Entropy penalization methods for Hamilton-Jacobi equations

ENTROPY PENALIZATION METHODS FOR HAMILTON-JACOBI EQUATIONS

DIOGO A. GOMES, ENRICO VALDINOCI In this paper we present a new entropy penalization problem and we dis uss its relations with approximate solutions of Hamilton-Ja obi equations, the

onvergen e of asso iated dis rete s hemes, as well as several appli ations, su h as: a generalization of the Hopf-Cole transformation whi h onverts non-linear HamiltonJa obi equations into linear evolution equations, the study of xed point problems, approximation of ertain linear evolution equations, and the onstru tion of entropy penalized Mather measures. Abstra t.

Contents

1. Introdu tion 2. Motivation for the s heme 3. Existen e of a minimizing measure and equivalen y between (4) and (7) 4. Generalized Hopf-Cole transformation 5. Some properties of 6. Some properties of and ^ 7. Formal Asymptoti s 8. Fixed point problems 9. Convergen e issues 10. Entropy penalized Mather measures 11. Convergen e to Mather measures Appendix A. A Bana h-Ca

ioppoli-type Theorem Appendix B. Some elementary properties of the semi on ave fun tions Referen es G

L

L

2 5 6 7 9 16 18 25 26 32 43 49 51 54

Supported in part by FCT, POCTI/FEDER, POCI/FEDER/MAT/55745/2004 and MIUR Variational Methods and Nonlinear Dierential Equations Subje t Classi ation: 70H20, 37M25, 37J50

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2

DIOGO A. GOMES, ENRICO VALDINOCI

``A quel tempo, di numeri e n'erano soltanto due: il numero e e il numero pi gre o. Il De ano fa un al olo a o

hio e ro e, e risponde: - Cres e di e elevato a ti. Bravo furbo! Fin l i arrivano tutti.'' (Italo Calvino, Le Cosmi omi he).

1. Introdu tion The obje tive of this paper is to study a new approximation pro edure for vis osity solutions of the Hamilton-Ja obi equations arising in optimal ontrol. This pro edure

onsists in the dis retization of the ontrol problem and in the addition of a suitable entropy fun tional. This fun tional regularizes the evolution of the dis rete value fun tion by a ting as a vis osity term. The entropy penalization method is motivated by

onsiderations in statisti al me hani s, as well as by the works of [JKO98℄, [FS86℄, 1 [Ana04℄, and [Eva04℄. In this paper, we introdu e an entropy penalized s heme related to Lagrangian and Hamiltonian dynami s and Hamilton-Ja obi-type equations, we prove its onvergen e, we introdu e a related generalized Hopf-Cole transformation, and we investigate the

onne tions between xed points of the s heme, the vanishing vis osity method, and the theory of Mather measures. In further detail, the organization of this paper and its main results are the following. In x 2, we introdu e and motivate the entropy s heme Z   7 ! G [℄ =  ln e dv RN

Z

hL(x;v )+(x+hv ) 

= inf [h L(x; v) + n(x + hv) +  ln (v)℄ (v) dv ; R where the in mum above is taken over all the probability densities on R N . Here, h plays the r^ole of a time step dis retization and  is the entropy penalization parameter. We noti e from the above formulas that there are two equivalent ways of de ning su h a s heme (see formulas (4) and (7) here below). The equivalen e between these two de nitions is shown in x 3. Then, a linear s heme Z 7 ! L[ ℄ = e (x + hv) dv N

RN

h L(x;v ) 

It may be useful to give a short sket h about the motivations of entropy methods in mathemati al physi s. Rougly speaking, in genetral, an entropy penalizedR metod onsists in perturbing a given problem by adding an \entropy term" of the type S (f ) = f log f . In statisti al me hani s, su h entropy S \ ounts", in a logaritmi s ale, the number of mi ros opi on gurations of a physi al system whi h are ompatible with a given ma ros opi behavior, while f has the meaning of the frequen y of any state. We refer to [Vil03℄ for a more detailed statisti al motivation of the entropy fun tional. 1

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

3

is introdu ed in x 4, whi h turns out to be equivalent to the (non-linear) entropy penalized method, via a generalized Hopf-Cole transformation of the type  7 ! e = : An analogous linear s heme has been introdu ed in [Ana04℄ for the study of Mather measures on path spa es, and, althought not de ned expli itly there, the s heme G also arises in that paper. In fa t, although our starting point is dierent, there are some deep onne tions between this work and the ones by Nalini Anantharaman; in this introdution we will try to spell out some similarites and dieren es between some of our results and the ones in [Ana04℄. We devote x 5 to the study of some properties of the non-linear s heme G . In parti ular, the s heme is proved to enjoy a ontra tion property in the spa e of fun tions \where

onstants are quotiented out" (see Theorem 9 for details). Though this feature is weaker than the standard stri t ontra tion property, it will be still suÆ ient to solve the related xed point problem by iteration. We also study the semi on avity properties of the s heme. In parti ular, a uniform semi on avity estimate is obtained in Theorem 13, from whi h a uniform Lips hitz bound follows. More pre isely, we show that the Lips hitz and semi on avity moduli of G [℄ are bounded by the maximum between the semi on avity modulus of  and a universal onstant. Some elementary properties of linear s heme L are outlined in x 6. We arry out some asymptoti s in x 7, a

ording to dierent hoi es of the parameters involved. Namely, the linear s heme is related to a paraboli equation (see Proposition 23), while the non-linear s heme asymptoti is related to the Hamilton-Ja obi equation, as shown in Proposition 24. In x 8, the nth iteration of the non-linear s heme is proved to onverge uniformly. This result is obtained by a xed point pro edure whi h nmakes use of the non-standard

ontra tion property of the s heme. We show that G [℄ onverges, as n ! +1, to a xed point of G \when onstants are quotiented out" (we refer to Theorem 26 for further details). In fa t, in this, our results improve the ones in [Ana04℄, as we an avoid S hauder's theorem for the onstru tion of a xed point, and, in parti ular, we obtain a very simple proof for the uniqueness of xed points. The onvergen e of the s heme for small time steps is dis ussed in x 9, by following [Sou85℄ and [BS91℄. We prove that, when the time step h goes to zero, the solutions of the suitably s aled linear and non-linear s hemes onverge uniformly to solutions of the heat equation and of the vis ous Hamilton-Ja obi equation, respe tively. The ellipti term in these equations is provided by the entropy (see Theorems 28, 29 and 30). In x 10, we investigate the onne tion between the xed points of our s heme and suitably penalized Mather measures. In parti ular, the generalized Hopf-Cole transformation provides a orresponden e between xed points of the s heme and minimizing measures for an entropy penalized a tion (see Theorem 32), whi h, in turn, onverge to the usual Mather measures when the penalization vanishes (see Theorem 40). We will also prove uniqueness of the entropy penalized Mather measures (see Theorem 36). We also remark that an \expli it" representation of entropy penalized Mather measures

4

DIOGO A. GOMES, ENRICO VALDINOCI

in terms of xed points of the non-linear s heme holds (namely, formula (108)). In parti ular, we show that, if  is the unique (up to onstants) solution of G [℄  =  2 R and  is the unique probability density solving Z

RN

then

(x hv )e (x; v )

hL(x hv;v )+(x) (x hv )  

dv

= (x) ;

= (x)e is the unique minimizer for the entropy penalized a tion. Our results in this se tion, although similar to the ones in [Ana04℄, avoid ompletely the use of path spa es, and only use elementary te hniques. In the paper [Ana04℄ the analog to entropy penalized Mather measures, Gibbs measures, are measures on the spa e of all paths on (Tn )Z, our approa h is simpler as we only need to work with the stationary version of these measures whi h are supported on Tn  R n , whi h simpli es onsiderably the problem. In parti ular, we present a self- ontained proof of the minimization property, as well as uniqueness of entropy penalized Mather measures. We devote x 11 to the analysis of the onvergen e of the penalized problem to the lass i al problem of Mather. In parti ular, both the penalized measure and the penalized \ee tive Hamiltonian" onverge to the ones of Mather's problem (see Theorems 38 and 40 for pre ise statements). In parti ular, both the minimizing measures and the minimal values of the entropy penalized a tion onverge to the analogous obje ts in the non-penalized setting. More pre isely, if L is the Lagrangian, the entropy penalized a tion of a \holonomi " (or \ ow invariant", see (91)) measure  is given by Z

hL(x;v )+(x+hv ) (x)  

 L(x; v )d(x; v ) + S [℄; : h TN RN

Then, we onstru t the densities whi h minimize su h penalized fun tional and we prove their onvergen e to Mather measures for small entropy. Again, a similar result was proved in [Ana04℄ for path spa e measures, but our approa h is onsiderably simpler and self- ontained. The paper ends with two appendi es. The rst one is devoted to the \abstra t" xed point argument whi h is used in the proof of Theorem 26. The se ond appendix olle ts some elementary results on semi on ave fun tions. The paper is, essentially, self- ontained, ex ept for some basi fa ts about Mather theory, for whi h we refer to [Mat91℄, [MF94℄ and [Gom05℄, and for some general fa t about vis osity solutions, whi h may be found, for instan e, in [CIL92℄. In general, we devoted some eort to using elementary arguments in the proofs, rather than other ones involving ner te hnologies.

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

5

2. Motivation for the s heme Let H (p; x) be a Hamiltonian, whi h has L(x; v) as onjugated Lagrangian (the pre ise assumptions on whi h will be listed in x 3): (1) L(x; v ) = sup p
v H (p; x): p Let (x; t) be a vis osity solution of the Hamilton-Ja obi equation t + H (r; x) = 0; with pres ribed terminal ondition (x; T ). The solution admits the following representation formula: Z T (2) (x; t) = inf L(q ( ); q_( )) d + (q (T ); T ) ; t for any T > t, where the above in mum is taken over all Lips hitz urves q : [t; T ℄ ! R N 2 satisfying q(t) = x (see, for instan e, Theorem 6.4.5 in [CS04℄). A dis rete s heme with time step h > 0 to approximate  onsists in the iteration n+1 (x) := inf [h L(x; v ) + n (x + hv )℄ : v2R By elementary onsiderations, theZ previous formula an be written as (3) n+1 (x) = inf [h L(x; v) + n(x + hv) ℄ d(v) ; 2P R where P denotes the spa e of the probability measures on R N . To make the s heme smoother, we repla e (3) byZthe \entropy penalized" s heme [h L(x; v) + n(x + hv) +  ln (v)℄ (v) dv; (4) n+1 (x) := inf 2D R where  > 0 (here and in the rest of the paper) is a small parameter and we denoted by D the set of probability densities on R N , i.e. Z   1 N (5) D := 2 L (R ) j (v)  0 a:e: ; (v) dv = 1 : R The idea of penalizing a linear optimization problem with a non-linear term an be tra ed ba k to [JKO98℄, where some ni e appli ations to Fo ker-Plank equations are presented. For a given (measurable) fun tion  : R N ! R , we de ne G [℄ as Z  (6) G [℄(x) :=  ln e dv : R Note that G is well de ned, for instan e, if  is bounded and G [℄ is also bounded. In Theorem 2 below, we show that (4) is equivalent to the expli it iteration s heme (7) n+1 := G [n ℄ : N

N

N

N

hL(x;v )+(x+hv ) 

N

2Beware of a sign hange both

in [CS04℄.

in the time dire tion and in (1) between our notation and the one

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DIOGO A. GOMES, ENRICO VALDINOCI

Then, in Theorem 3, we will establish the equivalen e with a linear s heme, whi h generalizes the Hopf-Cole transformation. In relation with that, we re all that if u is a solution to the Hamilton-Ja obi equation jDuj2 = 
u; ut + 2 then the Hopf-Cole transform v = e is a solution to the heat equation vt = 
v ; thus it is on eivable that similar exponential transformations happen to be useful in our framework. We now provide a formal setting of the above dis ussions. u 

3.

Existen e of a minimizing measure and equivalen y between (4) and (7)

From now on, we assume that the Lagrangian is a suitably smooth (e.g., Lips hitz) fun tion that has the form L(x; v ) = K (v ) U (x) ; for v 2 RN ; x 2 R N ; in whi h K , the \kineti enery", is stri tly onvex in v and superlinear at in nity, and U is the \potential energy" whi h is bounded, ZN -periodi and semi onvex, that is, there exists CU > 0 su h that (8) sup U (x + y) + Uj(yxj2 y) 2U (x)  CU : 2R 6

N x;y y =0

We suppose further that K is semi on ave, i.e., that there exists CK su h that (9) sup K (v + w) + Kjw(vj2 w) 2K (v)  CK : 2R 6

N v;w w=0

Proposition 1.

