideas behind their existence proofs will be sketched. Full details will appear in [6]. More complex solutions that are `near' formal concatenations of basic solutions will be treated in §3. In analogy with similar situations in the theory of dynamical systems, they will be referred to as multibump solutions. As mentioned above, both sets of solutions are obtained via elementary minimization arguments. This research was sponsored in part by the National Science Foundation under Grant #MCS8110556. Reproduction in whole or in part is permitted for any purpose of the U.S. government.

246

Paul H. Rabinowitz and Ed Stredulinsky

2. The basic solutions In this section the existence of two kinds of basic solutions of (PDE) will be sketched. The simpler family, consists of solutions, v, that are heteroclinic between

the pure states, say 0 to 1, in one independent variable and are periodic in the second independent variable. The second family of solutions is heteroclinic in both variables, say from 0 to 1 in x and from v to a phase shift of v in y. The simpler family will be treated first. Let

L(u) = 2

IVu12

+ F(x, y, u),

the Lagrangian associated with (PDE), and set 1

I(u) = jj L(u)dxdy

for u E 1'(O.1) where

1(0.1) = {u E 14,1, lie (R2,R) I u is 1-periodic in y,

u2dxdy -+ 0 as k - -oo and

11(l

Jk

- u)2dxdy -, O as k --, -oo}.

(2.2)

Finally set c(O, 1) =

inf

uer(0,1)

I(u).

(2.3)

Then the existence of solutions of (PDE) that are 1-periodic in y and heteroclinic in x from 0 to 1 follows from

Theorem 2.4. If F satisfies (F1) -(F3), there is a v E r (o, 1) such that I(v) _ c(0,1). Moreover v is a classical solution of (PDE) and IlVlIC2([k,k+1]x]0,1],R)

liv - II1C2flk,k+1]x1o,1].R)

-+ -+

0 as k -+ -co, 0 as k -+ oo.

Sketch of Proof. Note that if u E 1(0, 1), so is Tiu(x, y) ° u(x - .1, y) and

I(rju) = I(u).

(2.5)

Let (uk) C 1'(0,1) be a minimizing sequence for (2.3). Therefore there is an M > 0 such that

I(uk) < M. By the properties of F, it can be assumed that 0 < uk < 1 a.e.

(2.6)

Solutions of an Allen-Cahn Model Equation

247

Furthermore, using (2.5), it can be assumed that (Uk) satisfies the normalization i+1

1

r

1

ukdxdy <

2'

o

jEZ,j<0

(2.7)

1

1

fftLkdXdY >

2

The form of I implies (Uk) is bounded in Wig and therefore along a subsequence converges weakly in WW1, and strongly in L to v with 0 < v < 1 a.e. satisfying (2.8) and rj+1

J0 Hence v

1

v d xdy

<2

jE Z ,j< 0

,

(2.9)

.

fo

0, 1. Moreover weak lower semicontinuity arguments show 1(v) < M

(2.10)

which combined with (2.9) implies that (2.1) holds. To verify (2.2) requires a comparison argument as in [1] or [7]. Hence v E 1'(0,1) and it readily follows that 1(v) = c(0,1). This minimization characterization of v and standard elliptic arguments then imply that v is a C2 solution of (PDE) and the C2 convergence of v to 0, 1 as x -+ ±oo respectively.

Remark 2.11. Similar arguments show that there is a solution v of (PDE) in r(1, 0)

= {u E WW1, (1R2, R) I u is 1-periodic in y, k+1

Jk

1

f (1-u)2dxdy-0 ask --oo,and 0

u2dxdy

0 ask

oo}.

0

Likewise there are solutions of (PDE) which are 1-periodic in x and heteroclinic from 0 to l or 1 to 0 in y.

Remark 2.12. Let 0 E (0, 2ir) be such that tan 0 E Q. Then F is periodic in the directions obtained by rotating the x and y axes by 0 radians. Hence the argument of Theorem 2.4 produces solutions of (PDE) heteroclinic from 0 to 1 in the 0 direction and periodic in the orthogonal direction. Before describing the next family of solutions of (PDE), some preliminaries are required. Set M(0,1) = {v E 1'(0,1) 1 1(v) = c(0,1)).

Note that v E M(0,1) implies rev E M(0,1) for all j E Z. The next result is based on an argument of Moser [4] in a related setting. Proposition 2.13. M (0, 1) is an ordered set, i.e., v, w E M (0, 1) and v 0 w implies v(x, y) < w(x, y) or v(x, y) > w(x, y) for all (x, y) E 1R2.

Paul H. Rabinowitz and Ed Stredulinsky

248

Proof. Suppose not. Then cp = max(v, w) and 0 = min(v, w) belong to F(0,1). Therefore

I

1(0) > 2c(0,1).

(2.14)

But (2.15) I(p) + I(t[i) = I(v) + I(w) = 2c(0,1). Consequently I(;p) = I(ii) = c(0,1) and by Theorem 2.4, cp and W are solutions of (PDE). Since co coincides with v and w on an open set, a unique continuation argument or more simply use of the maximum principle shows this is impossible. Having obtained heteroclinic solutions of (PDE) that are 1-periodic in y,

and noting that F is j-periodic in y for any j E N, we can ask whether there are solutions of (PDE) that are j-periodic in y and heteroclinic from 0 to 1 in x. Towards that end, set

Ij(u) = jf L(u)dxdy, I'j(0,1)

(u E Wig (R2,R) I u is j periodic in y, rk+1 I.j

u2dxdy

Jx

0 as k -. -co, and

u

k+1

L

(1 - u)2dxdy

as k -s oo},

0

and

cj =

inf

uE1"(0,1)

I AU).

