This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
/ ipA}lkudx —> / ipvdx. Jo,1 Jw However / <^A^fcuda; = — I 1 uAl_hkipdx —> — /1 Jci' Ja Jo. where 6 is an arbitrary constant. Proof. € V(QT) as the test function in (3.1.6) and (3.1.7). 3.1.2 I in the an above identity. Then we obtain / \SJ ) dxdt = 0 ) / (fipdx - / • V(p) dx. Sometimes the weak subsolution (supersolution) is said to be a function in if 1 (fi) satisfying -Au + c(x)u < (>)/(x) + div/(x) in the weak sense. Proposition 4.3.1 Assume that f = 0, / = 0, c(x) > 0 andu e if 1 (f2)n L°°(Q) is a weak subsolution of equation (4-1.9). If g"(s) > 0, g'(s) > 0, 3(0) = 0, then w = g(u) is also a subsolution of equation (4-1.9). Proof. By the definition of weak subsolutions, for any nonnegative function tp G Cg°(fi), / (Vu • Vy> + cwp)dx < 0. n Since g'(s) > 0 and u <E tf^fi) n L°°(D,), we have 0 < g'{u) + g"(u)\Vu\2 ) / / JJQT 1 + a / 2 ( Q T ) . If u £ 2+a 1+a 2 C ' / (QT) is the solution of the first initial-boundary value problem (7.2.35), (7.2.36), then M 2 + a , l + < * / 2 ; Q T ^ C{\f\a,a/2;QT 1+a/2(QT)nC(QT). Proof. Without loss of generality, we assume that 1. Assume that the coefficients of equation (9.2.9) satisfy (9.2.10) and (9.2.11) and ay £ C{QT). If u £ W2,1(QT) is a strong solution of the first initial-boundary value problem (9.2.9), (9.2.12), then 0 depending only onn, X, A, ||s||L«(n), ||/i||L«.(n)) Ks)> \fU,a-,n, \a\i,a, \b\a andfl, such that for any solution u G C2'a(fl) of problem (10.1.1), (10.1.2), Ml,/3;fi < M; M) Problem (10.1.1), (10.1.2) admits a solution u G C2'a{ft). C Gk- Multiplying (13.2.21) by k = 0 outside BR X [0, r]. Since the extended function ipk = 0 may not necessarily be a sufficiently smooth function in QT, we use a function
uD^dx.
Thus /
•
Similarly we can prove Proposition 2.2.3 1,2--- , n - 1.
Lei -8^(0) = {a; G Bi(0);x n > 0},0 < p < l,i =
50
Elliptic and Parabolic
i) If u G Hl(Bf(0)), H^B+iO)) and
then for sufficiently small \h\ > 0, A{u G
HAfcullL»(B+(0» ii) IfuE
L2(Bf(0))
Equations
WDiu\\mB+(0)y
and for sufficiently small \h\ > 0, II^ U IIL2(B+(O)) ^
K
with constant K independent of h, then DiU G L2(B+(Q)) and HAu|lz,2(B+(0)) ^ 2.2.2
Interior
K
-
regularity
Now we proceed to discuss the regularity of weak solutions of Poisson's equation. First we discuss the interior regularity. Theorem 2.2.1 Let f G L2(Q), and u G iJ : (ft) be a weak solution of equation (2.1.1). Then for any subdomain fi' CC fl, u € H2{Q.') and \\U\\H'(Q')
< C (||u||jfi(n) + H/Hz^n)),
where C is a constant depending only on n and dist{ft',9ft}. Proof.
For fixed ft' CC ft, denote d = -dist(ft',dft). Choose a cut-off
function on ft relative to ft', i.e. a function r](x) G Co°(ft), such that 0 < r}(x) < 1,
rj(x) = 1 in ft', dist{suppr?, 9ft} > 2d.
To prove the conclusion of Theorem 2.2.1, by Proposition 2.2.2, it suffices to derive the estimate 2
\A{Vu\2dx
< C(\\u\\2H1{n) + ||/||£ a ( n ) )
(2.2.1) Jn for some constant C independent of h. To establish any estimate on weak solutions, the original starting point is the definition of weak solutions, i.e. the integral identity
/
V
/ Vu • Vydx = f fipdx,
V
(2.2.2)
L2 Theory of Linear Elliptic Equations
51
The crucial step is to choose a suitable test function
L
2 v
\AhVu\2dx
= f AhVu • V{ri2Ahu)dx Jn = f Vw • VAi*(r)2Ahu)dx Jn
-2 -2
[ A\Vu Jn
• r/A^Vrycfo
[ TjA{uVr] • AhVudx, Jn
(2.2.3)
it is natural to choose cp = Azh*(rj2Alhu) (clearly it belongs to HQ(Q,)) in (2.2.2). Thus we obtain / Vu • VAJ, V A ^ ) c f e = / fA\* Jn Jn
(r,2Ahu)dx,
which combined with (2.2.3) gives / r)2\AhVu\2dx Jn
= f fAh*(r,2Ahu)dx Jn
- 2 / T / A ^ V T ? • A^Vudx. Jn
Now we use Cauchy's inequality with e to the integrals on the right side of the above formula to obtain
/ r,'\AhVu\'dx
n Jn [ f2dx + 2e [\Ak*(r,2Aiu)\2dx <± Z£ Jn Jn ~2e + - f \Vri\2\A\u\2dx + e / r ^ l A ^ V u l 2 ^ . £ Jn Jn Using Proposition 2.2.2 i), we have / \Ai*(V2Aiu)\2dx
= f
(2.2.4)
\Ath(v2Aiu)\2dx
< / lA^A^)!2^ Jn < f Jn
\V(V2Ahu)\2dx
= [ I^VA^u + Jn
2rjAiuVv\2dx
<2 f \r]2VAiu\2dx Jn
+ 2 / \2rjAhuVr]\2dx Jn
52
Elliptic and Parabolic
Equations
<2 / r)2\VA{u\2dx Jo
+ 8 [ |VT ? | 2 |A^l 2 da? Jo
and f |V»/|2|A£u|2da; < C f JO
|A^r*|2da: < C f \DiU\2dx
^supp7j
JQ.
\Vu\2dx.
JO
Combining these with (2.2.4), we finally obtain (1 - 5e) / jj 2 |A^V«| a di < C(-v e + lfe) / \Vu\2dx + 2±f £ Jo ' Jo Jo
fdx
and the desired conclusion (2.2.1) follows by choosing e suitably small.
•
Corollary 2.2.1 Let u be a weak solution of equation (2.1.1). If f G Hk(Q) for some nonnegative integer k, then, for any subdomain Q' CC Cl, u G Hk+2(fl') and IMU*+2(fi') < C (||u|| H i(n) + ||/||if=(n)) , where C is a constant depending only on k, n and dist{fi',9fi}. Proof.
First consider the case k = 1. By the definition of weak solutions,
/ Vu • VDupdx = / fDt(pdx, ty> G Cg°(n) Jo JO
(i = 1, • • • , n).
Since, by Theorem 2.2.1, for any subdomain fi' CC Cl, u G H2(n') and / G if 1 (£2) is assumed, we can integrate by parts in the above formula to derive / VDiU • \>
Jo
Jo,
V
(i = 1, • • • , n).
This shows that DiU is a weak solution of the equation -Av
= Dif,
xeQ.