(10)

Let

D as in (5). Fix x 2 R N . Then, there exist ? 2 D realizing

inf

2D

Z 



h L(x; v ) + (x + hv ) +  ln (v ) (v ) dv :

More expli itly, ? is given by with and

? := e

^ (x;v )+ L 

;

L^ (x; v ) := hL(x; v ) + (x + hv );  :=  ln

Also, the quantity in (10) is equal to

Z

.

e

^ (x;v ) L 

dv :

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

7

By de nition, ? is a probability density satisfying (11) hL(x; v ) + (x + hv ) +  ln ? (v ) =  : We laim that ? is optimal. Indeed, by the onvexity of the fun tion t 7! t ln t, we have, for any Zother probability density , that (h L(x; v) + (x + hv) +  ln ) (v)dv  Proof.

N RZ

N ZR

(h L(x; v) + (x + hv) +  ln ?) ?(v)dv +

(h L(x; v) + (x + hv) +  ln ? + ) ( (v) ?(v))dv: From (11) and the fa t that both (v) and ? are probability densities, we on lude that the last integral vanishes, and so ? is optimal. The last laim in Proposition 1 follows from (11) and the fa t that ? is a probability density.  From Proposition 1, we easily on lude that: RN

Theorem 2.

We have that Z

inf [h L(x; v) + (x + hv) +  ln (v)℄ (v) dv

2D

=

 ln

Z

RN

e

h L(x;v )+(x+hv ) 



dv :

In parti ular, the s hemes in (4) and (7) are equivalent.

4. Generalized Hopf-Cole transformation In this se tion, we dis uss a hange of variables whi h transforms the non-linear evolution operator G into a linear evolution operator L, this operator has in fa t been introdu ed in [Ana04℄, without an expli it mention to G , although this one has in fa t been used there too. This pro edure is related with the lassi al Hopf-Cole transformation (see, for instan e, x4.4.1 of [Eva98℄). For this, we de ne the linear operator Z (12) L[ ℄(x) = e (x + hv) dv : R We will relate L and G through the exponential transformation: (13) E [℄(x) := e : Then, it is easy to see that: h L(x;v ) 

N

(x) 

Theorem 3.

(14)

We have

In parti ular, if

LÆE = E ÆG: n

(x) := e

n (x) 

;

8

DIOGO A. GOMES, ENRICO VALDINOCI

the s hemes in (4) and (7) are equivalent to the s heme

= L[ n ℄ : We omit the details of the elementary al ulation needed to prove (14) and hen e Theorem 3. It is also onvenient to introdu e the res aled linear operator L^[ ℄(x) := R L[ ℄(x) n+1

RN R RN

=

h L(x;v ) 

dv (x + hv) dv : R hK v RN e  dv

e e

h K (v ) 

( )

A

ordingly, we may onsider the iteration ^n+1 = L^[ ^n℄ : (15) Noti e that L^ is independent on the potential U . As an appli ation of the generalized Hopf-Cole transformation we now show the following property, whi h is somehow related with the fa t that the minimum of vis osity solutions is a vis osity solution. Proposition 4.

Assume that n and 'n are solutions to the iterative s heme

n+1 = G [n ℄

Then so is

'n+1 = G ['n ℄ :



n =  ln e



+e : Proof. This follows from (14) and the linearity of the s heme L.  We observe that    ln e + e ! minfn; 'ng; as  ! 0+, whi h is onsistent with the fa t that the minimum of vis osity solutions of 3 the terminal value problem is a vis osity solution. Note that the fa t that L[ ℄ = L[ ℄ re e ts into G [ + ℄ = G [℄ + , that is G

ommutes with addition of onstants. This elementary fa t will be further stressed and used in the sequel. n 

3Indeed, if 1 and 2 ZT t

n 

'n 

'n 

are solutions of (2), so is  := min

( ( ) _( )) d + i (q(T ); T )

L q  ;q 

Z



T

t

. To on rm this, note that

1 ; 2 g

f

( ( ) _( )) d + (q(T ); T ) ;

L q  ;q 

for i = 1; 2 and thus, by taking the in mum as in (2), ( ) inf

i x; t



Z

T t

( ( ) _( )) d + (q(T ); T ) ;

L q  ;q 

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

9

5. Some properties of G We now point out some features of G , su h as algebrai properties, smoothness, analyti bounds and semi on avity estimates. In parti ular, a ne ontra tion property is established in Theorem 9 and a uniform bound on the semi on avity modulus is given in Theorem 13. The bounds for the semi on avity modulus are essential to prove results su h as existen e of xed points and onvergen e of the s heme, among others. The rst three properties follow dire tly from the de nition and we thus omit the proof: Proposition 5. If 0 is ZN -periodi , then so is n := G n [0 ℄ for any n. If U and 0 belong to C 1 (R N ), then so does n for any n. Proposition 6 (Monotoni ity). If    , then G [℄  G [ ℄. Proposition 7. G [ + r ℄ = G [℄ + r , for any r 2 R . We now point out that G is a (non-stri t) ontra tion on L1(R N ): Proposition 8. For any ;  2 L1 (R N ), kG [℄ G [℄kL1(R )  k kL1(R ) : Proof. Fix x 2 R N . Without loss of generality we may assume that G [℄(x)  G [℄(x) ; thus N

N

R L(x;v )+ (x+hv )  dv RN e R L(x;v )+(x+hv )  dv RN e k  kL1 (RN )

jG [℄(x) G [ ℄(x)j =  ln R RN

  ln

L(x;v )+(x+hv ) 

e

R RN e  L1 (RN ) :

= k

k

e



L(x;v )+(x+hv ) 

dv

dv

The ontra tion property an be improved in the following way:



for i = 1; 2. On the other hand, xed any a > 0, there exists a suitable qa so that, possibly inte hanging the indexes of i , a

+ inf Z



=

Z

Z

T t T

t

T

( ( ) _( )) d + (q(T ); T )

L q  ;q 

( ( ) _ ( )) d + (qa (T ); T )

L qa  ; qa 

( ( ) _ ( )) d + 1 (qa (T ); T ) 1 (x; t) (x; t) : These omputations show that  is a solution of (2). t

L qa  ; qa 





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DIOGO A. GOMES, ENRICO VALDINOCI

Suppose that both  and  are ZN -periodi and Lips hitz fun tions, with Lips hitz onstant bounded by . Then, there exists a onstant C > 0, possibly depending on h, ,  and other universal quantities, su h that

Theorem 9.

(16)

kG [℄ G [℄k℄  (1 C k kN℄ ) k k℄ ;

(17)

k k℄ := inf k +  kL1(R ) : 2R

in whi h we used the norm Proof.

N

Let us observe that

(18) k k℄  k kL1(R ) and (19) k k℄ = k + rk℄ ; for any fun tion and any r 2 R . Also, there exists a onstant  2 R su h that k +   kL1(R ) = k k℄ : By (19) and the fa t that G ommutes with onstants (re all Propopsition 7), possibly repla ing  with  +  inf  and  with  inf , we may assume, without loss of generality, that (20) k kL1(R ) = k k℄ : and that (21) inf  = 0 : Noti e that, by (21) and the periodi ity assumption, k k℄  k (0) kL1 p (22)  k (0)kL1 + kkL1  2 N : Let x be a point maximizing (23) jG [℄(
) G [℄(
)j ; and let  the optimal probability measure on RN that yields: Z   (24) G [ ℄(x ) = hL(x ; v ) +  (x + hv ) +  ln   dv : R Re all indeed that, by Proposition 1,     ;

 = e with Z    (25)  =  ln e dv : R Also, due to (4), Z   (26) G [℄(x )  hL(x ; v ) + (x + hv ) +  ln   dv : N

N

N

N

hL(x ;v )+ (x +hv )+ 

hL(x ;v )+ (x +hv ) 

N

RN

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

De ne, for v 2 h1 TN , (27)  (x; v ) :=

X k2ZN

e





hL(x;v +k=h)+ (x+h(v +k=h))+ 

11

:

Note that, sin e  is bounded and L grows superlinearly in v by assumption, the series above onverges. Furthermore, Z (28)  (x; v ) dv = 1 ; 1

h

TN

sin e  is a probability measure on R N . We laim that there exists a positive onstant  su h that (29)    (x; v ) 1 N N for any x 2 R and v 2 h T . Indeed, rst note that, by the Lips hitz properties of  and (21), we have that (30) 0     C1 and therefore, by (25), (31) e =  C2 ; where Ci > 0 denote here suitablepquantities possibly depending on N , , h and . Also, if v 2 h1 TN (and thus jvj  N=h), using again (30), we see that  (32) e  C3 : Then, from (27), (31) and (32), we have that    C4 ;  (x; v )  e for any (x; v) 2 R N  h1 TN . This proves (29). Now, we use the ZN -periodi ity of  and  , (24), (26) and (27) to dedu e that G [℄(x ) G [ ℄(x ) Z  [(x + hv) (x + hv)℄ (v) dv hL(x;v )+ (x+hv ) 

hL(x;v )+ (x+hv )+ 

(33)

=

N ZR

[(x + hv)

N hT Z 1

= h1N

x +TN

[(w)

 (x + hv )℄  (x ; v ) dv 

 (w)℄  x ;

w

h

x



dw :

Note also that (34) inf   = sup   = k k℄ ; thanks to (20). We laim that there exists a set N  x + TN of measure larger than C^ k kN℄ , for a suitable C^ > 0, in whi h    k k℄ =8. For proving this, note that if  =  the laim is obvious; otherwise, by (34), there is a point x 2 x + TN so

12

DIOGO A. GOMES, ENRICO VALDINOCI

that ( )(x) < 0 and k k℄  2j( )(x)j. By the Lips hitz property of the fun tions involved, it follows that    k  k℄=8 in the ball of radius k k℄ =(4) entered at x. Thus, if we take N := Bk k =(4) (x) \ (x + TN ) ; the above laim follows from (22). Then, by su h a laim and (29), we dedu e that   Z w x  x ; dw  C k  kN℄ ; h N for a suitable C > 0. Combining this with (33) and (28), we on lude that G [℄(x ) G [ ℄(x )  Z    1 w x  hN dw (w)  (w)  x ; h (x +T )nN   Z w x k   k℄ dw  x;  ℄

N

  hN h (x +TN )nN " # Z k   k℄ N N h  hN  (x ; v ) dv C k  k℄ N hT = (1 C k  kN℄ ) k  k℄ ; Then, by the hoi e of x performed in (23), possibly inter hanging 1

for a suitable C > 0. the r^oles of  and , we gather that kG [℄ G [ ℄kL1(R )  (1 C k kN℄ ) k k℄ : This and (18) imply (16).  We now de ne Z dv : ;h := sup ln e x2R R Note that, in many

ases, ;h may be estimated expli itly; e.g., if L(x; v ) = K (v ) = jvj2, then ;h = (N=2)j ln(=h)j. By taking  = 0 in Proposition 8, we gather the following Proposition 10 (L1 -bound). kG [℄kL1(R )  kkL1(R ) +  ;h : N

h L(x;v ) 

N

N

In parti ular,

N

N

knkL1(R )  k0kL1(R ) + n ;h : N

N

We point out that, in general, it is not possible to obtain a bound of knkL1 whi h is uniform on n: even in the ase of L = jvj2 and 0 = 0, whi h an be worked out expli itly, one obtains  (35) knkL1 = Nn j ln j: 2 h

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

13

We now study the semi on avity of n. The semi on avity modulus of a fun tion  is (36)  := sup (x + y) + j(yxj2 y) 2(x) : 2R 6 N x;y y =0

We say that a ontinuous fun tion  is semi on ave if  < +1. We will prove that  is uniformly bounded (see Theorem 13). Sin e n is ontinuous if so is 0, this yields that n is uniformly semi on ave. The estimate on  will be obtained in three steps: we rst get a rst rough estimate on  ; we then prove a semi on avity \improvement" estimate (that is, knowing a bound on  , we show that a tually a slightly better bound is possible); nally, by iterating the pro edure, we will be able to obtain a uniform bound on  . Let us now x a semi on ave fun tion 0 and work out the details: n

n

n

n

n

Lemma 11.

Let CU and CK be as in (8) and (9). Then,

  C2hK + h C2 U ; n

for any n  1. Proof.