(2.16)

The argument of Theorem 2.4 shows there is a vj E Fj(0,1) minimizing the vari-

ational problem (2.16) and vj is a classical solution of (PDE). Of course any v E M (0, 1) lies in ri (0, 1) and is a candidate for (2.16). In fact:

Proposition 2.17. cj = jc(0,1) and if

Mj(0,1)={uEr'j(0,1)1Ij(u)=cj}, then Mj(0,1) = M(0,1). Prof. Let u E M j (0, 1). Then u(x, y + 1) E M j (0, 1). By the argument of Proposition 2.13, M j (0, 1) is an ordered set. If u(x, y) = u(x, y + 1) for all u E M j(0,1), the proposition is proved. Otherwise u(x, y) # u(x, y + 1) for some u E Mj(0,1). Hence either (i) u(x, y) < u(x, y + 1), or (ii) u(x, y) > u(x, y + 1). The argument is the same in either case so suppose that (i) holds. Then u(x, ,y) < u(x, y + 1) <

. < u(x, y + j) = u(x, y),

(2.18)

a contradiction.

Thus there are not solutions of (PDE) heteroclinic in x from 0 to 1 and j periodic in y except for j = I. Remark 2.19. By Proposition 2.17, if u E r'3(0,1), Ij(u) > jc(0,1). This fact will be required later.

Solutions of an Allen-Cahn Model Equation

249

To get the next class of basic solutions of (PDE), assume that (*) There is an adjacent pair v < w E M(0,1), i.e. v, w E M(0,1), v < w, and no other members of M(0,1) lie between v and to. If (*) fails, there is a continuum of minimizers of I joining v and r_ I v. This degenerate situation certainly can occur. E.g., if F is independent of x, v E X1(0,1) implies rev E M(0,1) for all 9 E R. However if (*) fails, one can always make a

small perturbation of F so that (*) is valid for the perturbed potential. Thus (*) holds generically. Assuming (*), we seek a solution of (PDE) that is heteroclinic from 0 to 1 in x and from v to w in y. The existence of related doubly heteroclinic solutions was the major contribution of Alessio, Jeanjean, and Montecchiari in [1]. They considered

the case of F(x, y, u) = a(x)G(u) where a(x) is positive, periodic in x and, e.g., smooth. The function G has multiple nondegenerate global minima at u = ai, 1 < i < n. For the simplest case of a, = 0, a2 = 1, under a stronger nondegeneracy condition than (*), they obtain a solution of (PDE) that is heteroclinic from 0 to 1 in x and from v to rjv in y for some j > 0 and some j < 0. Subsequently in [2], they extended the results of [1] to allow a also to depend periodically on y with the further requirement that a is even in y. The significance of this evenness condition will be discussed in Remark 2.24. A minimization argument will be employed to find the doubly heteroclinic solutions. As the class of admissible functions, set aju = u(x, y - j) and

I'(v,w)={uE Wo, Iv
-oo and to 1 as x - oo.

4r(u) = JR2 L( u)dxdy.

However it is not difficult to see that if u E F(v, w), there exists a constant ry > 0 and independent of u such that on each strip Qj = R x [j, j + 1], 1 L(u)dxdy > ry. n;

Hence ' (u) = oo for all u E r(v, w). Thus more care is required and a `renormalized' functional on f (v, w) will be introduced. For j E Z, set

aj(u) =

Jn;

L(u)dxdy - c(0,1).

For m :5 n E Z, set n

Jm.n(u) _ L aj(u)

M

Paul H. Rabinowitz and Ed Stredulinsky

250

and finally define the renormalized functional J via

J(u) = lim J,n.o(u) + lien J1.n(u) n- 00

712-+-04

Thus J(u) is certainly finite for any u which for some j- < j+ equals v in Sl; for j < j_ and equals w in Sl; for j > j+. The next proposition contains the properties of J that will be needed for

what follows.

Proposition 2.20.

1° There is a constant K > 0 such that for all u E f (v, w) and m < n,

-K < J,..,, (u) < J(u) + 2K.

(2.21)

2° If u E f (v, w) and J(u) < oo, then

J(u) = lim J,,,.,,(u) and

lim

rn-+-oo

Ilu-1111u'l.2(s:.,, )

lim Ilu-w111I.2(sa .' no° )

= 0, = 0.

(2.22) (2.23)

The proofs of 1° 2° are lengthy and can be found in [6]. Work of Bosetto and Serra [8] provided some important ideas. Remark 2.19 plays a key role in the proof of 1° and 2° involves several weak lower semicontinuity arguments. Observe that whenever u E I'(v, w) and J(u) < oo, (2.22)- (2.23) imply that u is asymptotic to

v and w as y -' ±00. Remark 2.24. If F is even in y, it is straightforward to show that aj (u) > 0 for all j E Z. Hence J(u) > 0 and this permits working in a broader class than r(v, w) and a simpler treatment of the doubly heteroclinic case. With the aid of the above preliminaries, define c(v, w) =

inf uEI'(i'.w)

J(u).

(2.25)

Then we have

Theorem 2.26. Let F satisfy (FI) (FI) and let (*) hold. Then there is a U E r(v, w) such that J(U) = c(v, w). Moreover U is a classical solution of (PDE). Sketch of proof. Let (Uk) be a minimizing sequence for (2.25). As in the proof of Theorem 2.4, (Uk) must be normalized. Note that if u E I'(v, w), for all j E Z,

o,u E I'(v, w) and J(a,u) = J(u).

(2.27)

Set rI

A(u) =

r1

JJ o

u

udxdy.

(2.28)

Solutions of an Allen-Cahn Model Equation

251

By Proposition 2.13, p is a strictly monotone function of u on .M(0,1), i.e., < b implies p(cp) < p(V,). Using this fact, (2.27), and the asymptotic behavior in y of u E f (v, w), it can be assumed that Uk satisfies

J P(a-muk) <_ pt E (P(v) +P(w))I2, m < 0

l P(a-muk) > p',

m > 0.

(2.29)

With this normalization, by (2.21), (Uk) is bounded in Wio?(R2,R) and therefore along a subsequence converges weakly in WW1, , strongly in LiC, and pointwise a.e.

to U with v < U < a-1U < w a.e. Moreover U satisfies (2.29) so U 0 v, U ; w. Therefore U E 17(v, w).