(i =
l,---,n)
and hence we can use Theorem 2.2.1 to assert Dtu G H2(Q') for any subdomain fi' CC O and obtain the estimate l|w|k 3 (n') < C (|M|tfi(n) + ||/||ifi(n)) • By induction, we can prove the conclusion of Corollary 2.2.1 for any positive integer k. •
L? Theory of Linear Elliptic
If f £ Hk(fl)
Corollary 2.2.2
Equations
53
with k > —, then the weak solution u
of equation (2.1.1) satisfies the equation —Au = f(x) in fl in the classical sense. Proof. By Corollary 2.2.1, for any subdomain Q.' CC Q, u G Hk+2(Q,'). Since k > - , by the embedding theorem, Hk+2{Cl') ^ C2'a(n') with
2.2.3
Regularity
near the
boundary
P r o p o s i t i o n 2.2.4 Let Cl C R n be a bounded domain with dfl € C2 and y = ty(x) be a local flatting mapping in a neighborhood of the given point x° G dfl (see §1.1.6). Then for any weak solution of Poisson's equation (2.1.1), u(y) —u{^~1(y)) is a weak solution of the equation
namely, for any ip G C^(B^),
there holds
^ l r • j£rdy = i + KvMv)dy, JB+ °Vi °Vj JB+ where a,ij(y) is the (i,j) element of the matrix [
+
(2-2.5)
(fiii(y))„xn = | J ( j / ) | * ' ( * - 1 ( y ) ) * ' ( * - 1 ( y ) ) T ,
f(y) = \J(y)\W-Hy)), ,_1
and x = ^ (y) is the inverse mapping of y = ^(x), J(y) the Jacobi determinant of the mapping x = * _ 1 ( y ) , W(x) the derivative matrix of the mapping y — $(x) and ^'(x)T the transposed matrix of^'{x). In this book, repeated indices denote summation from 1 to n if there is no other indication. The proof of the above proposition is left to the reader. R e m a r k 2.2.1 In the formula (2.2.5) and the formula before it, repeated indices imply a summation from 1 to n. Such summation convention will be adopted frequently in the sequel. T h e o r e m 2.2.2 Let f G L2(Q) and u G HQ(Q,) be a weak solution of the Dirichlet problem (2.1.1), (2.1.2). IfdCl G C2, then for any x° G dfi, there
54
Elliptic and Parabolic
Equations
exists a neighborhood U of x°, such that u G H2(U n 0) and IM|jj2(t/nn) < C (\\u\\m{n) +
\\fh*(n)),
where C is a constant depending only on n and SlnU. Proof. By Proposition 2.2.4, there exists a neighborhood U\ of x° and a C2 invertible mapping \P : U\ —> Bi(0), such that
*((7i nfl) = B+ = B+(o),
*(t/i n an) = 8B+ n {y e Rn; y„ = 0}
and -t*(y) = u(^!^1(y)) satisfies (2.2.5) for any
Tj 2 |A£v«| 2 dy,
where A£(fc = 1,2, • • • , n — 1) is the tangential difference operator. Similar to the derivation of (2.2.1), we choose
/ ^ 7 P • IT fA£VA^)1 dy = / f(y)tf(r,*Akhu)dy. Using Proposition 2.2.1 i), ii) further gives
Lt AJ (a« w) w, (,2A!:") * = L t /C'^VAS*)*. Since, using Proposition 2.2.1 iii), ii), we have Ah aij
\ dy-)-
h ij
~dy~+
ht3
dy0'
we are led to
o„ fc . 0 A U aA*<2 /. /
9Aju 5u 7 7 2 A^a i j —A- • — d y - 2 /
77—^-A^A^u—
L 2 Theory of Linear Elliptic Equations
-2f
JB{
AtaijAkhu^dy+ °Vj
°Vi
55
/ /(y)AJtVAJt«)d».
JB{
From this we can proceed similar to the proof of the interior estimate in Theorem 2.2.1 to derive
/
\kkhVu\2dy
\Wu\2dy + C f
JB+2
JB+
f2dy.
(2.2.6)
JB+
To do this, it is to be noted that, since dfl G C2, we have &ij G C1 and hence |A*ciij| < M for some constant M. Another more important fact to be noted is that there exist constants A > 0, ho > 0, such that ThaiMo > A KI 2 . provided 0 < \h\ < h0. Using Proposition 2.2.3 ii), from (2.2.6) we see that for any k = 1,2, •• • ,n — 1 and j = 1,2, • d2u
•lst/2
dy
dykdyj
*** dykdyj
(4
GL2(5+2)and
\\7u\2dy + J
+
f2dy
(2.2.7)
Now we rewrite (2.2.5) as
Lt &nnwn • S > = L t IiyMy)dy - £2n Lt &ij iji • 1 ^ or, after integrating by parts in the second term of the right side,
Li ^nWn • | > = Li {Iiy) + i+^2n W, ( % ^ ) J *{V)dyProm this and (2.2.7) it follows that ——\a n n -r—) G L2(Bt/0). Since it is l2 dyn\ dynJ '' 2 du easy to verify that ann ^ 0 in By2, we further have jr-j2 G L 2 (Sjy 2 ) and dy
L
d2u dVn
dy
\Vu\2dy + C / jBi JBi
•L
f'dy.
This combined with (2.2.7) implies u(y) G H2(B~f,2). Changing the variable y to the original one shows that u G H2(U fl fi) with U = }i~1(B^,2) and u satisfies the estimate in Theorem 2.2.2. •
56
Elliptic and Parabolic
Equations
Similar to the interior regularity, we also have Corollary 2.2.3 Let u G HQ (Cl) be a weak solution of the Dirichlet problem (2.1.1), (2.1.2). IfdCl G Ck+2 and f G Hk(Cl) for some nonnegative integer k, then for any x° G dCl, there exists a neighborhood U ofx°, such that u G Hk+2(U n Cl) and IM|jf*+2(t/nn) < C(H|tfi ( fj) + ||/||jj*(n)), where C is a constant depending only on k, n and CinU. Corollary 2.2.4 If dCl G Ck+2, f G Hk(Cl) and k > | , then for any x° G dCl, there exists a neighborhood U of x°, such that any weak solution u of the Dirichlet problem (2.1.1), (2.1.2) belongs to C2'a{U C\Cl) with 0 < Ik 2.2.4
Global
regularity
To prove the global regularity of weak solutions, we choose a finite open covering of Cl and decompose the solutions by means of the partition of unity (§1.1.5). Theorem 2.2.3 Let f G L2(Cl) and u G HQ(CI) be a weak solution of the Dirichlet problem (2.1.1), (2.1.2). If dCl G C2, then u G H2{Cl) and \HHHO)
< C (||«||ffi(n) + ||/||L*(n)) ,
(2-2.8)
where C is a constant depending only on n and Cl. Proof. By Theorem 2.2.2, for every x° G dCl, there exists a neighborhood U(x°) such that u G H2(U(x°) n 0 ) and \H\H2(U(x°)nu) < C (\H\HHQ)
+ ll/lk 2 (n)) •
Using the finite covering theorem we can choose such neighborhoods of N
finite number f/i, • • • , UN to cover dCl. Denote K = Cl \ (J [/*. Then K i=\
is a closed subset of Cl and there exists a subdomain UQ CC Cl, such that U0 D K. Theorem 2.2.1 shows that u G H2(U0), and NI/r*(c/0) < C (||u||ffi(n) + ||/l|z.3(«)) •
L 2 Theory of Linear Elliptic
Equations
57
Using the theorem on the partition of unity, we can choose functions ^o, ?7i> • • • i T)N> s u ch that 0<»/i(x)
VxeK
(i = 0,1, - - - ,N),
N »=i
Thus N
u
2
ll llfl" (n)
N
<£>•
^ViU j? 2 (n)
i=0
li?2(n)
i=0
< c ( H f f i ( n ) + ||/|| L a ( n ) ). R e m a r k 2.2.2
D
Under the assumptions of Theorem 2.2.3, we have hUnn)
< C||/||La(n),
(2.2.9)
where C is a constant depending only on n and fi. Proof.