Fix w 2 R N . By using (9) and the Cau hy-S hwarz Inequality, we get that n+1 (x)  h U (x)  Z dv  ln e (K (v+w)+K (v w) C jwj ) e h

=

h U (x)  ln

Z

RN

h U (x) Z  




=

RN

2 ln

Z

RN

e

RN

2

K

2

h CK e

2 2 jw j

h K (v +w)+n (x+hv ) 2

e

h K (v w)+n (x+hv ) 2

h CK e

2 2 jw j

h K (v +w)+n (x+hv ) 

h K (v w)+n (x+hv ) 

dv

dv



h CK

h U (x) Z  

2 ln ZR   2 ln

N

RN

n (x+hv ) 

e e

2 2 jw j



h K (v +w)+n (x+hv ) 

dv

h K (v w)+n (x+hv ) 

) dv



:

dv



14

DIOGO A. GOMES, ENRICO VALDINOCI

Thus, hanging variables of integration, h CK 2 n+1 (x)  h U (x) jw j 2 Z   e 2 ln 

= thus, from (8),

2 ln

RN

Z

RN h CK 2 w

e



h K (~ v )+n (x hw+hv~) 

dv~

h K (~ v )+n (x+hw+hv~) 

dv~



h

2 j j h U (x) + 2 U (x + hw ) + U (x + 21 n+1(x + hw) + n+1(x hw) ;



hw)





2n+1(x)  C2hK + h C2 U (hjwj)2 ;

n+1 (x + hw) + n+1 (x hw)

from whi h the desired laim follows.  Noti e that the bound proven in Lemma 11 does not really look satisfa tory for small h. We now show how to improve it. Lemma 12 (Semi on avity improvement). Fix n 2 N . Assume that (37)    : n

Then,4

  h CU + C CK+h :

(38)

n+1

K

Fix y 2 R N and  2 (0; 1) and set t := 1 . Then, hanging variable in the integration, we gather that n+1 (x  hy ) = h U (x  hy ) Z  K (wy)  (x+hwthy)  ln e e dw ; Proof.

RN

4Note that, if

h 

q



1

 n



+ h2 CU2 + 4 CU CK =2 ; then the right hand side of (38) is less than , thus we improved the semi on avity bound from the nth step to the (n + 1)th step. If, on the other hand, 







h CU



h CU

+

q

2 h2 CU



+ 4 CU CK =2 ;

then the right hand side of (38) provides a uniform bound for n . Observations of this type will play a r^ole in the proof of Theorem 13 here below. +1

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

15

so that, by means of (8) and using the Cau hy-S hwarz Inequality, we infer that n+1 (x + hy ) + n+1 (Zx hy ) 2n+1(x)    h CU (hjyj)2 + 2 ln e K (w) e  (x+hw) dw  ln




R

Z

R

Z

RN

e

N

hK 

e

hK 

(w y) e

(w+y) e

 h CU (hjyj) + 2 ln 2

 n

1

h 

1

 n

N

 n

1

(x+hw+thy) dw

(x+hw thy) dw

Z

h K

e

(w) e

RN h (K (w y )+K (w +y )) 2

Z

  n

1

(x+hw) dw



2 ln e R 
e ( (x+hw+thy)+ (x+hw hty)) dw : Therefore, by (9) and (37), n+1(x + hy ) + n+1 (x hy ) 2n+1 (x)  h CU (hjyj)2 + h CK (jyj)2 +  (htjyj)2 ; hen e, sin e t = 1 ,   h CU + ChK 2 + (1 )2 : The desired result follows now by hoosing  := h=(CK + h).  We now get the uniform ontrol on  by a suitable iteration of Lemma 12 (thus improving Lemma 11 in the ase of small h): Theorem 13 (Uniform semi on avity). Let p h CU + h2 CU2 + 4 CU CK ? := : 2 Then, for any n 2 N ,   max  ; h CU + ? : N

1 2

n

n

n+1

n

n

n+1

In parti ular,

  max  ; h CU + ? : Proof. The se ond laim follows from the rst one, by a simple iteration. To prove the rst laim, we distinguish two ases: either   ? or   ? . Let us deal rst with the ase   ?. Note that, if   ? , then C   : h CU + K CK + h This observation and Lemma 12 imply that   h CU + C CK+h   ; n

0

n

n

n

n+1

n

K

n

n

16

DIOGO A. GOMES, ENRICO VALDINOCI

proving the desired result when   ? . If, on the other hand,   ? , Lemma 12 implies that   h CU + C CK+h K   h CU +   h CU + ? ; whi h proves the desired result.  We are now in position to dedu e a uniform Lips hitz bound on n. Theorem 14 (Uniform Lips hitz bound). Let h 2 (0; 1℄. Assume that 0 is ZN periodi and semi on ave. Then, there exists C , depending only on  , N , CU and n

n

n

n+1

n

n

CK so that

0

jn(x) n(y)j  C jx yj ;

8n  1:

8x; y 2 R N ;

By Proposition 5 and Theorem 13, n is ZN -periodi and uniformly semi on ave. Then, the result follows from Theorem 44.  6. Some properties of L and L^ In analogy with the results in x 5, we now point out some properties of the linear operators L and L^. The following ones are quite easy to he k: Proposition 15. If 0 is ZN -periodi , then so is n := Ln [ 0 ℄ for any n. If ^0 is ZN -periodi , then so is ^n := L^[ ^0 ℄ for any n. If U and 0 belongs to C 1 (R N ), then so does n for any n. If ^0 belongs to C 1 (R N ), then so does ^n for any n. Proposition 16 (Monotoni ity). If  , then L[ ℄  L[  ℄. If ^  ^ , then L^[ ^℄  L^[ ^ ℄. R dv , and L^[ ^ + r℄ = L^[ ^℄ + r, for Proposition 17. L[ + r ℄ = L[ ℄ + r R e any r 2 R . Also, it is easy to see that L^ is a (non-stri t) ontra tion on L1(R N ): Proposition 18. For any ^; ^ 2 L1 (R N ), kL^[ ^℄ L^[ ^ ℄kL1(R )  k ^ ^ kL1(R ) : In parti ular: Proposition 19 (Uniform L1 -bound). kL^[ ^℄kL1(R )  k ^kL1(R ) : Analogously, ^[ ^℄ 2 C k (R N ) Proposition 20 (Uniform C k and W k;1-bound). If ^ 2 C k (R N ), then L and kL^[ ^℄kC (R )  k ^kC (R ) : Proof.

N

h L(x;v ) 

N

N

N

k

N

N

k

N

An analogous statement holds by repla ing C k with W k;1.

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

17

We1 now obtain a (non-uniform in h and ) C 1-bound of L[ ℄ for the dependen e of the L -norm of : Proposition 21 (Improvement of regularity). Assume that Z

(39)

RN

hK (v ) 

e

jDK (v)j dv < +1 :

Then,

D

for any Remark.

(L^[ ℄)



1 N L (R )

 k kL1(R

N

)

R RN

e  jDK (v )j dv ; R hK v  RN e  dv

2 L1(R N ).

hK (v )

( )

For K (v) = jvj2, the result in Proposition 21 yields that



^  onst kp kL1(R ) :

D (L[ ℄) 1 N

L

Proof.

h

(R ) N

By hanging variable, we have that L^[ ℄(x) =

Therefore, Dx



(L^[ ℄)  =

R RN

R RN

e  K ((w x)=h) (w) dw : R hK v hN RN e  dv h

( )

x)=h)j j (w)j dw dv e R h K (v ) jDK (v)j j (x + hv)j dv ; RN e  R hK v  RN e  dv e

hK 

((w x)=h) jDK R hN RN

((w

hK (v ) 

( )

whi h implies the desired result.  Further regularity results an be also dedu ed in an analogous way from Proposition 21. Also, it is straightforward to see that that L^ preserves the average over TN of ZN periodi fun tion, namely: Proposition 22. If 2 L1 (TN ), then Z Z ^L[ ℄ = : TN

TN

We remark that the above bounds and the Theorem of As oli imply the uniform onvergen e, up to a subsequen e, of the iteration L^n[ 0 ℄. In parti ular, if 0 is ZN -periodi , Propositions 20 (or 21) and 22 imply that L^n[ 0 ℄ onverges, when n ! +1, up to a subsequen e, to a xed point of L^ with the same TN -average of 0 .

18

DIOGO A. GOMES, ENRICO VALDINOCI

7. Formal Asymptoti s In this se tion, we dis uss the formal asymptoti behavior of the s hemes orresponding to L^, L and G . These omputations play a ru ial r^ole in the rigorous onvergen e results given in x 9. We deal with dierent hoi es of the parameters  and h. Ellipti , paraboli and Hamilton-Ja obi-type equations will show up in these asymptoti s. ^). Let Proposition 23 (Asymptoti s for L bi :=

R

R e R RN e N

hK (v ) 

vi dv ; hK v  dv

Suppose that jbi j + jaij j < +1. Let Then,

(40)

^n+1 ^n h

aij :=

( )

h

R RN e R RN

2

hK (v ) 

e

vi vj dv : dv

hK (v ) 

^n+1 = L^[ ^n℄ :

= bi Dx ^n + aij Dx2 x ^n + error terms : i

i j

The error terms an be estimated by:

onst

h2 kD3

^n kL1(R

R N

jv j3 e

) R

e

hK (v ) 

hK (v ) 

dv : dv

The left hand side of (40) may be thought as a dis rete approximation ^ of t as h ! 0. Though the oeÆ ients in (40) depend on the spe i form of the kineti energy and on the relative s ale of  and h, it may be useful to work out the ase 2 K (v ) = jv j somehow expli itly. In this ase, the expressions in Proposition 23 be ome bi = 0, aij = onst  Æij and the error term is bounded by onst h1=2 3=2 kD3 n kL1 . Proof. We have h i R ^ ^ ^n+1 ^n e n (x + hv ) n (x) dv = R h Remark.

hK (v ) 

Z

h e

hK (v ) 

dv

h = ;h e vi Dx ^n (x)dv + 2  Z + O h2 ;hkD3 n k1 jvj3e hK (v ) 

Z

hK (v ) 

with

;h =

e

i

Z

e

hK (v ) 

dv

hK (v ) 



vi vj

D2

xi xj

^n(x)dv



dv ;

 1

:



ENTROPY PENALIZATION FOR HAMILTON-JACOBI

19

As an appli ation of this theorem, we study the xed points5 of the s heme (15). Assume, for the moment, that h = . Observe that if we de ne Z 1 K (v) aij = 2 e vivj ; the operator aij i j

is (possibly degenerate) ellipti , as Z Z 1 1 K (v) K (v) 2 aij ij = 2 e vi ivj j = 2 e jv
 j  0: Furthermore, if K is even, then bi = 0. In this ase, the xed points of the s heme (15) satisfy aij i j = O(hkD3 k2 ); as it follows from the estimate of Proposition 23. Proposition 24 (Asymptoti s for G ). Let  2 C 3 (R N ). Assume that K (v ) = jv j2 and

that

(41)

h kD2 kL1(RN ) is smaller than a suitable onstant.