By (2.21) and (2.25), J,,,,,, (U) < F(v, w) + 2K

(2.30)

so letting m -* -oo, n -+ oo shows J(U) < oo. Hence U satisfies (2.22)-(2.23). It remains to show: (A) U minimizes J on f (v, w) and (B) U is a classical solution of (PDE). Again we defer to [6] for complete proofs of these facts. The proof of (A) involves some of the arguments used in the proof of 2° of Proposition 2.20 as well as Remark 2.19. The proof of (B) is somewhat nonstandard. Once one has a minimizer, e.g., u, of a variational problem of `elliptic' type, say V(u) = minimum E c, there is a standard approach (as in the proof of Theorem 2.4) to show that u is a classical solution of the associated partial differential equation. Namely let cp be a smooth function possessing compact support and let b E R. Then V(u + dip) > V(u) and generally this implies V'(u)cp = 0

(2.31)

where (2.31) is a weak form of the partial differential equation. Thus u is a weak solution of the equation and then elliptic regularity results show it is a classical solution. Unfortunately this approach does not work here because the minimization

problem (2.25) contains the-global constraints v < u < a_lu < w. Although U E I'(v, w), in general u = U + &p will not satisfy these constraints. Therefore a different argument is needed. It is based on the fact that in addition to its global characterization via (2.25), U also possesses a local minimization property. To state it,

4r(u) = Proposition 2.32. For any r E (0,

f

L(u)dxdy. ,-(z)

and z E R2, if

2) Z(Br(Z)) E {u E Wloc (IR2, R) ( u

= U in Bi (z) \ Br(z)},

then

c(Br(z)) E

_inf

uEZ(B,(z))

4?r(u) = 4?r(U).

252

Paul H. Rabinowitz and Ed Stredulinsky

The proof of the Proposition employs ideas from Proposition 2.13, the maximum principle, (2.35), and comparison arguments. Given this local minimization characterization of U, the standard argument mentioned above implies U E C2(Br(z)) (and even U E C2."(B,.(z)), i.e., U has second derivatives which are Holder continuous with exponent a for any a E (0,1)) and U is a solution of (PDE) in Br(z) for all such r and z E R2.

Remark 2.33. If I'(w, v) is defined in the natural fashion, under the hypotheses of Theorem 2.26, there is a V E P(w, v) such that c(w,v) =

inf

J(u)

uE!'(te. P)

is achieved by V, a solution of (PDE) heteroclinic in y from w to v.

Remark 2.34. There is an analogue of Proposition 2.13 in the current setting. Let

AV, W) = {u E r(v, w) I J(u) = c(v, w)}.

Then Jct(v, w) (and similarly M (w, v)) is an ordered set. The proof is related in spirit to the earlier one but is more technical since we are dealing with the renormalized functional. To conclude this section, some close connections between the results presented here and work of Moser [4) and Bangert [5] will be pointed out. Moser considered a class of functionals of the form

(u) = fit" F(x, u, Du)dx.

(2.35)

Here Du denotes first derivative terms. The function F(x, z, p) was assumed to be smooth and 1-periodic in the components of x = (x1,...,x") and in z. As a function of p E R", F was convex and satisfied various upper and lower bounds that are usually imposed in studying the regularity of weak solutions of elliptic variational problems. Moser was interested in quantitative and qualitative properties of the solutions of the Euler equations (EE)

F. (x, u, Du) - E(x, u, Du) = 0

corresponding to (2.35) that are minimal in the sense of Giaquinta and Guisti [91. Such solutions satisfy O(u + X) > $(u)

for all X E W l.2(R", R) having compact support. In the course of his work, Moser showed if in (2.35), R" is replaced by the n-torus, T", there are minimizers of 4+

which are 1-periodic in x1, ... , x and are minimal solutions of (EE). Moreover M, the set of such solutions, is an ordered set. Bangert further studied the existence of minimal heteroclinic solutions of (EE). Among other things, he showed that if tp and Vi are adjacent members of M, there is a minimal solution of (EE) heteroclinic in x 1 from cp to ili and 1-periodic

Solutions of an Allen-Cahn Model Equation

253

. , x,,. Furthermore this set of such minimal heteroclinics is ordered and if f and g are an adjacent pair of such solutions, there is a minimal solution of (EE) heteroclinic in x1 from p to V) and in x2 from f to g, and 1-periodic in x3, ... , x,,. Specializing to the case of n = 2, observe that gyp,,o correspond to our 0,1 and f, g to our v, w of M (0, 1). While our Lagrangian L(u) = I Vu]2 + F(x, y, u) does not z by a simple device, it can initially satisfy the periodicity in u required for F in [5], be made to do so. Namely extend F(x, y, z) evenly in z from [0,1] to [-1,1] and then further extend it to be 2-periodic in z. Then with the aid of the maximum principle, parts of the existence assertions of this section follow from Bangert's work. However the clever existence arguments given in [5] are not variational in nature. The variational characterization of the basic heteroclinic solutions given in Theorems 2.4 and 2.26 seem to be essential for the construction of the more complex multibump solutions that will be studied in §3. Therefore we cannot take direct advantage of the results of [4]-[5]. in X 2 ,--

3. Multibump solutions In this section it will be indicated how the basic heteroclinic solutions of §2 can be used to construct more complex solutions of (PDE). Beginning with v as given by Theorem 2.4 and v by Remark 2.11, suppose for simplicity that M(0,1) _ {r_jv I j E 7G} and

M(1,0)_{r_;vIjEZ}. Then one can seek solutions of (PDE) near the functions obtained by formally gluing r_jv and 7-0- Such solutions are spatially homoclinic to 0 in x. Constrained minimization arguments will be used to find such so-called 2-bump solutions of (PDE). These arguments are analogues of ones that have been used in the variational approach to chaos in the theory of dynamical systems by Mather [10] and others, e.g. [7]. Of course the technicalities in the PDE setting are more onerous.