We first set
Jn
V € H^{Q)
Jn
and use Cauchy's inequality with e and Poincare's inequality to obtain / \Vu\2dx = /
Jn
Jn
fudx
<1£- [ u2dx+±i£ Jn
Jn
[fdx
<£J± [\Vu\2dx + ± [ fdx, 2 Jn
2e Jn
where fi > 0 is the constant in Poincare's inequality. Choosing e = — then M gives / \Wu\2dx < (j. I Jn Jn
fdx.
and further by Poincare's inequality, / u2dx
Jn
l
Jn
fdx.
58
Elliptic and Parabolic Equations
Thus the desired inequality (2.2.9) follows by combining the above two estimates and substituting into (2.2.8). • In the proof of Theorem 2.2.2, the normal derivative of second order is estimated via the equation and the estimates for the tangential derivatives. Repeating this procedure we can estimate higher order derivatives in normal direction. Theorem 2.2.4 Let u G H&(Q.) be a weak solution of the Dirichlet problem (2.1.1), (2.1.2). Ifdn G Ck+2 and f G Hk{Q) for some nonnegative integer k, then u G Hk+2{Q) and llulltf*+2(fi) < C (||u||ffi(n) + ||/||ff*(n)), where C is a constant depending only on k, n and fi. As an immediate corollary of this theorem, we have Theorem 2.2.5 Let u G HQ(Q) be a weak solution of the Dirichlet problem (2.1.1), (2.1.2). IfdCl G C°° and f G C°°(Tl), then u G C°°(ty. Remark 2.2.3 Theorem 2.24 shows that 30, G Ck+2, f G Hk(Cl) imply u £ Hk+2(Q.) and Theorem 2.2.5 shows that <9fi G C°°, f G C°°(Ti) imply u G C°°(f2). This means that the conclusions on the regularity of weak solutions are complete both in Hk(Q) and C°°(fi). In Chapter 8 it will be proved that f G Ca(Q) (a G (0,1)) implies u G C 2,a (f2), which means that the conclusion on the regularity of solutions is also complete in Holder spaces. However it is impossible to assert u G C 2 (0) from f G C(Q). 2.2.5
Study of regularity
by means of smoothing
operators
Smoothing operators u£ = JEu=
/ je(x - y)u(y)dy, Jo.
(2.2.10)
instead of difference operators can also be applied to the study of regularity of weak solutions, where js(x) is an arbitrary mollifier. In (2.2.10), u is regarded as zero outside of CI. It is easy to check the following facts which are an analog of Proposition 2.2.1 i), ii). Proposition 2.2.5
Let Cl c Mn be a domain.
L 2 Theory of Linear Elliptic Equations
59
i) For any u, v £ L J (fi) vanishing outside ofQ, / uevdx = /
Jn
Jn
uv~dx,
where v~ = J~v=
/ j~(x - y)u{y)dy,
j~(x) =
j£(-x).
ii) For any Q' CC fi and sufficiently small e > 0, DiUe = {DiU)e
inQ'
(i = 1,2, • • • ,n).
However we do not have the analog of Proposition 2.2.1 iii) for smoothing operators. This will restrict the application of the present method in the study of regularity. We also have the following proposition which corresponds to Proposition 2.2.2 and can be proved similarly. Proposition 2.2.6 Let Q, CM" be a domain and i — 1,2, • • • , n. i)Ifu£ i? 1 (n) and Q' CC fi, then for sufficiently small e > 0, \\DiUe\\L2(ni)
< ||Diu|| L 3(n).
ii) Ifu e if 1 (fi), fi' CC fi and for sufficiently small e > 0, ||Aw£||L2(fi') < -^ wii/i constant K independent of e, then DiU £ L 2 (0') and ||Aw||L2(fl') < # •
The proof is left to the reader. We do not state the analog of Proposition 2.2.3, although it does hold. Now we proceed to use smoothing operators to establish the interior regularity, i.e. to prove Theorem 2.2.1. Choose a cut-off function r)(x) as in the proof of Theorem 2.2.1. By Proposition 2.2.6 ii), it suffices to establish the estimate 2
2
/ V \DiVue\ dx Jo.
< C (|M| 2 „ 1 ( n ) + | | / | | £ a ( n ) )
for some constant C independent of e. Using Proposition 2.2.5 and integrating by parts, we have
/ rf\DiS7ueYdx n Jn
(2.2.11)
60
Elliptic and Parabolic
Equations
= / A V u e • V(r}2DiUe)dx - 2 / •qDiueVn • A Vu e dx Jn Jn = - / Vu £ • V(A(r? 2 A« e ))da; - 2 / •qDiuEViq • A V u £ d x Jn Jn = - / Vu- V ( A ( l 2 A M £ ) ) £ " d i - 2 / T]DiU£Wr) • DiVu£dx. Jn Jn Choosing ip = (Di(r]2DiUe))~ in (2.2.2) and combining the resulting equality with the above formula lead to / r)2\DiVue\2dx Jn
= - I fiDiirfDiU^-dx Jn
- 2 / rjDiUeVri • DiVuedx. Jn
From this we may deduce (2.2.11) similar to the proof of (2.2.1). 2.3 2.3.1
L2 Theory of General Elliptic Equations Weak
solutions
Now we turn to the following general elliptic equations in divergence form Lu = -Dj(cnjDiu)
+ biDiU + cu = f + A / * ,
(2.3.1)
where ay, 6», c G L°°(fi), / G L2(il), /* G L2(f2), and ay = a^j satisfy the uniform ellipticity condition, i.e. for some constants 0 < A < A, A|£|2 < a y ( * ) ^ < A|^| 2 ,
V£ 6 i n , x € fi.
In this case, we call (2.3.1) uniformly elliptic equations. Here, repeated indices imply a summation from 1 up to n. As in the preceding sections, only the Dirichlet problem with the homogeneous boundary value condition u
= 0
(2.3.2)
an is discussed. Remark 2.3.1
If a nonhomogeneous boundary value condition u
an
= g
is prescribed with g G H1^), then, setting w = u—g, we can change (2.3.1) to an equation for the new unknown function w,
Lw = f + DJi,
L2 Theory of Linear Elliptic
Equations
61
where f = f - hDig - eg,
f =
f+aijDjg.
The boundary value condition which w satisfies is then a homogeneous one. If u £ C 2 (fi) is a solution of equation (2.3.1), then multiplying (2.3.1) with any ip £ CQ?(£1) and integrating over fi lead to, after integrating by parts, / (aijDiuDjip + biDiU(p + cwp) dx = (ftp — f%Ditp)dx. Jn Jn Conversely, if u £ C2{fl) and for any (p £ satisfies (2.3.1) in the classical sense.