Let

R

(42)

a

Then,

(43)

jwj2 jwj2 dw : e jwj2 dw

RNRe RN

:= 2

G [℄  = H (D(x); x) + a
 N ln  + error terms : h 2h h

The error terms an be estimated by

C? (h + 3=2 h1=2 ) ;

where C? is a suitable positive quantity depending only on kDj kL1 for

1  j  3.

that any onstant is a xed point for ^. Moreover, if K is even in any of ea h variables, given a quadrati polynomial p(x) = b + ` x + M x x ; N with b R, ` R and M Mat(N N ), one easily sees that p is a xed point for ^ if and only if aj j2 p = 0 ; where R 5Note

L




2

2

2



aj

In parti ular, for K (v) =

L

hK (v) 

RN e

:= R

RN e

vj2 dv

hK (v) 

dv

:

, is a xed point of ^ if and only if it is harmoni .

v2 p

j j




L

20

DIOGO A. GOMES, ENRICO VALDINOCI

Remark. There are several heuristi ways of interpreting the result in Proposition 24 a

ording to the s heme n+1 := G [n℄. The rst onsists in taking h ! 0 and =h ! 0, so that, formally, (43) goes to the Hamilton-Ja obi equation t  = H (Dx (x); x) : Another possibility is taking  > 0 to be xed and sending h ! 0. Then, if we de ne Nn  (44) n := n + 2 ln h ; it follows that (43) goes formally to t  = H (Dx  (x); x) + a
 ; that is, a Hamilton-Ja obi equation of \vanishing vis osity" type. The reader may

ompare this ase with the example dis ussed in (35). This fa t will be made rigorous in Theorem 30 below and it may also be rephrased as follows. A dire t omputation shows that G [0℄(x) = hU (x) N2 ln h : Therefore, if we de ne (45) G^[℄ := G [℄ G [0℄ ; then Proposition 24 gives that G^[℄  = K (D(x)) + a
 + error terms : (46) h The latter interpretation is related with Proposition 8, whi h gives that (47) kG^[℄kL1(R )  kkL1(R ) ; that is, a uniform bound on the G^-iterations. We will use these observations in Theorem 29 below, where a formal proof of the onvergen e of the G^-s heme for h ! 0 will be provided. Proof. We have Z G [℄ =  ln e (48) dv N

N

hL(x;v )+(x+hv ) 

Z

= hU (x)  ln e j j De ne v = v(x; h) to be a minimizer of the fun tion h jv j2 + (x + hv ) : Note that the minimizing property of v yields that h jv j2 + (x + hv  )  (x) and so h jv  j2  h kDkL1 jv j ;

h v 2 +(x+hv ) 

dv:

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

that is (49) jvj  kDkL1 : By the minimizing property of v, it also follows that (50) 2v + Dx(x + hv ) = 0 and thus hjv j2 + (x + hv ) (51)  [ hjv  j2 + (x + hv  )℄ + h jv v  j2 +  (x; v) ; = 1



21

with    (52) 1(x; v) := (x + hv) (x + hv )  h Dx(x + hv )
(v v ) : Note that, by onstru tion, 2 2 (53) j1 j  C h kDkL1 jv vj2 ; where C , here and in the remainder of the proof, we will denote an appropriate universal

onstant (possibly taking a dierent value at dierent steps of the omputation). We will also use the following short hand notation: R j j Z f (v ) dv e f (v ) dv := R : N

hv v 

R RN

2

jj

hv2 

e

dv

Let also Z (54) 2 (x) := (e  1 + 1 ) : Then, by (53) and (41), we get that Z X 1 j1jk j2j  k! k=2 Z 1 k X 2  j1j jk1!j k=0 Z (55) = j1j2e Z C h4 kD2 k2L1  j4 e k  j v v 2 1

1

k 1jv v j2

C h2 D 2  L 

R

 C h2 kD2k2L1  C h2 kD2k2L1 :

RN e

R RN

jwj2 8

jwj4 dw

e jwj dw 2

22

DIOGO A. GOMES, ENRICO VALDINOCI

We also de ne (56) 1(x; v) := hU (x) + hjvj2 + (x + hv) : Using this de nition, (48), (51) and (54), we obtain that G [℄ = 1(x; v)  ln

(57)

= 1

 ln

= 1

 ln

= 1

 ln

Z

Z 

1

Z

e

RN

e

j

h v v 

1 (x;v) dv



 j2

1 (x;v) dv

e

 ln

Z N R

(1 1) dv + 2 (x) Z



1 dv + 2(x)



e

jj

hv2 

dv

 ln 2 h N  2 ln h :



N

We now de ne Z (58) 3 (x) := 2(x) 1 (x; v) dv : Note that (59) j3j  C h kD2kL1 ; thanks to (53), (55) and (41). Let also (60) 4 := ln(1 + 3 ) 3 : Then, (61) j4j  C h2 kD2k2L1 ; due to (59). Also, using (57), (58) and (60), we on lude that  G [℄ = 1 N (62) ln 2 h  Z3  4 = 1 N2 ln h +  1 (2 + 4 ): Sin e 2 and 4 have the same order of magnitude, it is onvenient to de ne 5 := 2 + 4 ; so that (55) and (61) give (63) j5j  C h2 kD2k2L1 ; and we obtain from (62) that Z N  (64) G [℄ = 1 2 ln h +  1 dv  5 :

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

We now de ne 2 6 (x; v) := 1 (x; v) h2 ij (x) (vi Z (65) 7(x) := 6(x; v) dv : Note that, from (52) and (49), 2 j6j  1 h2 ij (x + hv) (vi 3

+ Ch jvj kD3kL1 jv

and so (66) Moreover,

vi )(vj

v  )(v i

vj )

23

and



j

v  ) j

v  j2

3  Ch kD3kL1 (jv vj + kDkL1 ) jv vj2 p

j7j  C h2 kD3kL1 ( =h + kDkL1 ) : Z

1 dv = ah
 + 7 ; due to (65), (42) and a parity argument. Therefore, we dedu e from (64) that  G [℄ = 1 N ln 2 h + ah
 +  7  5 : A

ordingly, if 8 := 7 5 , we have that  (67) G [℄ = 1 N ln 2 h + ah
 +  8 : Also, by (63) and (66), h i p (68) j8 j  Ch2 kD2kL1 + kD3kL1 ( =h + kDkL1 ) : We now de ne  h  2  2 2(x; v ) = 4 jD(x)j jD(x + hv )j 3(x; v) = (x) h4 jD(x + hv )j2 hjvj2 (x + hv) ; so that6, by (56), (69) 1 = hH (D(x); x) + (x) + 2 3 : Note that j2j  2h kDkL1 D(x + hv) D(x)  C h2 kDkL1 kD2kL1 jvj  C h2 kDk2L1 kD2kL1 ; 6Of ourse, if K (v ) = jv j2 , one has that H (p; x) = jpj2 =4 + U (x).

24

DIOGO A. GOMES, ENRICO VALDINOCI

thanks to (49). Furthermore, by using (49) and (50), one has that j3j  j(x) (x + hv) + h D(x + hv )
v j 1  2  2   +h 4 jD(x + hv )j + jv j + D(x + hv )
v  C h2 kD2kL1 jvj2 + 0  C h2 kDk2L1 kD2kL1 : Therefore, setting (70)  := 2 3 +  8 ; the estimates above and (68) yield that n

(71)

jj  C h2 kDk2L1 kD2kL1 h io p 3 2 + kD kL1 + kD kL1 ( =h + kDkL1 ) :

Also, by olle ting the identities in (67), (69) and (70), we have that  ln G [℄ = hH (D(x); x) + (x) N 2 h + ah
 +  :

This and (71) yield the desired result. The previous result immediately applies to a xed point7 problem: Corollary 25.



Suppose that

(72)

G [℄ =  + ? :

Then, we have

a
 H (Dx; x) =

? N + h 2h

ln Nh + error terms:

The oeÆ ient a and the error terms have the same form as in Proposition 24.

Heuristi ally, as h ! 0 and =h ! 0, the result in Corollary 25 means that ? =h plays the r^ole of the \ee tive Hamiltonian" H in the equation H (Dx ; x) + H = 0: A formal justi ation for this will be given in Theorem 38. Remark.

7For the

so does

G

existen e of fun tions satisfying (72), see Theorem 26. Note also that, if  satis es (72), for any n N , due to Proposition 7.

n [℄,

2

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

25

8. Fixed point problems We now derive from the above estimates a xed point result on our iteration s heme G . Namely, any iteration on the s heme G onverges, in the k
k℄ de ned in (17), to a fun tion whi h solves a xed point type problem, up to additive onstants. Theorem 26. There exists ? 2 R N and a ZN -periodi Lips hitz semi on ave fun tion ? so that

(73)

G [? ℄ = ? + ? :

Furthermore, given any ZN -periodi semi on ave fun tion 0 , we have that

(74)

lim kG n[0℄

n!+1

? k℄

= 0:

The Lips hitz onstant and the semi on avity modulus of ? are bounded by a onstant depending only on N , CU and CK . Also, ? and ? enjoy the following uniqueness properties: if there exist  2 R N and a Lips hitz ZN -periodi fun tion  so that G [℄ =  + , then  = ? + for some 2 R and  = ? . Finally, if

(75)

Z

RN

e

hK (v ) 

jDK (v)j dv < +1 ;

then the above uniqueness property holds even if we repla e the assumption that  is Lips hitz with the one that it is bounded.

Let us onsider the quotient spa e Y := C 0(R N )=R, that is, let us identify

ontinuous fun tions if and only if they dier by a real number. Then, Y is a Bana h spa e, with norm k
k℄ as de ned in (17): see, e.g., Theorem 3.14-A on page 105 of [Tay58℄ for a proof of this fa t. Note that the semi on avity de nition in (36) passes to the quotient in Y : thus we may take S to be the ( lass of the) fun tions in Y with semi on avity modulus bounded by an appropriately large . It is easily seen that S is

losed in Y . The results in Propositions 5 and 7 also show that G is well de ned on Y and, by Theorem 13, G sends S into itself. Fun tions in S are also uniformly Lips hitz (see Theorem 44), due to their semi on avity properties. Therefore, by Theorem 9 and Theorem 41, we dedu e that G has a unique xed point in S  Y and that (74) holds. Unfolding ba k the quotient spa e Y to the over spa e X , we get (73). Let us now deal with the uniqueness property. Observe that, by hanging variable of integration in (6), we have that   Z 1 G [℄ =  ln dw e

Proof.

hN

RN

hL(x;(w x)=h)+(w) 

26

DIOGO A. GOMES, ENRICO VALDINOCI

and therefore, if G [℄ =  + , it follows that  is Lips hitz, sin e so is K (re all also (75)). Therefore, by Theorem 9, k ?k℄ = kG [℄ G [? ℄  + ?k℄ = kG [℄ G [? ℄k℄    1 C k ?kN℄ k ?k℄ ; showing that k ?k℄ = 0 and thus  = ? + . But then, by Proposition 7,  ? = G [℄  G [? ℄ + ? = 0 :  We remark that a xed point result may also be proved applying the S hauder Fixed Point Theorem (see, e.g., x 9.2.2 in [Eva98℄) to G [℄(x) G [℄(0) ; seen as a ting on the spa e of fun tions with suitably bounded L1 and Lips hitz norm (and su h spa e is onvex and ompa t by the Theorem of As oli). The estimates needed for making su h an argument work are given by Proposition 5 and Theorem 14. This approa h, however, only gives the xed point property (73) but it does not give information either on the onvergen e of the iterations of G or on the uniqueness of the xed points, while Theorems 26 and 41 do. Corollary 27. Fix x 2 R N . Under the assumptions and the notation of Theorem 26, let the measure ? be de ned by

(76)

d? := e

h L(x;v )+? (x+hv ) ? (x) ? 

dv :

Then, ? is a probability measure on R N . Furthermore, if ?

?

:= E [?℄, then

(x) = e L[ ?℄ : ? 

Straightforward from (73).  Other features of the xed points for G will be dis ussed in x 10, where we will relate the term ? in (73) and the measure ? in (76) with an entropy penalized Mather measure. Proof.

9. Convergen e issues In this se tion, we use a variation of the results by [Sou85℄ and [BS91℄ to prove the

onvergen e of the entropy penalized s heme for small time step h. We will prove a rigorous relation between our linear (resp., non-linear) s heme and the heat equation (resp., the Hamilton-Ja obi equation).

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

27

We rst deal with the onvergen e of the L^-s heme as h ! 0, for a xed8  > 0. For this, we stress that L^ depends onN h by writing L^h (the dependen e on  is not expli itly 1 ; 1 written). Consider u 2 W (T ) and assume, for simpli ity,9 that 1=h 2 N . We de ne uh : TN  [0; 1℄ ! R in the following way. Let uh (x; 0) := u(x) and then, iteratively, uh(x; t) := L^t ih [uh(
; ih)℄(x) if t 2 (ih; (i + 1)h℄, for i = 0; : : : ; (1=h) 1. Then, we have the following onvergen e result: Theorem 28. Let K (v ) = jv j2 and R e jvj jv j2 dv R R (77) a := 2 R e jvj dv : Let u = u(x; t) be the unique (vis osity) solution of (78) t u(x; t) = a
x u(x; t) in TN  (0; 1℄, with u(x; 0) = u(x). Then, uh onverges uniformly to u. Proof. The proof is a minor variation of the one given in Theorem 2.1 of [BS91℄ (see also [Sou85℄). Sin e, in our ase, the arguments involved are elementary, we provide full details, for the reader's onvenien e. N We note that, if t 2 (ih; (i + 1)h℄ and x 2 T juh(x; t)j  sup juh(x; ih)j; x2T and so sup juh(x; t)j  sup juh(x; t)j 


 kukL1(T ) ; x2T ; t2(ih;(i+1)h℄ x2T ; t2((i 1)h;ih℄ that is kuhkL1(T [0;1℄)  kukL1(T ) : Therefore, we may de ne (79) u (x; t) := lim inf uh (y; s) : (y;s)!(x;t); h!0 We show that u is a vis osity supersolution of (78). For this, x (x0 ; t0) 2 TN  (0; 1) and let  beNa smooth fun tion, so that u  has a stri t minimum at (x0 ; t0 ). Let (xh; th) 2 T  [0; 1℄ be minimizers for the fun tion uh . We show that we may

hoose a sequen e hk ! 0 su h that (80) lim (xh ; th ) = (x0 ; t0) : k!+1 2

N

2

N

N

N

N

N

N

N

k

k

8The ase when  ! 0 too is a tually easier and an be dealt with by a modi ation of the proof of

Theorem 28. In this ir umstan e, Theorem 28 holds with  = 0 in (78), namely u(x; t) = u(x). We omit the details of the proof, sin e it losely follows the one of Theorem 28. 9The reader may onvin e herself that, with minor modi ations, it would be possible to onsider more general partitions of an interval, instead of the uniform h-mesh that, for simpli ity, we deal with.