To describe how to obtain 2-bump solutions, let m E Z4 and e E N with met - e > m2i_ 1 + e, i = 1, 2 and m3 > m2. Choose pi E (0, e), 1 < i < 4 such that for all cp E M(0, 1), rm,+1 pl

Jm mz

P2

1m2 -t

1

cpdxdy; fo 1

f (1 -
(3.1)

Paul H. Rabinowitz and Ed Stredulinsky

254

and similarly for all yb E M(1, 0), my+f

Ps

1

J0

1. 3 M.1

P1

(1 - ij')dxdy;

(3.2)

1

( '+bdxdy. "t.3 -f 0

54

Let Y,,, denote the set of u E Wig (R2, R) which are 1-periodic in y and satisfy the constraints (1)

juldxdy < P1

11-ujdxdy< (iii)

P2

11-ufdxdy 5 rr1.1

P:1

U 1

(iv)

/ juldxdy < p,4.

Finally define

b,,, = uin I (u). Then we have

Theorem 3.5. For pr E (0, .1] satisfying (3.1)-(3.2) and t, m,+i - mi sufficiently large, there is a U,,, E Y,, such that 1(U m) = bm Moreover Um is a classical solution of (PDE) homoclinic to 0 in x. .

Sketch of proof. (A detailed proof of Theorem 3.5 can be found in [6].) It is not difficult to use arguments as in the proof of Theorem 2.4 to show that even without the size restrictions on t and m;+1 - mi, there is a U,,, E such that I(um) = b,,,

and0
1j+1

1

fo IUmI2dxdy - 0 as I j I oc follows from the fact that t is appropriately large and some elementary estimates using the form of I. Furthermore by the usual argument mentioned in the proof of Theorem 2.26, aside from the constraint intervals of (3.3): [mi, mi +t], [m2 - t, m2], etc., Um is a solution of (PDE). The same argument shows Um is a solution of (PDE) in a constraint interval provided that it is interior to the constraint set, i.e., it satisfies the constraint with a strict inequality. That this is the case follows from comparison arguments when the requirements that the m,+. i - m, are large is satisfied.

Solutions of an Allen-Cahn Model Equation

255

Remark 3.6. Theorem 3.5 provides an infinitude of 2-bump solutions of (PDE). Similarly there are infinitely many `k-bump' solutions of (PDE) which are heteroclinic in x from 0 to 1 (or 1 to 0) or homoclinic to 0 (or to 1) depending on whether k is odd or even. The proof of this fact requires working with m E 7L2k where again

e and m;+1 - m; are large. For the existence argument, this size restriction does not depend on k. Therefore since 0 < Urn < 1 and (PDE) provides uniform (in k) C ° bounds on this collection of solutions, one can let k oo and get `oo-bump' solutions of (PDE) as in related dynamical system settings [7]. Having found multibump in x solutions of (PDE) corresponding to the set of basic solutions of Theorem 2.4 and Remark 2.11, now one can ask about multibump solutions in y corresponding to those obtained in Theorem 2.26. There is a greater

variety of possibilities for multibump solutions in y than in x. This is due to the fact that in x, there are only two limiting states: 0 and 1 while in y there is all of M(0,1). Continuing to restrict ourselves to the nicest case of M(0,1) _ {r_jv I j E 7L}, one can seek four types of 2-bump solutions which are asymptotic to v as y -- -oo. Namely there are (A) a pair of possible homoclinics from v to v which are close resp. to r_ 1 v and to rl v for a large intermediate range of y's as well as (B) a pair of possible heteroclinic solutions, one from v to r_2v (and near r_lv for an intermediate range of y's) and one from v to rev (and near r1v for an intermediate range of y's). To obtain any of these solutions, one needs hypotheses like (*), i.e., there are appropriate gaps in M(v, w), M(w, v). Solutions in the spirit of type (A) as well as related k-bump solutions were obtained by Alessio, Jeanjean, and Montecchiari in [2] for F as in Remark 2.24 and in particular a even in y. More general solutions of type (A) as well as heteroclinics of type (B) were obtained in [6], also under the assumption that F is even in y. Without this assumption, the existence of heteroclinics of type (B) which were also monotone in the sense that o_,U > U (or likewise a1U > U) was proved in [6]. Quite recently, we have succeeded in eliminating the evenness condition for multibumps and will treat this case in a future paper. One can also try to find multibump solutions in x near concatenation of U of Theorem 2.26 and its analogue V_ of Remark 2.33. Likewise one can seek combined multibumps in x and y for U and its analogues. These questions also will be explored in a future paper. One of the main difficulties is the construction of appropriately renormalized functionals to treat these problems.

Paul H. Rabinowitz and Ed Stredulinsky

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References [1] Alessio, F., L. Jeanjean, and P. Montecchiari, Stationary layered solutions for a class of nonautonomous Allen-Cahn equations, Calc. Var. Partial Differential Equation, 11 (2000), 177-202. [2] Allessio, F., L. Jeanjean, and P. Montecchiari, Existence of infinitely many stationary layered solutions in R2 for a class of periodic Allen-Cahn equations. Preprint. [3] Alama, S.; Bronsard, L. and Gui, C. Stationary layered solutions in R2 for an AllenCahn system with multiple well potential, Calc. Var. Partial Diff. Eq., 5 (1997), 359390.

[4] Moser, J., Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincar6 Anal. NonLineaire, 3 (1986), 229-272.