CQ°(Q),
(2.3.3)
(2.3.3) holds, then u
Definition 2.3.1 A function u € Hl(Vi) is said to be a weak solution of (2.3.1), if for any ip £ C^(Q), (2.3.3) holds. If, in addition, u £ H^{Cl), then u is said to be a weak solution of (2.3.1), (2.3.2). 2.3.2
Riesz's
representation
theorem
and its
application
Riesz's representation theorem applied to Poisson's equation can be carried over to equation (2.3.1) with 6j = 0 (i = 1, • • • , n), i.e. the equation -DjiaijDiu)
+cu = f + Dif*.
To this purpose, we define a new inner product in
(2.3.4) HQ(CI)
as follows:
(u, v) = / (aijDiuDjV + cuv) dx,
Ja
whose corresponding norm is denoted by ||| • |||. It is easy to verify that if c > Co for a certain constant Co, then (•,•) possesses all properties of inner product. For example, using the ellipticity condition and Poincare's inequality, we have, for u £ HQ{Q),
Mil2 = (u, u) = / {aijDiuDjU + cu2) dx Jn
>A / |Vu| 2 dx + co /
Jn
>a]\u\\2,
Jn
u2dx
62
Elliptic and Parabolic
provided
Equations
\- c0 > 0, where ||u|| is the norm in #o(Q), M > 0
constant in Poincare's inequality and a = min { - ,
is
tne
\- CQ \. From this it
I 2 2fi
)
follows that (u, u) = 0 implies u = 0. Clearly, for a certain constant /3 > 0,
IIMII2
(2.3.5)
which implies, in particular, that F[v] = / (fv - fDiv) Jn
dx
is a bounded linear functional in HQ(Q), the same space as H Q ( O ) endowed with the inner product (•, •). Hence, by Riesz's representation theorem (see §2.1.2), there exists a unique u G HQ{£1) such that (u,v) = / (dijDiuDjV + cuv)dx Jn = * > ] = f {fvJn
fDiv)
dx,
\fv e
H&n).
From (2.3.5) we have u £ HQ(Q) and the above formula implies / {dijDiuDjtp + cu
My e
C^(Q),
which means that u is a unique weak solution of (2.3.4), (2.3.2). Theorem 2.3.1 There exists a constant CQ such that for any f £ L2(Cl), p £ L2(Cl)(i = l,--- ,n), the Dirichlet problem (2.34), (2.3.2) admits a unique weak solution provided c > CQ .
2.3.3
Variational
method
Theorem 2.3.1 can also be proved by means of variational method. The functional corresponding to (2.3.4) is J v
\ \ = o / (aijDivDjV + cv2) dx2 Jn
I (fv - fD^) Jn
dx.
L2 Theory of Linear Elliptic
63
Equations
In fact, arguing as in §2.1.3, we may prove that if u G H^ft) is an extremal of the functional J[v] in HQ(CI), then u is a weak solution of the Dirichlet problem (2.3.4), (2.3.2). To prove the existence of a minimizer of J[v] in HQ(£1), we first establish the boundedness from below of J[v] in HQ(CI). Using the ellipticity condition and Cauchy's inequality with e, we have 2 J[v] >7? / v2dx - | / v2dx 2 / \Vv\ dx + ^ Ja 2 Jn 2 JQ
-—
f fdx
- -
f \Vv\2dx - -J- / ffdx.
(2.3.6)
This and Poincare's inequality further give
M
> i (A_I + c o _£) j ^ i x _ ^ ^ _ ijf
/(/Ml , (aAT)
where /x > 0 is the constant in Poincare's inequality. If Co satisfies — + c o > 0, then we may choose e > 0 so small that - (
f- Co — e) > 0. The
2 V
/Li
/
boundedness from bellow of J\v] in # o ( ^ ) i s thus proved. Combining (2.3.6) with (2.3.7), we obtain / v2dx + / \Vv\2dx
C2J[v],
Vv G H^(il),
(2.3.8)
where C\, Ci are constants independent of v G HQ(Q). Let {ufc} be a minimizing sequence of J[v] in #o(fi). From (2.3.8) it follows that both {uk} and {Vufc} are bounded in L2(Q) and hence there exist a subsequence {MA:I} of {uk} and a function u G i / ^ f i ) such that Ufc, —»• u,
Vwfc, —*• Vu
in L2(f2) as / —> oo.
In particular, we have ;
lim / fuk,dx = / fudx, ^ ° ° JQ JO,
lim / fiDiukldx=
f fDiudx
(2.3.9) (i = 1, • •• ,n).
Moreover, from / [a,ijDi(uk, - u)Dj(uk, -u) + c(v,k, - u)2] dx > 0
(2.3.10)
64
Elliptic and Parabolic
Equations
i.e.
/ (cLijDiUktDjUkt Jn
+cul)dx — / (aijDiuDjU + cu2)dx, Ja
>2 / (aijDiU^DjU + cuk^dx Jn it follows that lim (aijDiiiktDjUkt+cull)dx i—>oo Jn
(aijDiuDjU +cu2)dx. Jn
>
(2.3.11)
Combining (2.3.9), (2.3.10) and (2.3.11), lim J[uk] = lim J[ufc,] > J[u). k->oo
j^oo
Thus u is a minimizer of J[v] in #o (fi) of (2.3.4), (2.3.2) is proved. 2.3.4
Lax-Milgram's
theorem
an(
^ * n e existence of weak solutions
and its
application
It is to be noted that not any elliptic equation of the form (2.3.1) has its corresponding functional so that the variational method can be applied to the Dirichlet problem. If bi (i = 1, • • • ,n) are not all equal to zero, then Riesz's representation theorem also can not be applied. In this case, we need to slightly extend the representation theorem. Lax-Milgram's theorem is one of the very useful results obtained in this direction. Definition 2.3.2 Let a(u, v) be a bilinear form in the Hilbert space H, i.e. a(u, v) is linear in u and in v respectively. i) a(u, v) is said to be bounded, if for some constant M > 0, \a(u,v)\ < M\\u\\\\v\\,
Vu,v £ H;
ii) a(u, v) is said to be coercive, if for some constant 8 > 0, a(u,u)>S\\u\\2, VueH. Lax-Milgram's Theorem If a(u,v) is a bounded and coercive bilinear form in the Hilbert space H, then for any bounded linear functional F(v) in H, there exists a unique u £ H, such that F(v)=a(u,v),
Vv£H
(2.3.12)
L2 Theory of Linear Elliptic Equations
65
and
IMI <
]\\n
Proof. Since a(u, v) is bilinear and bounded, for any fixed u G H, a(u, •) is a bounded linear functional in H. By Riesz's representation theorem, there exists a unique Au G H, such that a(u,v) = (Au,v),
VveH.
(2.3.13)
It is easy to verify from the bilinearity of a(u, v) that the operator A is linear. The boundedness of a(u, v) implies the same property of A: \\Au\\ < M\\u\\,
VuG if.
In addition, since a(u, v) is coercive, we have S\\u\\2 < a(u,u) = (Au,u) < ||Au||||u||,
Vu € H.