28

DIOGO A. GOMES, ENRICO VALDINOCI

For this, let (yk ; sk ) ! (x0 ; t0) and hk ! 0 be a sequen e so that u (x0 ; t0 ) = lim uh (yk ; sk ) ; k!1 a

ording to (79). We may also assume, up to subsequen es, that (xh ; th ) ! (x; t) 2 N T  [0; 1℄ as k ! +1. Then, (80) is proved if we show that (81) (x; t) = (x0 ; t0 ) : For this, let us observe that, by onstru tion, (u )(x0; t0) = klim (u )(yk ; sk ) !1 h  klim (u )(xh ; th ) !1 h  (u )(x; t) : Therefore, sin e (x0 ; t0) was assumed to be a stri t minimum for u , the above estimate proves (81) and, then e, (80). We now onsider the sequen e hk in (80), and we denote h := hk ! 0 for short. The fa t that uh uh(xh ; th )   (xh ; th ) ; together with Propositions 16 and 17, implies that L^[uh(
; t)℄(x) uh(xh; th)  L^[(
; t)℄(x) (xh; th) for any x 2 TN , any t 2 [0; 1℄ and any  > 0. Then e, if ih 2 f0; : : : ; (1=h) 1g is so that th 2 (ihh; (ih + 1)h℄ and we set h := th ih h ; we have that lim h = 0 h!0 and 0 = L^ [uh(
; th h )℄(xh) uh(xh; th) (82)  L^ [(
; th h)℄(xh) (xh; th) : Also, sin e  is smooth, we infer from Proposition 23 that L^ [(
; th h)℄(xh) (xh; th) = a
(x ; t )  (x ; t ) : (83) hlim x 0 0 t 0 0 !0 h By olle ting (82) and (83), we get that a
x (x0 ; t0 ) t (x0 ; t0 )  0 if u  has a minimum at (x0 ; t0 ). That is, u is a vis osity supersolution of (78). Analogously, if we de ne (84) u+ (x; t) := lim sup uh (y; s) ; k

k

k

k

h

h

h

(y;s)!(x;t); h!0

we have that u+ is a vis osity subsolution of (78).

k

k

k

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

29

Also, if s 2 (0; h), juh(y; s) u(x)j = jL^s[u℄(y) u(x)j R jj ju(y + sv) u(y)j dv + ju(y) u(x)j e R  R jj dv e R p   kDuk 1 s + jx yj ; s v2 

N

s v2 

N

L

and so

We laim that (85)

u (x; 0) = u+ (x; 0) = u(x) :

(resp., u+) is lower semi ontinuous (resp., upper semi ontinuous). For proving this, x z := (x; y) 2 TN  [0; 1℄ and onsider a sequen e zj 2NTN  [0; 1℄, so that zj ! z as j ! +1. For ea h j 2 N , let also hj;k > 0 and j;k 2 T  [0; 1℄ be so that lim hj;k = 0 ; k!lim+1 j;k = zj and k!+1 u (zj ) = k!lim u (j;k ) ; +1 h for any xed j 2 N . Fix now  > 0. Let k0(; j ) be so that juh (j;k) u (zj )j   for any k  k0(; j ). Let also k1 (j ) su h that jhj;kj + jj;k zj j  1j ; for any k  k1(j ). We de ne n o k? (; j ) := max k0 (; j ) ; k1 (j ) ; hj := hj;k (;j ) and j := j;k (;j) : Then, by onstru tion, hj ! 0 and j ! z as j ! +1. Therefore, u (z )  j !lim u ( )  j !lim u (zj ) +  : +1 h j +1 Sin e  is arbitrary, we on lude that u (z )  j !lim u ( zj ) ; +1 for any sequen e zj ! z, and thus u is lower semi ontinuous. Analogously, one sees that u+ is upper semi ontinuous, thus on rming (85). u

j;k

j;k

?

?

j

30

DIOGO A. GOMES, ENRICO VALDINOCI

Thanks to (85), the Comparison Prin iple (see, e.g., Theorem 8.2 in [CIL92℄) yields that u+  u . Sin e, by (79) and (84), the opposite inequality also holds, we obtain that u = u+ =: u and thus u is the unique10 ( ontinuous vis osity) solution of (78). To omplete the proof of the desired result, we show the uniform onvergen e of uh to u, by arguing as follows. If, by ontradi tion, uh did not onverge uniformly to u, there would exist  > 0, and zh 2 TN  [0; 1℄ so that juh(zh) u(zh)j   for in nitely many h ! 0. Then, either uh(zh) u(zh)   or u(zh) uh(zh)   for in nitely many h ! 0. Let usN assume the latter (the other ase being analogous) and assume also that zh ! z 2 T  [0; 1℄ for this set of h's. Then, u(z ) = u (z )  hlim u (z ) !0 h h  u(z)  ; whi h is a ontradi tion.  We now deal with the s heme G as h ! 0. We de ne G^ as in (45). We will x  > 0 and take h ! 0. We will expli ilty write G^h to stress the dependen e on h in G^. 3 N We x w 2 C (T ), we assume11 that 1=h 2 N and we de ne wh : TN  [0; 1℄ ! R as follows: rst, we set wh (x; 0) := w(x) and then, re ursively, wh (x; t) := G^t ih [wh(
; ih)℄(x) if t 2 (ih; (i + 1)h℄, for i = 0; : : : ; (1=h) 1. Then, the following onvergen e result holds: Theorem 29. Let a be as in (77). Then, as h ! 0, wh onverges uniformly to w , where w satis es

(86)

t w(x; t) + K (Dx w(x; t)) in the vis osity sense, with w(x; 0) = w(x).

= a
x w(x; t)

As in Theorem 28, the proof is a variation of the arguments in [Sou85℄ and [BS91℄. First, we observe that (87) kwhkL1(T [0;1℄)  kwkL1(T ) ;

Proof.

N

N

10The results we use about vis osity sub/super/solutions may be found, for instan e, in [CIL92℄. 11For the sake of simpli ity, we assumed w to be smooth enough, in order to use (46) in estimate (88)

here below. We remark that, by applying the lower order arguments in Proposition 24 dire tly to w, less smoothness may be required.

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

31

due to (47). Thus, we may de ne w (x; t) := lim inf wh (y; s) : (y;s)!(x;t); h!0 We show that w is a vis osity supersolution of (86). For this, x (x0 ; t0) 2 TN  (0; 1) and let  be a smooth fun tion, so that w  has a stri t minimum at (x0; t0 ). Let (xh; th) 2 TN  [0; 1℄ be minimizers for the fun tion wh . As shown in (80), we may and do assume, up to a subsequen e, that (xh ; th) ! (x0 ; t0). Sin e wh wh(xh; th)   (xh; th), by exploiting Proposition 6, we gather that G^[wh(
; t)℄(x) wh(xh; th)  G^[(
; t)℄(x) (xh; th) for any x 2 TN , any t 2 [0; 1℄ and any  > 0. Then e, if ih 2 f0; : : : ; (1=h) 1g is so that th 2 (ihh; (ih + 1)h℄ and we set h := th ihh, 0 = G^ [wh(
; th h)℄(xh) wh(xh; th)  G^ [(
; th h)℄(xh) (xh; th) : Then, by means of (46), we have that ^ lim G [(
; th h)℄(xh) (xh; th) h

h

Therefore,

=

h

h K (Dx (x0 ; t0 )) + a
x (x0 ; t0 ) t (x0 ; t0 ) : h!0

a
x (x0 ; t0 ) t (x0 ; t0 )  K (Dx (x0 ; t0 ))  has a minimum at (x0 ; t0 ). That is, w is a vis osity supersolution of (86).

if w Analogously, if we de ne

w+ (x; t) :=

lim sup

(y;s)!(x;t); h!0

wh (y; s) ;

we have that w+ is a vis osity subsolution of (86). Also, if s 2 (0; h), (88) jG^s[w℄ wj  C h ; thanks to (46), where C depends only on kwkC (T ), and so jwh(y; s) w(x)j = jG^s[w℄(y) w(x)j (89)  jG^s[w℄(y) w(y)j + jw(y) w(x)j  C h + jx y j : This implies that w (x; 0) = w+(x; 0) = w(x). Thus, by using the Comparison Prin iple and arguing as in the proof of Theorem 28, we obtain that wh onverges uniformly to w = w+.  3

N

32

DIOGO A. GOMES, ENRICO VALDINOCI

Theorem 29 may also be adapted to obtain a Hamilton-Ja obi equation with a potential term, a

ording to the following s heme, related to (44). We de ne  ln Ge[℄ := G [℄ + N 2 h ^ = G [℄ hU (x) : We then de ne wh(x; 0) := w(x) and then, re ursively, wh (x; t) := Get ih [wh(
; ih)℄(x) if t 2 (ih; (i + 1)h℄, for i = 0; : : : ; (1=h) 1. Then, the following onvergen e result holds: Theorem 30. Let a be as in (77). Then, as h ! 0, wh onverges uniformly to w ,

where w satis es

t w(x; t) + H (Dx w(x; t); x) in the vis osity sense, with w(x; 0) = w(x).

Proof.

= a
x w(x; t)

One sees by indu tion that sup jwh(x; t)j  kwkL1(T ) + jh kU kL1(T N

2TN t2[0;jh℄

N

x

);

for j = 0; : : : ; 1=h. From this a uniform estimate as in (87) follows. The proof of Theorem 29 may then repeated verbatim, but substituting K (
) with H (
; x), G^ with Ge, and taking C in (88) and (89) to be also depending on kU kL1(T ) .  N

10. Entropy penalized Mather measures In Mather's theory (see,N e.g., [MF94℄ and referen es therein), one looks for probability N measures  on T  R that minimize the a tion Z (90) L(x; v )d(x; v ) T R and satisfy the \holonomy" (or \ ow invarian y") onstraint Z (91) ['(x + hv) '(x)℄ d(x; v) = 0 ; N

N

TN RN

for all ' 2 C (R N ). In this se tion, we dis uss an entropy penalized version of Mather's problem, and we present a solution in terms of xed points of the operator G . The entropy penalized Mather problem onsists in minimizing Z  (92) L(x; v )d(x; v ) + S [℄; h T R in whi h Z (x; v ) dxdv S [℄ = (x; v ) ln R T R R (x; w )dw N

N

N

N

N

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

33

is the \entropy term". The minimization in (92) is performed over the spa e of probability densities onTN  R N  Z 1 N N (93)  2 L (T  R ) ;   0 a:e: ; d = 1 T R that satisfy12 the onstraint (91). In this setting, we study the minimizers of the fun tional in (92) in the spa e given by (93) under the onstraint in (91), that is the \entropy penalized Mather problem". This problem is the stationary version of the problem studied in [Ana04℄, whi h avoids the use of measures on path spa es. We will show that these penalized minimal measures always exist (see Theorem 32 here below) and that they are unique (see Theorem 36). An expli it formula for this measure will be provided by using the xed point stru ture of NG (see formula (108)). In proving these results, an important r^ole is played by the T -proje tion of penalized measures and on their analyti bounds (see Propositions 34 and 35). For this, we need the following preliminary result: Lemma 31. Let  2 W 1;1 (TN ) satisfy (94) G [℄ =  + : 1 N Then, there exists  2 L (T ) satisfying (95)   0; N

(96) and

Z

RN

(x hv )e

hL(x hv;v )+(x) (x hv )  

Z

(97)

N

dv

= (x)

= 1: Proof. The following elementary observation will be used throughout this proof: if  2 L1 (TN ) and  2 R N , then Z Z =  (x +  ) dx : T T Further, we note that Z dv = 1 ; (98) e R for any x 2 TN , due to (94). ZNow, de ne for # 2 L1 (TN ), dv : (99) F [#℄(x) := #(x hv)e R Let us observe that (98) and (99)Zimply that Z (100) F [#℄ = # : TN

N

(x) dx

N

hL(x;v )+(x+hv ) (x)  

N

hL(x hv;v )+(x) (x hv )  

N

12As

TN

TN

ustomary, with a slight abuse of notation, given a fun tion d(x; v ) = (x; v ) dx dv .