[5] Bangert, V., On minimal laminations of the torus, Ann. Inst. H. Poincare Anal. NonLineaire, 6 (1989), 95-138. [6] Rabinowitz, P.H. and Stredulinsky, E. Mixed states for an Allen-Cahn type equation, Comm. Pure Appl. Math., to appear. [7] Rabinowitz, P.H., Heteroclinics for a reversible Hamiltonian system, Ergodic Theory and Dynamical Systems, 14 (1994), 817-829. [8] Bosetto, E. and E. Serra, A variational approach to chaotic dynamics in periodically forced nonlinear oscillators, Ann. Inst. H. Poincar6 Anal. NonLin(.aire, 17 (2000), 673-709.

[9] Giaquinta, M. and E. Guisti, On the regularity of minima of variational integrals, Acta. Math., 148 (1982), 79-107.

[10] Mather, IN., Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43, (1993), 1349-1386.

Paul H. Rabinowitz Department of Mathematics University of Wisconsin Madison Madison, WI 53706, USA Ed Stredulinskv Department of Mathematics University of Wisconsin-Richland Richland Center, WI 53581, USA E-mail address: [email protected]. edu

Progress in Nonlinear Differential Equations and Their Applications, Vol. 54, 257-267 © 2003 Birkhauser Verlag Basel/Switzerland

Some Equations of Non-geometrical Optics Giorgio Talenti 1. Suppose an isotropic, non-conducting, non-dissipative medium and a monochromatic electromagnetic field interact in the absence of electric charges. Let n and v denote the refractive index and the wave number, respectively. Here n is a scalar real-valued field, whose reciprocal is proportional to the relevant velocity of propagation through the medium, and v is a large positive parameter, whose reciprocal is proportional to the length of waves involved. The following Helmholtz equation

AU+v2n2U=0

(1)

is an archetype of those partial differential equations that ensue from the Maxwell

system and model the affair mathematically. A distinctive feature of (1) is its stiffness - the order of magnitude of v is significantly greater than that of the other coefficients involved.

An expansion, which represents solutions to (1) asymptotically as v -* +oo, originates from the WKBJ method and reads thus +00

U ^ exp(ivS) E Ak(iv)-k.

(2)

k=o

Here S and Ak are scalar fields, independent of v. The former, named eikonal, is real-valued and governed by

Ivsl2

= n2;

the latter is complex-valued and governed by

2VS VAo + (AS)A0 = 0 2AoVS V(Ak/Ao) + AAk-1 = 0,

(k = 1, 2,3,...),

the so-called transport equations. See, e.g., [LK]. Inference built upon the expansion (2) amounts to geometrical optics. Though successful in describing both the propagation of light and the concurrence of caustics via the mechanism of rays, geometrical optics is inherently unable to account for those phenomena, such as the development of evanescent waves, that take place past a caustic. Recall the following:

(i) the geometric optical rays can be characterized as those paths that render the travel time a minimum, i.e., the geodesics that belong to the Riemannian metric

n x (Euclidean metric);

Giorgio Talenti

258

(ii) the characteristic lines of the eikonal equation, which portray the structure of any eikonal and coincide with the relevant lines of steepest descent, are rays;

(iii) the caustics are envelopes of rays, therefore convey the singularities of eikonals.

A more powerful asymptotic expansion, which is apt to represent solutions to (1) on both sides of a caustic, simultaneously in the region covered by geometric optical rays and in the opposite region where geometrical optics breaks down, is provided by a theory of Kravtsov and Ludwig. See [Kr], [Lu], or [BM]. When the caustic involved is smooth and convex, such an expansion reads as

x

x

U

Bk(iv)-k.

Ak(iv)-k+iv-113Ai' (Y2/3u)

Ai (V2/`;u)

(3)

Here u, v. Ak, Bk are scalar fields, independent of v; u and v are real-valued, Ak and Bk are complex-valued. Ai denotes the Airy function defined by the differential equation d ( d z2 z) Ai(z) = 0

(4)

and the initial conditions Ai(0) =

3-2/3

r(2/3)'

'3-1/3

-Ai (0) = r(1/3).

Properties of Ai inform us that the right-hand side of (3) oscillates rapidly where u is negative, approaches smoothly a limit if u approaches 0, quenches fast where u is positive. Therefore (3) matches geometrical optics in the region where u is negative, predicts the occurrence of damped waves in the region where u is positive, and causes the level surface where u = 0 to model a caustic. Assembling (1), (3) and (4) results in uIVuI2 - IVv12 + n2 = 0 (5)

- a fully nonlinear, first-order partial differential system governing u and v. In the present paper we sketch some lineaments of (5) in the case where the space dimension equals 2, i.e., we let x and y denote rectangular coordinates in the Euclidean plane and investigate the following system u(ur + uY) - vi - vy + n2(x,y) = 0 uxv= + uyvy = 0.

(6)

2. We have

2uu 2uuy -2v= -2vy -2 [u(uydx - u,.dy)2 + (vydx - v=dy)2] _

dx

dy

0

0

v=

vy

u=

UY

0

0

dx

dy

Some Equations of Non-geometrical Optics

259

the characteristic determinant of (6). Therefore a solution (u, v) to (6) is elliptic if u is positive, and is hyperbolic if u is negative. The characteristic curves of (6) keep away from the region where u is positive, and obey

dx: dy= (V--U. ux±vx) : (-,/---u - U, fvv) where u is either negative or zero - i.e., coincide with the lines of steepest descent of 2 (-u)3/2 ± v. 3

Observe that (u, v) is a hyperbolic solution to (6) if and only if u is negative and both the following equations 2

12 (-u)3/2 ± v)

=n2 ,

hold - in other words, hyperbolic solutions to (6) are in one-to-one correspondence with pairs of solutions to the eikonal equation. Differentiations and algebraic manipulations allow one to decouple (6). Eliminating v from (6) results in either IVvI = 0 and

uIVuI2 + n2 = 0,

or

(uI Vul' + n2u2) . uxx - 2n2u=uy uxy + (uIDuI4 + n2ux) U" +

+nIVuI2(Vn Vu) = 0,

(7)

a second-order semilinear partial differential equation having polynomial nonlinearities. Eliminating u from (6) results in either

u = 0 and IVvI2 = n2, or

IVul = 0 and

IVvI2

= n2,

o r e lse

(IVvI4 - n2vy) . vxx + 2n2vxvy vx, + (IVvI4 - n2vi) vyy

-nIVu I2(Vn Vv) = 0

(8)

another second-order semilinear partial differenti al equa tion. The following equation r l IVuIVv = f uIDuI2 + n2 I 0 O1 Vu

(9)

J

results from (6) too: it defines a Bdcklund transformation that pairs any solution u to equation (7), such that IVul # 0 and uIDuI2 + n2 > 0, with a solution v to equation (8) such that (u, v) obeys (6).