Hence j||u|| < ||A«||,
Vu G H,
which shows the existence of A~l. Now we prove that the range of A, denoted by R(A), is the whole space H. First of all, R(A) is a closed subset. In fact, if {Auk} C R{A) is a convergent sequence: lim Auk = v, k~*oo
then, from S\\UJ
- Ufc|| <
\\AUJ
- Auk\\
we see that {uk} is a Cauchy sequence and hence {uk} is also a convergent sequence: lim Uk = u. k—too
Hence, by the continuity of the operator A, lim Auk = Au. fc—»oo
Therefore Au = v, i.e. v G R(A).
66
Elliptic and Parabolic
Equations
Suppose R(A) ^ H. Then there exists a nonzero element w e H, such that (Au, w) = 0 ,
Vu € H.
In particular, if we choose u = w, then (Aw,w) = a(w,w) = 0, which and the coercivity of a(u, v) imply w — 0, a contradiction. Thus R(A) = H. For any bounded linear functional F(v) in H, by Riesz's representation theorem, there exists a unique w € H, such that \\w\\ = \\F\\ and F(u) = (w,v),
v e H.
Choose u = A~1w. Then
Hl^ll^lllhll^illFH and F{v) = {Au,v),
veH
which combined with (2.3.13) leads to (2.3.12).
•
As an application of Lax-Milgram's theorem, we have Theorem 2.3.2 There exists a constant CQ, such that for any f € L 2 (fi) and p € L2(Q.) (i = 1, • • • , n), the Dirichlet problem (2.3.1), (2.3.2) admits a unique weak solution provided c> CQ. Proof.
Denote
a(u,V) = f ( ^ ^ H A ^ + H ^ , Jn
V.,,
G
^(O).
Obviously, a(u,v) is bilinear and the boundedness of a,j, bi, c implies the same property of a(u, v): \a(u,v)\
+ IMU2(n)IMlL=(n))
L 2 Theory of Linear Elliptic
67
Equations
Moreover, using the ellipticity condition and Cauchy's inequality with e, we derive, for u € HQ(Q), a(u,u) >A||Vu||| 2 ( n ) -C J |Vu||«|da; + co||u||^ ( n ) >A||Vu||| 2 ( n ) - | | | V n | | | 2 ( n ) - 2^||«||i a ( n) +co||«|ll3 ( n)
c2 Choosing e such that 0 < e < 2A and then taking CQ > ——, it follows that 2s for some constant S > 0, a{u,u) > 5\\u\\2Hi{n), Vu € H*(Q), i.e. a(u, v) is coercive. Now we can apply Lax-Milgram's theorem to the bounded linear functional F(v) = f (fvJa
fDiv)
dx,
to conclude that there exists a unique u €
IMIW <
v e Hi (Q)
HQ(CI),
such that
-5\\F\\
and a(u,v) = F(v),
Vu€Hl(£l),
which means that u £ HQ (fl) is the unique weak solution of the Dirichlet problem (2.3.1), (2.3.2). D 2.3.5
Fredholm's
alternative
theorem
and its
application
Theorem 2.3.2 merely affirms the weak solvability of the Dirichlet problem (2.3.1), (2.3.2) for the case c > CQ (some constant). To investigate the general case we need the following result (see [Zhong, Fan and Chen (1998)]). Fredholm's Alternative Theorem Let V be a linear space endowed with a norm, A : V —• V a compact linear operator and I the identity operator. Then there is exact one of the following alternatives:
Elliptic and Parabolic
Equations
i) Either the equation x - Ax = 0
(2.3.14)
has a nontrivial solution x G V; ii) Or the equation x-Ax
=y
(2.3.15)
admits a unique solution x G V for any y € V. In other words, if the homogeneous equation (2.3.14) merely has a trivial solution, then the nonhomogeneous equation (2.3.15) admits a unique solution for any y G V, or if for some y G V, the solution of the nonhomogeneous equation (2.3.15) is unique, then for any y G V, the nonhomogeneous equation (2.3.15) has a unique solution. As an immediate application of this theorem, we have Theorem 2.3.3 There is exact one of the following alternatives: i) Either the boundary value problem Lu = 0,
u
an
= 0
has a nontrivial weak solution; ii) Or the boundary value problem Lu = f + DJ\
u
an
= 0
has a unique weak solution for any f G L2(il) !,••• ,n)-
and p
G L2{Q) (i =
Proof. According to Theorem 2.3.2, there exists a constant vQ, such that the equation Lu + vu = f + Dif has a unique weak solution u G HQ(CI) for any / G L 2 (fi) and /* G L2(f2) (i = 1, • • • ,n) provided v > v0, i.e. the operator L + vl has its inverse (L + vI)~l. Thus Lu = h = f + Dif is equivalent to u = (L + u l ^ h
+ v{L + J / / ) _ 1 U
L 2 Theory of Linear Elliptic Equations
69
or u-v{L
+ vl)~lu
={L +
vl)~lh
i.e.
u - Au = w, where A = v(L + i/J)" 1 , w = (L + vl)~lh G #£(fi). To apply Fredholm's alternative theorem, it suffices to prove the compactness of the operator A : HQ(Q) —» HQ(Q). In fact, A can be regarded as a linear operator from l? (fi) to H\ (fi). If we use E to denote the embedding operator from H&(Cl) to L 2 (fi), then we have A = AE : H%(Sl) -> H&(ft). Since the embedding operator from HQ(SI) to L2(Q) is compact and A is a bounded linear operator as shown in the proof of Theorem 2.3.2, we can assert that the operator A — AE : HQ (fi) —+ HQ (0) is also compact. Thus the conclusion of the theorem follows from Fredholm's alternative theorem.
D
Exercises 1. Introduce the definition of weak solutions of the boundary value problem ' A2u = / , du dv
x € 0, 0,
x G 80,
and prove the existence and uniqueness, where CI C K n is a bounded domain, / G L2(Cl) and v is the unit normal vector outward to dfl. 2. Define weak solutions of the Neumann problem for Poisson's equation —Au = / , du . du
0,
ie(], x£
dfl,
where Cl C K™ is a bounded domain, / G L 2 (fl), v is the unit normal vector outward to dfl. And prove that the problem has a weak solution if and only if
/ f{x)dx = 0. Jn
70
Elliptic and Parabolic Equations
3. Assume A > 0. Define weak solutions of the equation - A u + Xu = f,
ieR"
and prove the existence and uniqueness, where / G L 2 (R ra ). 4. Prove Proposition 2.2.1. 5. Prove Proposition 2.2.4. 6. Let B be the unit ball in Rn and u G HX(B) be a weak solution of Laplace's equation - A u = 0,
x £ B.
i) Prove u G C°°(B); ii) Prove that if there exists a function v € C°°(M.n) such that u — v G H^(B), then u G C°°(B). 7. Let u G CQ(IR") be a weak solution of the semi-linear equation
- A u + up = /,
xeW1
where / G L 2 (R"), p > 0. Prove u G iJ 2 (R"). 8. Establish the theory of regularity for weak solutions of general elliptic equations.
Chapter 3
L2 Theory of Linear Parabolic Equations This chapter is a description of the L2 theory of linear parabolic equations parallel to the previous chapter. As in treating elliptic equations, we first discuss a typical equation, i.e. the heat equation in greater detail and then discuss the general linear parabolic equations in divergence form in a brief fashion. 3.1
Energy Method
In this section we introduce the energy method, one of the basic methods available to parabolic equations. Let fl C i " be a bounded domain with smooth boundary dQ, and T > 0 be a constant. Consider the heat equation — - Au = f{x, t),
(x, t)£QT
= nx
(0, T).