2

L1

(T R), we denote 

34

DIOGO A. GOMES, ENRICO VALDINOCI

Let now 0 := 1 and, re ursively, (101) n+1 := F [n ℄ ; for any n 2 N . Note that n is ZN -periodi for any n 2 N , be ause so are 0, U and . Analogously, n  0, be ause so is 0 . Then e, by (100), we have that dn(x) = n(x) dx N is a sequen e of probability mesures on T , that is Z (102) dn = 1 ; T for any n 2 N . We may thusN suppose that, up to subsequen e, dn weakly onverges to a Radon measure d on T (see, e.g., page 55 of [EG92℄). Also, d is a probability measure, sin e TN is ompa t (see, e.g., Theorem 1-(ii) on page 54 of [EG92℄). Our obje tive now is to show that dN is absolutely ontinuous with respe t to the Lebesgue measure. For this, let  2 C (T ) be so that 0    1. Then, exploiting (101), we get that Z  (x) dn+1 (x) dx N

= (103)

=

N ZT

N ZT

 (x) n+1 (x) dx  (x) F [n ℄(x) dx

N ZT Z

=  (x) n (x hv ) e dv dx : T R We now denote by i some appropriate positive quantities, whi h depend only on N , , L, ,  and h (but not on  ). Then, by the fa t that K is superlinear and (102), we dedu e that Z N n (x hv ) e h dv N

hL(x hv;v )+(x) (x hv )  

N

= =

RN

Z

n ( w ) e

RN X Z

N k2ZN T

 1  3

hK (v ) 

n ( w ) e

X Z

k2Z

Z

N

TN

hK ((x w)=h) 

T

N

hK ((x w k)=h) 

n (w) e

2 jkj

dw

dw

n (w) dw

= 3 : This estimate, together with (103), gives that Z Z  (x) dn+1 (x)  4 TN

dw

TN

 (x) dx :

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

35

Taking now the limit as n ! +1, using the weak onvergen e of n, we gather that (104)

Z

TN

 (x) d(x)

 4

Z

TN

 (x) dx ;

for any  2 C (TN ). Take now any measurable set A  TN , with small Lebesgue measure, say jAj  , for  > 0. We also denote the -measure of A by (A). By standard results on the Radon measure approximation with open and ompa t sets (see, e.g., Theorem 4 on page 8 of [EG92℄), we have that there exist a ompa t set K and an open set U so that K  (A)   (K ) + 

 A  U;

and jU j  jAj +   2 :

Exploiting the lassi al Urysohn's Lemma (see, e.g., Theorem 2 on page 15 of [DS58℄), we seeN that there exists a fun tion  2 C (TN ) so that 0    1, (x) = 0 for any x 2 T n U and  (x) = 1 for any x 2 K . A

ordingly,  (A)

 Z(K ) +    (x) d(x) +  TNZ

 4 (x) dx +  T  4 jU j +   (24 + 1)  ; N

by means of (104). Then e, d is absolutely ontinuous with respe t to the Lebesgue measure. Thus, we write d(x) = (x) dx ;

with (x) 2 L1(TN ). Sin e d is, by onstru tion, a probability measure on TN , we have that  satis es (95) and (97).

36

DIOGO A. GOMES, ENRICO VALDINOCI

Moreover, given any  2 C (TN ), we dedu e from (103) and the weak onvergen e of dn that Z  (x) (x) dx ZT

N

=

TN

 (x) d(x)

= n!lim+1 = n!lim+1 = n!lim+1 = n!lim+1 = =

Z

 (x) dn+1 (x)

N ZT Z

N N ZT ZR

ZR ZT N

N

N N Z R T

Z

N N ZR ZT N N ZR ZT

 (x) n (x hv ) e

hL(x hv;v )+(x) (x hv )  

 (y + hv ) n (y ) e

hL(y;v )+(y +hv ) (y )  

 (y + hv ) e

 (y + hv ) e

hL(y;v )+(y +hv ) (y )  

hL(y;v )+(y +hv ) (y )  

 (y + hv ) (y ) e

dv dx

dy dv

dn (y ) dv

d(y ) dv

hL(y;v )+(y +hv ) (y )  

dy dv

=  (x) (x hv ) e dx dv : R T Sin e  here above is arbitrary, we get that  satis es (96), as desired.  Su h  plays a de isive r^ole in the onstru tion of penalized Mather measures, as we are now going to show. We also remark that the regularity and uniqueness of the fun tion  will be dealt with in Propositions 34 and 35 below. Let us now deal with the entropy penalized Mather measures: Theorem 32. Let  2 W 1;1 (TN ) satisfy (105) G [℄ =  + : Let  2 L1 (TN ) be so that13   0, Z dv = (x) (106) (x hv )e N

and

hL(x hv;v )+(x) (x hv )  

N

hL(x hv;v )+(x) (x hv )  

RN

Z

(107) Let

(108)

TN

(x; v )

(x) dx

:= (x)e

= 1:

hL(x;v )+(x+hv ) (x)  

:

Then,  minimizes the fun tional (92) over the spa e (93) under the onstraint (91). 13The existen e of a  satisfying (106) and (107) is assured by Lemma 31 here above.

(and essential uniqueness) of  and  satisfying (105) is assured by Theorem 26.

The existen e

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

37

The result in Theorem 32 here above thus says that d(x; v ) = (x)e dx dv is an \entropy penalized Mather measure". The uniqueness of the measure  will be proved in Theorem 36. Proof. Exploiting (106) and (98), we dedu e that Z '(x + hv ) d(x; v ) Remark.

hL(x;v )+(x+hv ) (x)  

= = = =

N N ZT R

N N ZT R N N ZT R N ZT

'(x + hv ) (x) e

hL(x;v )+(x+hv ) (x)  

'(x) (x hv ) e

hL(x hv;v )+(x) (x hv )  

dx dv dx dv

'(x) (x) dx

N N ZT R

'(x) (x) e

hL(x;v )+(x+hv ) (x)  

dx dv

= '(x) d(x; v ) ; T R for any fun tion ' 2 L1(TN ), whi h shows that the onstraint (91) is satis ed by . In addition, from (108) and (98),Z we get that (109) (x; v ) dv = (x) ; R for any x 2 TN . Therefore, by (107), Z Z d(x; v ) = (x) dx = 1 ; T R T and so  belongs to the spa e given by (93). To prove the minimizing property of  it suÆ es to show that for any other density ~ in (93) that satis es (91) and any 0Z    1, one has that the fun tion  (110) I [ ℄ = Ld + S [ ℄; h T R with  := (1  ) +  ~, is onvex and that I 0[0℄= 0. Denote by _  := ~  Then,   Z  
_  I 0 [ ℄ = Ld_  +
ln R  h  dw N

N

N

N

N

N

N

TN RN

N

Z





 R 
(111) _  dw ; h  dw where we have used theZ fa t that Z _ t = (~ ) = 1 1 = 0 : TN RN

TN RN

38

DIOGO A. GOMES, ENRICO VALDINOCI

Further, noti e that, sin eZ both  and ~ areZprobability measures,   R (~ )dw dx dv dw = (112)

=

RN TN RN RN R Z Z N dv RR  RN TN RN dw  Z Z



(~

N ZT

RN

(~

)dw dx

)dw dx

= (~ ) T R = 1 1 = 0: Moreover, making use of Z(109) and (108), we dedu e that hL d(~ ) N

N

TNZRN

 (~ ) dx dv dw N TN RN R   Z  ( x; v ) hL(x; v ) +  ln R TN RN RN (x; w ) dw
Z(~(x; v) (x; v)) dx dv  (x; v ) hL(x; v ) +  ln (x) TN RN
(~(x; v) (x; v)) dx dv

+

=

(113)

= = =

Z

ln R



TN RN



(x + hv ) + (x) + 


Z(~(x; v) (x; v)) dx dv 

+



(x) (x + hv ) d~

Z 



(x + hv ) (x) d

Z

Z



+ d~ d = 0 + 0 + (1 1) = 0 ; where we have also used again that both  and ~ are probability measures satisfying (91). Colle ting (111), (112) and (113), it follows that I 0 [0℄ = 0: Thus, to prove that  is minimal, it is suÆ ient to show that I 00 [ ℄  0:

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

39

To he k this, we take a further derivative in (111), thus obtaining
2 R R Z _ R _  dw  _  dw h 00 _ 2 + (114) I [ ℄ = 2 

=



Z

TN RN

TN RN





_  

R

 dw

R R

_  dw  dw

2

 dw


2

 0: 

As a onsequen e of Theorem 32, we provide the following variational hara terization for the xed point problem of Theorem 26. Corollary 33. Let  2 W 1;1 (TN ) and  2 R satisfy G [℄ =  + . Then, the in mum of

Z

TN RN

h L(x; v ) d(x; v ) + S [℄

when  is in the spa e given by (93) and satis es the onstraint (91), is equal to  (whi h, in turn, is equal to G [℄ ).

Theorem 32 says that the above in mum is attained at the measure  given by (108), for whi h, by a dire t omputation (and re alling (91)), we have: Z h L(x; v ) d(x; v ) + S [℄

Proof.

= = =

N N ZT R 

N N ZT R 

TN RN



(x) (x + hv ) +  d(x; v ) 

(x) (x + hv ) d(x; v ) + 

:



We now further investigate some properties of the fun tion , whi h was introdu ed in Lemma 31 and played an important r^ole in the onstru tion of the penalized Mather measure  in Theorem 32. First of all, we have the following regularity result: Proposition 34. Consider the setting of Lemma 31. Then,  2 L1 (TN ) and there   exist onstants 1 and 2 su h that

(115)

for any x 2 TN . Moreover, if

(116)

0 < 1  (x)  2

Z

RN

e

h K (v ) 

jDK (v)j dv < +1 ;

then,  2 W 1;1(TN ) and there exists a onstant 3 su h that (117) kk 1;1 N   : W

(T )

3

40

DIOGO A. GOMES, ENRICO VALDINOCI

1  i  3, depend only on N , , L, ,  and h. Proof. In the ourse of this proof we denote by  a onvenient positive quantity, possibly depending only on N , , L, ,  and h, and whi h may be dierent at dierent steps of the omputation. By hanging variable in (96), we have that Z 1 (118) (x) = N  (w )e dw : h R Therefore, re alling (95) and (97), Z 1 dw (x)  N  (w )e h The onstants i , for

hL(w;(x w)=h)+(x) (w)  

N

(119)

 

hL(w;(x w)=h)+(x) (w)  

N Z T

TN

(w) dw

= : On the other hand, sin e L is superlinear in v, one also dedu es from (118) and (97) that X Z (x)   (w) e jkj dw N k2ZN T

 

=

Z

:

TN

(w) dw

This and (119) yield (115). Furthermore, by dierentiating (118), we get that jDZ (x)j i h  x w     (w )e + jD(x)j dw DK h hL(w;(x w)=h)+(x) (w)  

ZR

N

  e R  ; N

h K (v ) 

h

i

jDK (v)j + 1 dv

thanks to (115) and (116). This and (115) yield (117).  We now apply the Hopf-Cole transform method to , that is, we de ne (120) & (x) :=  ln (x) : Note that, in the setting of (13), we have that  = E [& ℄. Also, (120) is a bona de de nition, sin e  > 0 by (115). Moreover, & is ZN -periodi , sin e so is . We now show that & is the solution of an entropy penalized s heme analogous to (6). Namely, let (x) (x + hv )  L(x; v ) := L(x + hv; v ) + h

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

and, for any fun tion ', onsider (121)

G ['℄(x) :=  ln

Z

RN

hL(x;v )+'(x+hv ) 

e

41



dv :

The reader may ompare (6) and (121) and note that the fun tion & +  is a solution to a time-reversed G s heme with Lagrangian L(x + hv; v), that is & is the dieren e between a forward and ba kward xed point. In this framework, the following result holds:

Consider the setting of Lemma 31 and let & be as in (120). Then, & 2 L1 (TN ) and its L1 -norm is bounded by a quantity depending only on N , , L, ,  and h. Also, G [& ℄ = & . Moreover, if (116) holds, then & 2 W 1;1(TN ), its W 1;1-norm is bounded by a quantity depending only on N , , L, ,  and h. Also, in this ase, the fun tion  in Lemma 31 is unique. Proposition 35.