Giorgio Talenti

260

Equations (7) and (8) might be recast in a divergence form. In fact, 1

l.h.s. of (7) = lu + IV 2

2

sgn (u IVu12 + n2) IVu14 div { lu + Ipul2

and 2

l.h.s. of (8) _

1-

n

sgn (IVvI2

- n2)

IVvI" - div

1-

n2

z

IVv12

provided u and v are free from critical points. Therefore any sufficiently smooth solution to the following equation 1

u+

n2

2

Vul =0,

IVuI2

(10)

such that I Vul is different from 0, satisfies (7); any solution u of (7), such that both IVul and uIVuI2 + n2 are different from 0, satisfies (10). Any sufficiently smooth solution to the following equation n2

1- IVvI2

2 Vvl =0

such that IVvl is different from 0, satisfies (8); any solution v of (8), such that both IVvI and IVvI2 - n2 are different from 0, satisfies (11). However, perfectly nice solutions to (7) exist, whose gradient vanishes exclusively in a set of measure 0, that do not satisfy (10) in the sense of distributions they make the left-hand side of (10) a well-defined distribution, which is supported by the set of the critical points, but is not zero. In the case where n =_ 1, one such solution is built by selecting a constant C such that 0 < C < 1 (e.g. C = 10-10) and letting

domain of u = ((X, y) : x2/(1 - C2) - y2/C2 < 1) 1l2

,

(((1 -x2-y2)2+4y2)+1-x2-y'-)

i/s

- apropos arguments can be found in [MT1, Section 2.2]. Equations (10) and (11) might be viewed as the Euter equations of variational integrals. Let f be the real function defined by

f(0)=0 and

2t f'(t) = It(1 + t)I1/2

Some Equations of Non-geometrical Optics

261

for every real t. Explicitly,

f(t)=2 t(1+t)+2log(f+ 1+t) _ =

ift>0,

if -1
(-t)(1+t)+2aresin(f)

1

+2 t(l+ t)-2log(v,'--t +

-1-t) ift<-1.

Let a functional J be defined by J(u ) =

f (u]-Y2) nzdxdy I

for any u from a suitable class of sufficiently smooth functions that have neither zeroes nor critical points; let K be defined by

K(v)=

n

nzdxdy

for any v from a suitable class of functions that are free from critical points. Then (10) and (11) are the Euler equations of J and K, respectively. The following facts are easily derived from the definition of f. Let p and q be real variables. The mapping (p, q) r-. f (u(pz + qz))

is convex if either u is positive, or u is negative and the range of p and q is specified by (-u)(pz + qz) > 1. If u is negative and the range of p and q is specified by

(-u)(pz+qz) < 1, the mapping in question is neither convex nor concave. In other words, equations (7), (8), (10) and (11) exhibit a mixed elliptic-hyperbolic character. A solution u to either (7) or (10) is elliptic if IvuI 54 0

and either

U>0. or

u < 0 and (-u)IDuI2 > n2; it is hyperbolic if

IVul 54 0, u < 0 and (-u)IVulz < nz. A solution v to either (8) or (11) is hyperbolic or elliptic depending on whether

0
Giorgio Talenti

262

3. In the present section we consider a close relative of equation (7). Relevant details and proofs can be found in [ATT1], [MT2]. Replacing s 1 u13t2 signu by u, then assuming that u is nonnegative turn (7) into the following equation

(Ivula + n2uY) u11 - 2n2uxuy uxy + Qvul'' + n2u2) uyy+ + nlvul2(Vn Vu) = 0.

(12)

Theorem 1. Suppose n is smooth and strictly positive. Suppose u is real-valued and smooth, and satisfies (12) in every open subset of its domain where u2 + uy > 0. Assertions:

(i) if u7 = u, = 0 and u2x + 2u2y + uyy > 0 at some point, then ux = uy = 0 everywhere on a smooth curve passing through that point. (ii) If ux = uy = 0 and u2z + 2u2y + uyy > 0 at every point of a smooth curve, then this curve is a geometric optical ray. Theorem 1 basically shows that (12), unlike more conventional second-order partial differential equations, prevents its solutions from having isolated critical points. Equation (12) is elliptic-parabolic or degenerate elliptic. a solution u to (12) is elliptic if u2 + u2y > 0, a degeneracy occurs where u., = uy = 0. The degeneracy at critical points is a feature of (12) that causes such points to cluster. Another relevant feature is the architecture of (12), which exhibits geometric ingredients. If critical points are ignored and h is defined by either 32 (uyurx - 2uuxuxv + u2uyy) h = - ('ur + ui,) or

/vu

h = -divlIDuI then (12) reads both

IVu(Ou-n2{h -Vlogo Ioul }=0 and U,'

0

U,

a

(ii 8x + Ivul 8y) log

n2 + Ivuls = h.

Observe the following. First, the principal normal to the level lines of u is

(1/h) IvuI - in other words, the value of h at any point (x, y) is a signed curvature at (x, y) of the level line of u crossing (x, y). Secondly, the value of

V logn (©u

Some Equations of Non-geometrical Optics

263

at (x, y) equals a signed curvature at (x, y) of the geometric optical ray which is tangent at (x, y) to a level line of u. Thirdly, ux

a

a

uy

Ivulax+Ivulay is a directional derivative along the lines of steepest descent of u.