(3.1.1)
Different from elliptic equations, we are not permitted to prescribe the condition on the whole boundary of QT- One of the typical conditions to determine the solution is OPQT
where dpQr is the parabolic boundary of QT, i.e. dpQT=dQT\(Clx
{t = T}).
The problem of finding solutions of equation (3.1.1) satisfying this condition is called the first initial-boundary value problem. For simplicity, we merely = 0, i.e. the condition consider the case g{x, t) d,QT
u(x,t)=0,
(x,t) edQx 71
(0,T),
(3.1.2)
72
Elliptic and Parabolic
u(x,0)=uo(x),
Equations
xetl.
Sometimes we even assume g(x,t) boundary value condition
(3.1.3)
= 0, i.e.
u
consider the zero initial-
=0.
(3.1.4)
opQr
If g(x,t) is appropriately smooth in QT, then we may introduce a new unknown function w = u — g to transform the original initial-boundary value condition into the latter.
3.1.1
Definition
of weak
solutions ° 11
Definition 3.1.1 A function u £W2 (QT) is said to be a weak solution of the first initial-boundary value problem (3.1.1), (3.1.2), (3.1.3), if for any
there holds //
(ut(p + Vu-Wip)dxdt=
JJQT
fipdxdt
(3.1.5)
JJQT
and ju(x, 0) = uo{x) a.e. on Cl. o
° in
Remark 3.1.1 Since C°°(QT) *S dense in W2
(QT),
the test function
Sometimes, we merely discuss the equation itself and no initial-boundary value condition is concerned. In this case the following definition is needed. Definition 3.1.2 A function u £ W2' (QT) is said to be a weak solution of equation (3.1.1), if for any tp € CQ°(QT), the integral identity (3.1.5) holds. Remark 3.1.2
It is not difficult to prove that ifu S W2' (QT) is a weak o
solution of equation (3.1.1), then for any (p O .
&C°°(QT)
an
d hence for any
n
tp £W2 (QT), the integral identity (3.1.5) holds. The following propositions provide some equivalent descriptions of Definition 3.1.1, which are frequently used in the sequel. °II Proposition 3.1.1 A function u €W 2 ' (QT) satisfies (3.1.5) for any o
£C°°(QT)
^
an
d
// JJQT
on
ly tfu satisfies
(uttpt + Vw • Vipt)dxdt = II JJQT
ftptdxdt
(3.1.6)
L2 Theory of Linear Parabolic Equations
73
foranyipeC°°(QT)Proof.
Suppose u eW^iQr)
satisfies (3.1.5) for any ip
Then, for any
then, since tp
A
(3.1.6) holds.
satisfies (3.1.6) for any ip £C °°(QT)>
1 (QT)
_
o
/"*
&C°°(QT)
eC^iQr)-
implies / ip(x,s)ds
_
we
&C°°(QT),
may choose
Jo
ip(x,s)ds as a test function in (3.1.6) to derive (3.1.5). Jo
I 1 A function u GWJ (QT) satisfies (3.1.5) for any
Proposition 3.1.2 o
rj
0
if and only tfu satisfies
&C°°(QT)
(uttpt + Vu-V
if
[[
JJQT
fipte~9tdxdt
(3.1.7)
JJQT o
for any
Suppose u et
Then, since y>te~ //
&WI'1(QT)
&C°°(QT)I
(ut(pte'et
satisfies (3.1.5) for any
eC^iQr)-
have
+ Vu • S/
JJQT
f
JJQT
namely, (3.1.7) holds. o . ,
o
Inversely, suppose u £W 2 ' (QT) satisfies (3.1.7) for any (p eC°°(Qr)Then, since ip GC°°(QT) implies ip(x,t)eet
ft
-6
ip(x,s)eesds
GC°°(QT), Jo we can choose the latter as a test function in (3.1.7) to derive (3.1.6) and also (3.1.5) by Proposition 3.1.1. • Remark 3.1.3 Since C°°(QT) C V{QT) CW\'1(QT) and C°°(QT) is °II dense in W2 (QT), we may choose any
A modified Lax-Milgram's
theorem
To prove the existence of weak solutions of the problem considered, we will apply a modified Lax-Milgram's theorem. First we prove Lemma 3.1.1 Let H be a Hilbert space, V C H a dense subspace of H and T : V —> H a bounded linear operator. If T _ 1 exists and is bounded,
74
Elliptic and Parabolic
Equations
then the range of the conjugate operator T* of T is the whole space H, i. e. R(T*) = H. Proof. We want to prove that for any h € H, there exists u € H, such that T*u = h. To this purpose, we consider the linear functional F(z) = (h,T~1z), defined in R(T) = D{T~l)
Vz£R(T)
(domain of T~l).
Since
||F|| = sup \F(z)\ < \\h\\\\T-% IMI=i F(z) is bounded. Now we extend F(z) to be a bounded linear functional in R(T) which is a Hilbert space and apply Riesz's representation theorem (§2.1.2) to assert the existence of u S R(T) satisfying (u,z) = F(z) = (h,T-1z),
VzER(T),
i.e. (u,Ty) = (h,y),
VyeV
(T*u,y) = (h,y),
VyeV.
or
This and the density of V in H lead to (T*u,y) = (h,y),
Vy£H.
Thus T*u = h.
U
Modified Lax-Milgram's Theorem Let H be a Hilbert space, V C H a dense subspace and a(u, v) a bilinear form in HxV satisfying the following conditions: i) For some constant M > 0, |o(u,w)| < M||u|| ff ||i;||v,
Vu £H,\/v£
V;
ii) For some constant S > 0, a(v,v)>6\\vfH,
V«GV.
Then for any bounded linear functional F(v) in H, there exists u G H, such that F(v)=a(u,v),
WeF.
(3.1.8)
L2 Theory of Linear Parabolic
Equations
75
Proof. Since for any fixed v £ V, a(-,v) is a bounded linear functional in H, whose boundedness follows from the condition i), Riesz's representation theorem can be applied to assert the unique existence of Av G H, such that a(u,v) = (u,Av)H,
Mu&H.
(3.1.9)
Prom the bilinearity of a(u,v) and the condition i), it follows immediately that the operator A : V —• H thus denned is bounded and linear. Condition ii) and (3.1.9) imply (v,Av)H>S\\v\\2H,
VveV
\\Av\\H>S\\v\\H,
VueV.
and
Thus A~l exists and is bounded. Therefore we can apply Lemma 3.1.1 to assert that the range of A*, the conjugate of A, is the whole space H : R{A*) = H. Now Riesz's representation theorem is applied, from which it follows that there exists a unique h £ H, such that F(v) = (h,v)H,
VveH.
(3.1.10)
Since R(A*) = H, there exists u G H, such that A*u — h, and hence («, Av)H = (A*u, v)H = (h, v)H,
VU G V,
which combined with (3.1.9), (3.1.10) leads to (3.1.8). 3.1.3
Existence
Lemma 3.1.2
and uniqueness Let u ew\'1 {QT).
of the weak
• solution
Then for almost all t G (0,T),
h(t) - I (lu(x,Q)))2dx = 2 / / Ja Jo Jo.
u-^dxds, dt
where h(t) = / u2(x,t)dx, Ja
t<= (0,T).
76
Proof.