The L1-bound (resp., the W 1;1-bound) plainly follows from (120) and (115) (resp., and (117)). The fa t that & is a xed point of G is a onsequen e of (96). Let us now show the uniqueness of . Assume that ^ also ful lls the thesis of Lemma 31 and let &^ be so that ^ = E [^& ℄. By what we have just proved, we have that both & and &^ are uniformly Lips hitz and that they are both xed points of G . Then, using Theorem 9 (applied to14 G instead of G ), we have that k& &^k℄ = kG [& ℄ G [^& ℄k℄  (1 C k& &^k℄) k& &^k℄ ; for some C > 0. This shows that & = &^ + , for some 2 R . But then, re alling (97), Proof.

Z

=

N ZT

TN

e

& 

^ =

=

Z

Z

TN

TN

e

=1 &^ 

=e



Z

TN

e

& 

;

then e = 0 and, therefore, & = &^. Consequently,  = ^.  We are now in position to prove the uniqueness of the penalized Mather measure

onstru ted in Theorem 32:

Assume (116). Then, the measure  onstru ted in Theorem 32 is the unique minimizer of the fun tional in (92) over the spa e given by (93) under the

onstraint (91). Theorem 36.

that L is superlinear, sin e so is L. Also, L W 1;1 (TN RN ), be ause  No further regularity of L was used in the proof of Theorem 9. 14Note

2



2

W 1 ;1

(TN ).

42

DIOGO A. GOMES, ENRICO VALDINOCI

Assume that ~ is also a minimizer. Let  := (1  ) +  ~, for  2 [0; 1℄. Let also _  := ~  and Z (122) ~(x) := ~(x; v ) dv : R These de nitions and (109) give thatZ Z _   dw  _  dw R R (123) = ~ ~ : Let now I be as in (110). Sin e both  and ~ are minimizers for the fun tional in (92), we have that both  = 0 and  = 1 minimize I . On the other hand, we know from the proof of Theorem 32 that I is onvex. Therefore, I must be onstant for any  2 [0; 1℄. In parti ular, I 00[0℄ = 0. By (114), this says that  R 2 R Z _  R  dw  R _  dw : 0= R 2 T R 
R  dw This and (123) yield that (124)  ~ = ~  almost everywhere in TN  RN . Then e, in the light of (108), we on lude that ~(x) = ~(x; v ) e ; and therefore (125) ~(x hv ) = ~(x hv; v ) e : Thus, given any ' 2ZC (TZN ), we gather from (91), (122) and (125) that dv dx '(x) ~(x hv )e Proof.

N

N

N

N

N

N

N

N

hL(x;v )+(x+hv ) (x)  

hL(x hv;v )+(x) (x hv )  

ZT ZR N

= = = = = =

N

N N ZT ZR N N ZR T N N ZT R N N ZT ZR

ZT

N

TN

hL(x hv;v )+(x) (x hv )  

RN

'(x) ~(x hv; v ) dv dx '(y + hv ) ~(y; v ) dy dv '(y + hv ) d~(y; v ) '(y ) d~(y; v ) '(y ) ~(y; v ) dv dy

'(y ) ~(y )dy :

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

43

Sin e ' here above is arbitrary, this means that ~ satis es (96). Also, ~ satis es (95) and (97), due to (122) and the fa t that ~ is a probability measure. Then, by the uniqueness result in Proposition 35, we dedu e that ~ = . Then, (124) yields that ~ = .  11. Convergen e to Mather measures In this se tion, we use duality methods to study the entropy penalized Mather problem and its onvergen e as =h ! 0 (that is, when the entropy vanishes). We will prove a variational identity whi h relates the a tion fun tional with the xed points of the non-linear s heme (see Theorem 37). This identity turns out to be useful in showing that the minimal a tion value, the xed point of the s heme and the entropy penalized Mather measure onverge to the analogous features in Mather theory as the entropy vanishes (see Theorems 38, 39 and 40). Theorem 37.

We have the following identity:

(126)

inf =

Z

hLd + S [℄

inf sup  ln 2C (T )

Z

x2TN

N

RN

e

hL(x;v )+ (x+hv ) 

(x)

dv ;

where the rst in mum is taken over all measures  in the spa e given by (93) under the onstraint in (91). Furthermore, if  2 W 1;1(TN ) and  2 R satisfy15 G [℄ =  + , then  is an optimal fun tion for the right-hand-side of (126) and  is equal to the quantity given in (126). Proof.

(127)

Clearly,16

inf sup  ln 2C (T ) N

Z

x2TN

 sup  ln

Z

x2TN

RN

e

RN

hL(x;v )+ (x+hv ) 

e

hL(x;v )+(x+hv ) (x) 

(x)

dv

dv =  :

On the other hand, from (6), Z dv inf sup  ln e 2 C (T ) x2T R (128) = 2inf sup G [ ℄ : C (T ) x2T Thus, take any fun tion 2 C (TN ) and hoose a point x0 in whi h  have a ^ ^ maximum. Let  :=  (x0) + (x0 ), so that   . Therefore, in the light of Propositions 6 and 7, G [℄ (x0 ) + (x0 ) = G [^℄  G [ ℄ : N

hL(x;v )+ (x+hv ) 

N

(x)

N

N

N

15Again, we re all that the existen e of  and  as desired follows from Theorem 26. 16Note that, in general, a proof of statements as the ones in Theorem 37 may require the use of the

Legendre-Fen hel Duality Theorem or some advan ed te hnology. However, in our ase, as we know the optimal measure by Theorem 32, we manage to give an elementary proof.

44

DIOGO A. GOMES, ENRICO VALDINOCI

Consequently, sup

G [ ℄  (x0) G [ ℄(x0 )  (x0 ) G [℄(x0 ) =  :

x2TN

Thus, by taking the in mum over all fun tions 2 C (TN ), we dedu e from the latter estimate, (128) and (127) that inf sup  ln 2C (T ) N

x2TN

Z

RN

e

hL(x;v )+ (x+hv ) 

(x)

dv =  :

Then, the laim in (126) follows from Corollary 33.  At this point, it is onvenient to re all that the Mather problem, whi h onsists in minimizing (90) under the onstraint in (91), also has a dual problem. Namely, if H 0 is the in mum of Z TN RN

Ld ;

over all the probability measures on TN  R N satisfying the onstraint in (91), one has that (129) inf sup [hL(x; v) + (x + hv) (x)℄ = hH 0 : 2C (T ) N

(x;v)2TN RN

We refer to [Gom05℄ for further details about this fa t. We just mention here that H 0 (possibly up to a sign onvention) is what is alled the \ee tive Hamiltonian" in the literature. Furthermore, there exists an optimal 0 su h that   (130) inf hL(x; v ) + 0 (x + hv ) 0 (x) hH 0 = 0; v2R N

for all x 2 TN (see, e.g., [Gom05℄). For onvenien e, we de ne ;h :=  and ;h := , where  and  are as in Theorem 37, and we set Z  H =h := inf Ld + S [℄;  h

where the in mum is taken over all measures  belonging to the spa e given by (93) and that satisfy the onstraint (91). We will see in Theorem 38 below that H =h may be seen as an approximate ee tive Hamiltonian, whi h will tend to H 0 when =h ! 0. It follows from Theorem 37 that  (131) H =h = ;h : h Theorem 38.

as =h ! 0.

We have

H =h ! H 0 ;

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

45

We will make use of the following elementary17 fa t: if f 2 C (R N ) is superlinear,

Proof.

then (132) Also, sin e

lim

Z

!0+

R

N

e

f (v ) 

dv



=e

inf RN f :

lim H =h = lim H = lim H ; !0  !0 =1 we may and do x h := 1 in this proof. Let 0 be as in (130). By (131) and Theorem 37, we have that Z H   sup  ln dv : e x2T R Therefore, using (132) and (130), (133) lim H   H 0 : !0 To prove the reverse inequality, we observe that ;1 is a uniformly Lips hitz fun tion, thanks to Theorem 26, and, therefore, we may take the in mum on the right hand side of (126) over the spa e of Lips hitz fun tions with Lips hitz onstant bounded by some suitably large number . In parti ular, by Theorem 37, xed any Æ > 0, there exists a in su h a Lips hitz spa e, so that Z H  + Æ = ;1 + Æ  sup  ln dv : e (134) x2T R Given su h a , we de ne (x ; v ) to be su h that [L(x ; v ) + (x + v ) (x )℄ = sup [L(x; v) + (x + v) (x)℄ : (x;v)2T R We denote by  the above supremum. Let us now lo alize v : sin e is Lips hitz, the de nition of (x ; v ) implies that p K (v )  K (0) +  N =h!0+

+

+

L(x;v )+0 (x+v ) 0 (x) 

N

N

+

L(x;v )+ (x+v ) 

N

N

N

17For

Take R

( x)

N

ompleteness, we give a sket h of the proof of (132). Let f (v) v if R0 , then Z Z Z

v f f f e  e  + e 

onst ( R N e  + N e N 

j j

v

j j 

R0

, for > 0.



inf

R



inf

j j

jvjR



jvjR

In parti ular, if R := 1= and  is small enough, Z f

onst  e  N 

R

Ne

inf



f

R 2

):

;

whi h gives one inequality in (132). For the other one, x Æ > 0. Then, there exists a ball B so that f (v ) inf f + Æ for any v B . Then, 

2

Z

R

N

e

f  

Z

B

e

then take the -power to both sides, then send 

f  

!

e

inf f +Æ



j

Bj ;

0 and then Æ

!

0.

46

DIOGO A. GOMES, ENRICO VALDINOCI

and so, sin e K is superlinear, we have that jv j  ^ , for a suitably large universal number ^ . Let now ~ := sup jDK (v)j jvj^ +2 and onsider the universal quantity  :=  + ~ : Then, if jv v j   < 1, we have that jvj  ^ + 1 and so j + L(x ; v) + (x + v) (x )j (135)  jK (v ) K (v )j + j (x + v ) (x + v )j    ; for any v 2 B(v ). By olle ting the estimates in (134) and (135), we obtain Z H  + Æ  sup  ln dv e x2TN

B (v



)

  ln jB(v )j e

 

L(x;v )+ (x+v ) 



(x)



=  + onst  ln( onst ) : Sin e, by (129) and the de nition of  , we have that H0   ; it follows from the above omputation that Æ lim H   H 0 : !0 Then e, sin e Æ is arbitrary, lim H   H 0 : !0 This and (133) omplete the proof of the desired result.  Having ompleted the proof of the onvergen e of the ee tive Hamiltonian, we now address the onvergen e of the \normalized" fun tion (136) (x) := ;h(x) ;h(0); as  ! 0, for a xed h > 0: Theorem 39.  ! 0 uniformly as  ! 0+ , in whi h 0 2 W 1;1 (TN ) is a solution +

+

to (130). Also, 0 is semi on ave. The Lips hitz onstant and the semi on avity modulus of 0 are bounded by a onstant depending only on N , CU and CK .

Along this proof, we will make expli it the dependen e on  of G by denoting it by G and we de ne G0 to be the operator G0 [ ℄(x) := infv hL(x; v) + (x + hv): Proof.

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

47

Noti e that, from (131) and Theorem 38, (137) lim ;h = h H 0 : !0 Moreover, due to Theorem 26, we know that  is uniformly Lips hitz ontinuous and 1 ; 1 semi on ave and so, by (136),  onverges uniformly to a suitable 0 2 W (TN ), and both the Lips hitz onstant and the semi on avity modulus of 0 are universally bounded. What is more, by Proposition 8, k 0kL1(T ) = 0 : (138) lim kG[℄ G[0 ℄kL1(T )  lim !0 !0 Noti e also that, by (132) and the ontinuity of 0, we have that (139) lim G[0 ℄(x) = G0[0 ℄(x) ; !0 for any x 2 TN . Therefore, we use (137), (138) and (139) to dedu e that h H 0 = lim ;h !0 = lim G [ ℄  !0       = lim G [  ℄ + G [  ℄ G [  ℄  +      0 0 0  !0  0 = G0 [0℄ + 0 0 + 0 = G0 [0℄ 0 ; and so 0 is a solution of (130).  We now onsider the penalized Mather measures  := ;h onstru ted in Theorem 32 and we prove its onvergen e as  ! 0+, for a xed h > 0: Theorem 40. As  ! 0,  onverges weakly to a (dis rete) Mather measure, that is, a probability measure on TN  R N that minimizes +

N

+

+

N

+

+

+

+

Z

TN RN

Ld;

under the onstraint in (91). Moreover, the support of 0 is ompa t.