Theorem 2. Assume that 12 = an open non-empty subset of R2 and that fI is essentially different from R2, i.e.,

measure of (R2\ 1) > 0; assume also

n belongs to L2(f2), On belongs to Li (12) x Li C(f2)

j = a real-valued member of Assertions: a function u, which enjoys properties (i) and (ii) below, exists and is unique; the same function enjoys properties (iii) and (iv) too. (i) u is real-valued and belongs to j + W01'2(1l). (ii) u is a viscosity solution to equation (12) in 12. In other words, let uE belong to j + W01'2(12) and satisfy the restored, uniformly elliptic version of (12) that results from adding the extra term

to the left-hand side of (12) - such a uE exists for any positive e, is unique, is locally twice differentiable, and can be obtained via a standard variational process. Let e approach 0. Then uE converges to u uniformly on every compact subset of 12, and Vu, converges to Vu in LP (Q) x L P(12) for every p larger than, or equal to 1. (iii) u is twice differentiable in a generalized sense and obeys

if

IVU2

112

I

(x.y):dist ((x.y).R2\K))>r}

n2 + IVUI (

6

{IVn12dxdy112 fK

+ 2r'' {

l

2

J

K

(u=x + 2u=y + u22.) dxdy

(n2 + IVuI2) dxdy

l! 1 111}

provided K is a nice compact subset of 12 and r is a positive number. (iv) u satisfies equation (12) almost everywhere in H.

Theorem 2 shows the existence and the uniqueness of solutions to equation (12) that live in a prescribed domain and obey a prescribed Dirichlet boundary condition. According to usage, we denoted W01'2(11) the closure of Co (12) in i.e., the subset of W',2(12) consisting of those functions that vanish on aft in a generalized sense. However, since 0 need not be bounded in the present context,

Giorgio Talenti

264

we depart slightly from a standard notation and call W1'2(0) the completion of C" (S2) under the norm defined thus

IIuIIw,,a(sa) = 4f u2 (x2 + y2 +4)-2 dxdy + f IVul2dxdy. a

a

4. Hyperbolic solutions to equation (8), i.e., those obeying 0 < IDvI < n, are characterized by the following theorem.

Theorem 3. A real-valued smooth function v satisfies 0 < IVvI < n and equation (8) if and only if two real-valued smooth functions W and v) exist such that v = (w + 0)/2, IV V,12

= n2, (Jacobian determinant of W and O) 0 0. Proof. Propositions (i) to (vii) below correlate as stated in the next paragraph. (ii)

0)/2, IVWI2 = n2, IV'2 = n2. v= Iov12 (n2 - IVvl2) = j (Wxt&y - (Py?,)x)2 .

(iii)

OXOY - WyOx T 0.

(i)

(iv) (v)

0 < lovI < n. VW and V are given by the following Backlund transformations Fn2, 1

V = Vv ±

VIO= Vv

O11

1

V: -

I

Vv,

_1

2 Ivnvl2

1 LL

(vi)

1

0

01J Vv.

div { 'VT,=

- 1 Vv} = 0. (IDvI" - n2vy) vxx + 2n2vxvyvxy + (Ivvl' - n2vX) vy, = nIVvl2(Vn Vv). lI

(vii)

Proposition (i) implies (ii). (iii) and (iv) are equivalent, if (i) is in force. (i) and either (iii) or (iv) imply (v); (iv), (v) and a renormalization imply (i). (iv) and (v) imply (vi). (iv) and (vi) ensure that the Backlund transformations displayed in (v) are locally integrable. (vi) and (vii) are equivalent, if (iv) is in force. In the special case where n equals 1 identically, (8) reads (13) (IVvl' - vy) v,,., + 2vxvy - v..,, + (IVvI4 - vz) . vyy = 0. Such an equation is linearized by the classical Legendre transformation, whose traits are recalled in the next paragraph. Let v be a smooth real-valued function defined in some open subset of the Euclidean plane. Suppose Vv is a one-to-one mapping and that vxyvyy - v2 0 0

Some Equations of Non-geometrical Optics

265

everywhere. V is the Legendre transform of v if (i) the domain of V is the range of Vv; (ii) for any (p, q) from the range of Vv, the negative of V (p, q) is the height above the origin of the tangent plane to the graph of v whose normal parallels (p, q, -1). The following equations v(x, y) + V (p, q) = xp + yq,

vxx(x,y)

p = vx(x,y),

q = vy(x,y),

x=VP(p,q),

y=Vq(p,q),

vxy(x,y)

[vxy(x, y)

fVPP(p,q)

VPq(p,q)

vyy(x, y)] LVPq (p, q)

Vqq (p, q)

_

1

0

0

1

(14)

relate v, the Legendre transform of v, their first-order and second-order derivatives, and their arguments. Theorem 4. Let v and V be a pair of Legendre transforms. Then v satisfies equation (13) if and only if V satisfies (15)

[(p2 + q2)2 - 1°2] VPP - 2pgVPq + [(p2 + q2)2 - q2] Vqq = 0.

Proof. Combine equations (13), (14) and (15). 5. Equation (15) is elliptic in the exterior region where p2+q2 > 1, and is hyperbolic in the disk where p2 + q2 < 1. Its characteristic lines are the circles specified by

and C = constant, and the circle specified by p2 + q2 = 1.

Equation (15) reads (p2 - 1)

82V

a2V

OV

ape + p 8p + awe = 0,

(16)

if p and w are the polar coordinates defined by

0
(17)

Equation (15) can be recast in the following divergence form

div { p-3 Ip2 - 11-1/2. [p4

[VVp]

pq 2

p4

0,

pq 2]

and is therefore the Euler equation of the following variational integral

fr

j c (p

4

/ 8V) 2

-p2). k,:

8V 8V

-2pq. ap '

q

+(p

4 -q2).