Elliptic and Parabolic
Equations
11
According to the definition of W2' (QT), there exists a sequence
{um} CC°°(QT),
such that mlimJ|um-u||
^1,1(^=0.
Prom this and Fubini's theorem, it follows that for almost all t € (0,T), lim hm(t) = h(t), m—MX)
where hm(t) = / u2n(x,t)dx.
Letting m —> oo in
Jn
ft
ft
hm(t) - hm(0) = / h'm(s)ds = 2 /
/»
Q
/
um—^dxds
and using lim
/
/ Um—^p-dxds =
™^°°Jo Jn
°t
u—dxds
J0 Jn dt
and (see Remark 1.5.1) lim hm(0) = lim / uL(x,0)dx m^oo
m->oo Jn
= /
{~iu(x,0))2dx,
Jn
we are led to the conclusion of the lemma.
•
Theorem 3.1.1 For any f € L2(QT), the first initial-boundary value problem (3.1.1), (3.1.2), (3.1.3) admits at most one weak solution. Proof. Let u\, un. be weak solutions of problem (3.1.1), (3.1.2), (3.1.3). Then by the definition of weak solutions (Definition 3.1.1) and Remark 0
i i
3.1.1, u = u\ — ui €W 2 ' (QT), JU(X, 0) = 0 and u satisfies //
(utip + Vu • Vip)dxdt = 0,
V
&W12°{QT)-
JJQT
Choosing
2 (uut + \Vu\ )dxdt = 0,
iQs JJQ,
where X[o,s](t) is the characteristic function of the segment [0, s] (0 < s < T). Hence //
uutdxdt = -
\Wu\2dxdt < 0.
L2
Theory of Linear Parabolic
Equations
77
From this, using Lemma 3.1.2 and noticing ju(x, 0) = 0, we deduce / u2(x, s)dx < 0,
a.e. s € (0, T).
Jn
Therefore u = 0 a.e. in QT, i.e. ui = u
•
Now we are ready to prove the existence of weak solutions. In this section, we consider the problem with zero initial-boundary value condition (3.1.4) and apply the modified Lax-Milgram's theorem stated above to the existence for this problem. Other methods will be introduced in §3.2 and §3.3 to the existence of weak solutions for problem (3.1.1), (3.1.2), (3.1.3) with general initial values. For any f £ L2(QT),
Theorem 3.1.2
the first initial-boundary value * i i
problem (3.1.1), (3.1.4) admits a weak solution u £W2 Proof.
(QT)-
Denote
a(u, v) = / / JJQT
[utvt + Vu • Vvt)e-etdxdt,
u £WI'X(QT),
V£
V(QT),
where 0 > 0 is a constant. Obviously Hu,v)\
< \\U\\WI,I{QT)\\V\\V{QT),
On the other hand, for v G //
Vv •
U ew^iQT),
V(QT),
v e V(QT).
(3.1.11)
we have
Vvte-etdxdt
=\IL«-et>v?dxdt ^t(\Vv\2e-et)dxdt+9-JJ
=\ j l IQ -6T
f IVd 2
Jn
t=T
dx--
f
2 Jn
\W\h-°*dxdt
7|Vd
2
t=o
dx + - [f 2
JJQT
\Vv\2e~etdxdt,
denotes the trace of | Vu| 2 when t = 0. Prom this, noticing
where7|Vv| t=o
that v € V(QT) implies -yS/v
0 and hence t=o
Jn
dx = 0, it=o
78
Elliptic and Parabolic
Equations
we obtain //
-8T
Vv • S7vte-etdxdt
JJQT
2
• Since C°°(QT) is dense in
V(QT),
(3.1.12)
JJQT
Poincare's inequality of the form
v2dxdt < fi / /
//
\Vv\2dxdt.
> ——- / /
JJQT IQT
\Vv\2dxdt
JJQT JJQT
still holds. From (3.1.12) we are led to Vi> • Vvte~etdxdt
// JjQT
-6T Qp-vT
rr rr
Qa-8T
flp-^
rr
Therefore a(v,v)>S\\v\\2wi,1{QT),
VveV(QT),
(3.1.13)
where 5 = min < 6
9e~eT ee~eT
eT
'
4
'
4/i
Choose # =wl'\QT), V = V(QT)- Then, by Proposition 1.4.1 of Chapter 1, V C H is a dense subspace of # and (3.1.11), (3.1.13) show that the conditions i), ii) in the modified Lax-Milgram's theorem are satisfied. Obviously, //
fvte
et
dxdt is a bounded linear functional of v in
JJQT
* 11
H. Therefore there exists a u G H =Wi a(u,v)=
[f
(QT), such that
fvte-9tdxdt,
^v£V{QT),
JJQT IQT
namely, ff JJQT
(utvt + Vu-Vvt)Q-6tdxdt=
ff
fvte-etdxdt,
Vv G V(Qr).
JJQT
This means, by Proposition 3.1.2, u is a weak solution of problem (3.1.1), (3.1.4). •
L2
3.2
Theory of Linear Parabolic
Equations
79
Rothe's Method
In this section we present another important method, called Rothe's method or semi-difference method, which is available to the study of existence for parabolic equations. The basic idea is to difference the equation with respect to the time variable, solve the obtained elliptic equations to construct approximating solutions and use the estimates for approximating solutions to complete the limiting process to arrive at the desired solution. Let Q C M.n be a bounded domain with piecewise smooth boundary. Consider the first initial-boundary value problem (3.1.1), (3.1.2), (3.1.3). Theorem 3.2.1
Let f G L2(QT)
anduQ G # o ( n ) - Then problem (3.1.1),
(3.1.2), (3.1.3) admits a weak solution u GWy (QT)Proof. Suppose for the moment, / G C(QT). We proceed to prove the existence of weak solutions in several steps. Step 1 Difference the equation with respect to the time variable t to construct the approximating solutions. For any positive integer m and function w(x,t), denote wm'j(x)=w{x,jh) where h = T/m.
(j = 0,l,--- ,m),
Consider the approximating equation of (3.1.1) Aum,3=fm,:
{j = lj2,...,m).
(3.2.1)
According to the condition of the theorem, um'° = UQ G HQ(Q). Suppose that u m '-J -1 G HQ(Q) is known. We want to prove that equation (3.2.1) admits a weak solution u m j G HQ(Q). Denote v = u m j . Then (3.2.1) can be written as 1
vm,j-l
-Av + TV = fm<3 + — T —
(3.2.2)
which is an elliptic equation. It follows from Theorem 2.3.1 of Chapter 2, (3.2.2) admits a unique weak solution v = u m j G HQ(Q,). Thus, by induction, we obtain um'\um'2,--in
HQ(Q),
,um'm
which is the weak solution of (3.2.1) for j = 1,2, • • • ,m, succes-
80
Elliptic and Parabolic
Equations
sively, namely, for j = 1,2, • • • , m and for any ip £
/ {^^
HQ(£1),
there holds
~ " m '^ 1 )^ + Wum,i " V < ^) dx = / / m 'Vz-
(3.2.3)
So far we merely obtain an approximation of the required solution on the line t = jh = jT/m (j = 1,2, • • • , m). In order to obtain an approximating solution in the whole domain QT, we define m
wm{x,t)
=Y,Xm'j(t)um'j(x)
(3.2.4)
and
« m ( i , t ) = 5 ] f J ' ( t ) P ( t ) i ' , " J W + (1 - AmJ(t))uroj'-1(i)]
= 5>mj'(*K,J'-1(*) m
+ 53x m , j i (*)A" , J (t)(« T , , , , '(x) - u r o J - 1 ( x ) ) , where x m ' J ' is the characteristic function of the segment [(j — l)h,jh) AmJ(4)=n-o--D, (^ 0,
(3-2-5) and
*€[(,--i)M/o, otherwise.