We will drop the h dependen e in some of the indi es of this proof, sin e no

onfusion an arise. We re all that, by (108),  =  (x)e with Z (x; v ) dv =  (x) : R Furthermore,  ! 0 uniformly, where 0 satis es (130), and  ! hH 0, a

ording to Theorems 38 and 39. Proof.

hL(x;v )+ (x+hv )  (x)  

N

48

DIOGO A. GOMES, ENRICO VALDINOCI

In parti ular, kkL1 and jj are bounded uniformly in . Therefore, from the fa t that K is superlinear, Z   N N  T  ( R n B R ) =  (x) e (140)

 e

hK (v ) hU (x)+ (x+hv )  (x)  

TN (RN nBR ) Z

1 2 hR 

 3 N e

1 

TN (RN nBR ) Z TN

 (x)e

jj

2 h v 

dx dv

 (x) dx

= 3 N e ; provided that R  4 1 =( 2h), where the i 's are suitable positive quantities independent of . Also, sin e  is a probability measure, we have that  * 0 , for some measure 0 on N N T  R (see, e.g., page 55 of [EG92℄). We now show that 0 is a probability measure, and that it has ompa t support. It is standard (see, e.g., Theorem 1-(ii) on page 54 of [EG92℄) that 0 (TN  R N )  lim inf (TN  R N ) = 1 : !0 On the other hand (see again Theorem 1-(ii) on page 54 of [EG92℄),   N N N 0 (T  BR )  lim sup (T  BR )  lim sup 1 3  e = 1; !0 !0 provided that R is large enough, thanks to (140). Therefore, 0 is a probability measure with ompa t support. Also, 0 satis es the onstraint in (91), sin e  does. We now prove that if (x; v) is a point in the support of 0, (141) then v is a minimizer of the fun tion v 7! hL(x; v ) + 0 (x + hv ) 0 (x): To prove this, take a point (x1 ; v1) 2 TN  R N whi hnis not a minimizer of the aboven fun tion. Choose a small neighborhood V1 of x1 in T and and a small ball B1  R

entered in v1 . Then, by (130), if these neighborhood are small enough, there exists a suitably small Æ > 0 su h that (142) hL(x; v ) + 0 (x + hv ) 0 (x)  hH 0 + Æ for any (x; v) 2 V1  B1. Now hoose a point v0 so to minimizing the fun tion v 7! hL(x1 ; v ) + 0 (x1 + hv ) 0 (x1 ) : Thanks to (130), we may now take a small ball B0  R n entered in v0 and a small neighborhood V0 of x1 , in su h a way that Æ (143) hL(x; v ) + 0 (x + hv ) 0 (x)  hH 0 + 4

1 

1 

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

49

for any (x; v) 2 V0  B0. We now onsider the neighborhood of x1 given by V := V0 \ V1 : By the uniform onvergen e of , (142) and (143), we have that, if  is small enough, 3 hL(x; v ) + (x + hv )  (x)  hH 0 + Æ; 4 for any (x; v) 2 V  B1 and Æ hL(x; v ) +  (x + hv ) (x)  hH 0 + : 2 for any (x; v) 2 V  B0 . Then e, Z  (V )e  (x) e (V  B1 ) = V B and Z  (x) e (V  B0 ) =  (V )e V B Thus, (V  B1 )  e ! 0; (V  B0 ) as  ! 0. In parti ular, (V  B1 ) ! 0 as  ! 0. This implies that 0 (V  B1) = 0 (see, e.g., Theorem 1-(ii) on page 54 of [EG92℄). Therefore, the proof of (141) is

omplete. It follows from (141) and (130) that L(x; v ) + 0 (x + hv ) 0 (x) = H 0 ; for any (x; v) in the support of 0. Thus, integrating and using the onstraint in (91), we obtain that Z Z Ld0 = [L + 0 (x + hv ) 0 (x)℄ d0 = H 0 : This ompletes the proof of the desired result, by the de niton of H 0 given on page 44.  hL(x;v )+ (x+v )  (0)  

h(H 0 H  )+ 34Æ 

hL(x;v )+ (x+v )  (0)  

Æ h(H 0 H  )+ 2 

1

0

Æ

4

A. A Bana h-Ca

ioppoli-type Theorem We state and prove a variation of the standard Bana h-Ca

ioppoli Theorem about the existen e and uniqueness of xed points of stri t ontra tions. The result we present is given in a form whi h is onvenient for the proof of Theorem 26. Theorem 41. Let S be a losed subset of a Bana h spa e, endowed with a norm k
k. Assume that g : S ! S is so that (144) kg(x) g(y)k  (1 kx yk ) kx yk ; Appendix

50

DIOGO A. GOMES, ENRICO VALDINOCI

for any x; y 2 S and some given onstants ; > 0. Then, there exists a unique x? 2 S so that g (x? ) = x? . Furthermore, given any x0 2 S , we have that (145) x? = n! lim+1 gn(x0) :

Let x0 2 S . For any n 2 N , we de ne xn := gn(x0 ) and dn := kxn+1 xn k. Note that xn 2 S for any n 2 N , be ause g(S )  S . Also, by (144), (146) kg(x) g(y)k  kx yk ; for any x; y 2 S and therefore dn = kg (xn) g (xn 1 )k  kxn xn 1k = dn 1 ; for any n  1, that is dn is a non-in reasing sequen e. Thus, there exists d  0 so that d = n!lim d : +1 n We now show that d = 0. Indeed, let us assume, by ontradi tion, that (147) d > 0: Then, using (144), one has that d = n! lim+1 kxn+1 xnk = n!lim+1 kg(xn) g(xn 1)k  n!lim+1(1 kxn xn 1 k ) kxn xn 1 k = n!lim+1(1 d n 1)dn 1 = (1 d )d ; that is, by (147), 1  1 d ; whi h gives d  0, ontradi ting (147). This shows that d = 0 or, equivalently that (148) lim kg(xn) xnk = 0 : n!+1 We show that xn is a Cau hy sequen e in S . Indeed, x  > 0 and let Proof.

+1

 (149)  := 2 : Thanks to (148), we may assumne that kg(xn) xn k  

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

51

provided n  n0 , for a suitable n0 = n0(). Combining this with (144), we gather that kxn xm k  kxn g(xn)k + kg(xn) g(xm)k + kg(xm) xm k  2 + (1 kxn xm k ) kxn xm k = 2 kxn xm k +1 + kxn xm k provided that n; m  n0 . That is, 0  2 kxn xm k +1 and so kxn xm k   due to (149). This?shows that xn is a Cau hy sequen e. Sin e S is omplete, we have that there exists x 2 S so that (150) lim xn = x? ; n!+1 proving (145). Also, ombining (146) and (150), lim g(xn) = g(x?) : n!+1 This, together with (150) and (148), gives that kg(x?) x? k = n!lim+1 kg(xn) xnk = 0; then e x? is the desired xed point. We now?show? that su h xed? point? is unique. If, by ontradi tion, there existed y? 2 S with kx y k 6= 0 and g(y ) = y , we would derive from (144) that kx? y?k = kg(x?) g(y?)k  (1 kx? y?k ) kx? y?k ; and thus that 1  1 kx? y?k : This would give that x? ?= y?, ontradi ting our assumptions. This shows the uniqueness of the xed point x .  Appendix B. Some elementary properties of the semi on ave fun tions The purpose of this se tion is to re all some standard properties of the semi on ave fun tions, and to express su h properties in a framework onvenient for our proofs. First of all, we remark that  is semi on ave if and only if     (1 ) jx yj2 ; (151) (x) + (1 ) (y )  x + (1 )y   2

52

DIOGO A. GOMES, ENRICO VALDINOCI

for any  2 [0; 1℄. Indeed, (151) is due to (36) and the ontinuity of ; see, e.g., Proposition 1.1.3 in [CS04℄.18 We also re all the following standard result, the proof of whi h may be found, for instan e in Theorem 2.1.7 and Remark 2.1.8 of [CS04℄: A bounded semi on ave fun tion  is Lips hitz ontinuous. Moreover, its Lips hitz onstants depends only on N ,  and kkL1 (RN ) .

Lemma 42.

Also, by modifying the proof of Theorem 2.1.7 in [CS04℄, the following result follows: Lemma 43. Let  be semi on ave and ZN -periodi . Then,  2 L1 (R N ) and sup j(x) (0)j x2RN

is bounded by a onstant whi h depends only on N and Proof.

Let us de ne

 .

u(x) = (x) (0) ;

so that u is ZN -periodi and semi on ave, with u =  byN (36), and u(0) = 0. Let now x1 and x2 be two onse utive verti es of the ube [0; 1℄ . Then, u(x1 ) = u(x2 ) = u(0) = 0 by the periodi ity of u and p jx1 x2 j  N by onstru tion. Thus, in the light of (151), we have that     (1 ) jx x j2   N ; u x1 + (1 )x2   1 2 2 8 for any  2 [0; 1℄. That is, if x belongs to any of the one-dimensional fa es of the ube [0; 1℄N , we have that u(x)  b1 ; with N b1 :=  : 8 This pro edure an be iterated. Namely, assume that for any x belonging to any of N the (n 1)-dimensional fa es of [0; 1℄ , we have that (152) u(x)  bn 1 ; for a suitable bn 1  0. Then, take any n-dimensional fa e of [0; 1℄NN , and any x on this fa e. Let x1 and x2 belong to a (n 1)-dimensional fa e of [0; 1℄ so that x = x1 + (1 )x2 ; 18As a notation remark, we note that what we all here \semi on ave fun tions" are alled \semi-

on ave fun tions with linear modulus" in [CS04℄.

ENTROPY PENALIZATION FOR HAMILTON-JACOBI

53

with  2 [0; 1℄. Then, by using (151) and (152), we on lude that   u(x) = u x1 + (1 )x2  u(x1) + (1 ) u(x2)   (12 ) jx1 x2 j2  bn 1 (1 )bn 1 8N ; that is u(x)  bn ; with N bn := bn 1 +  : 8 Note that the iteration shows that  Nn bn =  8 and so, taking n = N , we dedu e that (153) u(x)  b ; with  N2 b :=  ; 8 N for any x 2 [0; 1℄ and thus, by periodi ity, for any x 2 R N . This will be a bound on  from below. We now perform a bound from above. For this, we take x 2 [0; 1℄N , with x 6= 0 (re all that u(0) = 0), and we apply again (151), as follows: jxj u x  + 1 u(x)  jxj 1 + jxj jxj 1 + jxj 2 and therefore, by means of (153), (154) u(x)  2N ( + b) ; for any x 2 [0; 1℄N . By periodi ity, (154) holds for any x 2 RN , whi h is the desired upper bound. Colle ting the estimates in (153) and (154), we obtain that j(x) (0)j  4N 3  :  Putting together the results in Lemmata 42 and 43 we thus obtain:

If  is semi on ave and ZN -periodi , then it is Lips hitz ontinuous. Moreover, its Lips hitz onstant depends only on N and  . Theorem 44.

Possibly repla ing  with  (0), we may and do assume that (0) = 0. Then, by Lemma 43, kkL1(R ) is uniformly bounded. Therefore, thanks to Lemma 43, the Lips hitz onstant of  is also uniformly bounded, as desired.  Proof.

N

54

[Ana04℄ [BS91℄ [CIL92℄ [CS04℄ [DS58℄ [EG92℄ [Eva98℄ [Eva04℄ [FS86℄ [Gom05℄ [JKO98℄ [Mat91℄ [MF94℄ [Sou85℄ [Tay58℄ [Vil03℄

DIOGO A. GOMES, ENRICO VALDINOCI

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Diogo A. Gomes Departamento de Matem ati a, Instituto Superior Te ni o, Lisboa, 1049-001, Portugal e-mail: dgomesmath.ist.utl.pt Enri o Valdino i Dipartimento di Matemati a, Universit a di Roma Tor Vergata, Roma, I-00133, Italy e-mail: valdino imat.uniroma2.it