(8V 12 apJ

dpdq P3IP2

-

11

1/2

If polar coordinates (17) are in use, the last equation and variational integral

{(p2_1)Ip2_1I_1/2.+Ip2_1l_I/2. 9V I

2

=0

Giorgio Talenti

266 and

ap/ +I\aw

Ip2-ll-112dpdw,

{(P2-1)(a .)

Jl respectively.

The following theorem appears in [MT3] and suggests that equation (15), despite its mixed elliptic-hyperbolic character, can be uniquely solved under a Dirichlet boundary condition, provided the relevant boundary lies in the elliptic region.

Theorem S. Let R be any number strictly greater than 1 and let DR = {(p, q) : p2 + q2 < R2 } ,

the disk having center at the origin and radius R. Let D be a real function, squareintegrable in [-7r, 7r].

Assertions: a function V, which enjoys properties (i), (ii) and (iii) below, exists and is unique; the same function enjoys properties (iv) and (v) too. (i) The first-order derivatives of V are square-integrable in any subset of VII that has a positive distance from both the boundary and the center of DR. (ii) V is a weak solution to (15) in DR, i.e.,

JRIP2_lI_I/2dPJT do J{(2_

aV az

aV aZ 1)apap+awaw}=0

p

for every Z from C °(DR). (iii) V equals D on the boundary of DR, i.e.,

OasptR. (iv) V is infinitely differentiable in DR\Jorigin). (v) If ak and bk denote the Fourier coefficients of D, i.e., ?r

ak = f D(w) cos(kw)dw and a bk = n7

J

D(w) sin(kw)dw,

then V is represented by the following series a2

+

0" k=1

!LP! {ak cos(kw) + bk sin(kw)} . Tk(R)

Here Tk stands for the Chebyshev polynomial defined by Tk(p)

= cos(k arccos p)

2 [(+ p2-1)k+ (p-

if IpI < 1,

p2-1)k]

if IPI>1.

Some Equations of Non-geometrical Optics

267

References [1]

[21

[3]

[4]

D. Bouche & F. Molinet, bfethodes asymptotiques en electrvmagnetisme. SpringerVerlag (1994). Yu.A. Kravtsov, Asymptotic solutions of Maxwell's equations near a caustic. Radiofizika 7 (1964) 1049-1056. R.M. Lewis & J.B. Keller, Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell's equation. New York University, Courant Institute of Mathematical Sciences (1964). D. Ludwig, Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19 (1966) 215-2.50.

[51

[6]

[71

R. Magnanini & G. Talenti, On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties. Contemporary Math. 283 (1999) 203-229. R. Magnanini & G. Talenti, On complex-valued solutions to a 2D eikonal equation. Part two: existence theorems. Submitted. R. Magnanini & G. Talenti, Approaching a partial differential equation of mixed elliptic-hyperbolic type. Submitted.

Giorgio Talenti University of Florence Department of Mathematics., viale Morgagni 67A 1-50134 Firenze, Italy

E-mail address: talenti®math.unifi. it

Speakers Antonio Ambrosetti (Trieste) Marino Badiale (Torino) Elves Alves Barros e Silva (Brasilia) Thomas Bartsch (Giessen) Vieri Benci (Pisa) Hebe Biagioni (Campinas) Lucio Boccardo (Roma) Marta Calanchi (Milano) Paolo (Torino) Italo Capuzzo Dolcetta (Roma) Gabriella Caristi (Trieste) Isabelle Catto (Parigi) Giovanna Cerami (Palermo) Silvia Cingolani (Bari) Monica Clapp (Mexico City) Philippe Clement (Delft) Jean-Noel Corvelec (Perpignan) Daniel Cordeiro de Morais (Campina Grande) David Costa (Las Vegas) Mabel Cuesta (Calais) Francesca Dalbono (Torino) Djairo de Figueiredo (Campinas) Marco Degiovanni (Brescia) Manuel Del Pino (Santiago del Cile) Flavio Dickstein (Rio de Janeiro) Joao Marcos do O (Joan Pessoa) Maria J. Esteban (Paris) Donato Fortunato (Bari) Jean-Pierre Gossez (Bruxelles)

Rafel Iorio (Rio de Janeiro) Otared Kavian (Paris) Olga A. Ladyzhenskaya (St. Petersburg)

Enrique J. Lami Dozo (Bruxelles) Felipe Linares (Rio de Janeiro) Orlando Lopes (Campinas) Alessandra Lunardi (Parma) Antonio Marino (Pisa) Vladimir Maz'ya (Linkoping) Jean Mawhin (Louvain la Neuve) P.J. McKenna (Storrs) Anna Maria Micheletti (Pisa) Olimpio Miyagaki (Vicosa) Mohameden Ould Ahmedou (Bonn) Filomena Pacella (Roma) Simone Paleari (Milano) Paul H. Rabinowitz (Madison) Elisabetta Rocca (Pavia) Mikhail Safonov (Minneapolis) Maria Scialom (Campinas) Tatiana Shaposhnikova (Linkoping) Didier Smets (Louvain la Neuve) Michael Struwe (Zurich) Giorgio Talenti (Firenze) Gabriella Tarantello (Roma) Susanna Terracini (Milano) Pedro Ubilla (Santiago del Chile) Giuseppina Vannella (Bari)

Nonlinear Equations: Methods, Models and Applications Daniela Lupo. Carlo D. Pagani and Bernhard Ruf, Editors

This volume contains survey and research articles originating from the Workshop on Nonlinear Analysis and Applications held in Bergamo in July 2001. Classical topics of nonlinear analysis are considered, such as calculus of variations, variational inequalities, critical point theory and their use in various aspects of the study of elliptic differential equations and systems, equations of Hamilton-Jacobi, Schriidinger and Navier-Stokes, and free boundary problems. Moreover, various models are focused upon: travelling waves in supported beams and plates, vortex condensation in electroweak theory, information theory, non-geometrical optics, and Dirac-Fock models for heavy atoms.

ISBN 3-7643-0398-0

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