For fixed x e Cl, (3.2.4) is a step function of t, which equals um'*{x) on [{j — l)h,jh), and (3.2.5) is a broken line function of t, which equals umJ-l{x){um>i{x)) at t = {j- l)h(t = jh). Denote
m
771
f (x,t) =
j2xm'jwm,jw-
Then from (3.2.3) we see that for ip € H&(Q,), t € (0,T), j (^-
+ Vw m • V j dr = /" /"Vdar.
Estimate the approximating solutions.
(3.2.6)
I? Theory of Linear Parabolic Equations
81
We need to prove the following estimates dum dt
2
\\^m\\h(QT) m
m
(3.2.7)
<Mm,
LHQT)
(3.2.8)
<4TMm, 2
\\™ -u \\lHQT)
(3.2.9)
where Mm = ||Vuo||£ a( n) + ll/ m ll! 2 (Q T ). To this purpose, we choose ip — u m j — u m , J _ 1 in (3.2.3) to obtain ))dx
= / fm'j{um>i
-um'j-l)dx.
Prom this we can deduce i . | | „ m J _ •>,mJ-1|l2 u
u
h\\
\\\7n,m<3\\'i 2 L (Q)
4-
llL2(n) + l l V u
\\L
lli,2(n) •
_ll.,in,j
.,m,j
2h"
—1||2
"L'W
I +
h"
II
2"* 21
fm,}\\2 L2
"
(")•
Hence ^ll u
u
llL2(n) + l l V u
llL2(fi) (j = l , 2 , . . . ,m).
(3.2.10)
In particular,
iiv«mj'iiia(n) < i i v t ^ - 1 ! ^ + h\\rm*{ay
(3.2.11)
Iterating (3.2.11) j times yields l|V« m J ||£ a ( n ) < ||V«o||i a ( n ) + / i ^ | | / m ' i | | 2 ( n ) < M m and summing (3.2.10) on j from 1 up to m yields 1 1
m V II | inm , j _ u m , j - l | | 2 2 L (Q.) J= l
(3.2.12)
82
Elliptic and Parabolic
Equations
+ Am,J'(Vum,J' - Vu m j ' - 1 )).
Thus, using (3.2.13) leads to dun dt
•.
Z
Til
n
LHQT)
j =
l
and using (3.2.12) leads to HVu m ||| 2(QT) pT
m
= J2 Xm,j / 1(1 - A n , J ')V« m j ' - 1 + •=1 Jo Jn <2E / j=l
\m'jVum'j\2dxdt
*m'j (llV« m - , '- 1 |li a( n) + HVu r o 'ii 2 ( n ) ) dt
0
m
^JXIIVI^-1!!^)
+ ||VumJ||ia(n))
<4TM m . (3.2.7) and (3.2.8) are then proved. By the definition of wm and u m , we have ^ X m , i ( l - Am'J')(um'J' - um>j~l) which and (3.2.13) imply
and prove (3.2.9). Step 3 Complete the limiting process.
(3.2.13)
L 2 Theory of Linear Parabolic
Since / £ C(QT), fm converges to / in
83
Equations
L2(QT)-
Hence
Jirn^M™ = ||V«o||£ a{n) + | | / | | £ 2 W T ) , which implies, in particular, that {Mm}m=i i s bounded. Therefore, it follows from (3.2.7), (3.2.8) and Poincare's inequality m
•u
ll
2
that {um}^=1 is bounded in W2' (QT), which implies the existence of a subsequence of { u m } ~ = 1 , supposed to be {um}^=1 itself, and a function u £ W2' (QT), such that um converges to u, and —-— and V u m converge dii
weakly to — and Vu respectively, in
L2(QT)-
(JTI
Now we proceed to prove that u is a weak solution of problem (3.1.1), (3.1.2), (3.1.3). First prove u
£W\'1(QT)
and -yu(x,0) = u0(x) a.e. in £2. ° 1 1
To this end, it suffices to check um £ W y (Qr)> ryurn(x)0) = UQ{X) a.e. in fl. For every positive integer m, choose {u^}(^L1 limJuZ-um\\wui(QT)=0, lim / \uf(x,0)
-uo(x)\dx
o
CC°°(QT),
such that (3.2.14)
= 0.
(3.2.15)
For example, we can construct u™ as follows: first choose {u™'3}'%L1 C C§°(Cl), such that lim \\u™'j - umJ\\^m
=0
(j = 1,2, • • • ,m),
K—>00
and then replace u m J by u™'-1 in the expression (3.2.5) of um, followed by a °II mollification with respect to t. um SWV (QT) then follows from (3.2.14), m and 'yu (x, 0) = UQ(X) a.e. in Cl follows from (3.2.15), Remark 1.5.1 and Remark 1.5.2 of Chapter 1. In order to verify that u satisfies the integral identity in the definition o
of weak solutions, we integrate (3.2.6), which holds for any
('^
jf
fmipdxdt.
GC°°(QT),
(3.2.16)
Since from (3.2.9) we see that the fact that um converges to u in L2(QT) implies the convergence of wm to u in L2(QT), we can let m —> oo in
Elliptic and Parabolic
84
Equations
(3.2.16) to deduce ( "PT^ ~ U^(P ) dxdt = II
//
ftpdxdt,
which is equivalent to ' du tp + Vu • Vip I dxdt dt
JJQ.
ftpdxdt,
JJQ-
° 11
due to u £W2 (QT)- Summing up, we have proved that u is a weak solution of problem (3.1.1), (3.1.2), (3.1.3). Moreover, letting m —> oo in (3.2.7), (3.2.8), we obtain du L2(QT)
(3.2.17)
HV«||£aWr) <4T (||V«o||i a(n ) + ll/|li a( Q T) ) Now we turn to the general case / S L2(QT)such that lim || fk - f\\L*(QT)
Choose {fk}^!
(3.2.18) C C(QT)
= 0.
fc—>oo 0
1 1
Let uk GW2 (QT) ( dukbe the weak solution of the problem Aufc = fk, (x,t)eQT, dt uk(x,t) = 0, (x,t)e9Slx (0,T), [uk(x,0)
= u0(x),
xeQ,
as constructed above. Prom (3.2.17), (3.2.18), duk dt
l|V«fc||ia(Qr) <4T (||Vu 0 ||| 2(n) + \\fk\\lHQT)) 0
11
These show that {wfc}^i is bounded in W2' (QT)- Hence we can choose a subsequence of {ufc}£Li, supposed to be {uk}^ X
u GWl' (Qr),
itself, and a function
such that uk converges to u, and -^— and Vuk converge
L2 Theory of Linear Parabolic
weakly to — and Vu respectively, in L2(QT)-
//
l-zrlP
+ 'Vuk-'V(p)dxdt=
Equations
85
Letting k —> oo in
fkipdxdt,
we see that u satisfies the integral identity in the definition of weak solutions. Since 7Ufc(:r,0) = uo(x), from / \u(x,t) — uo(x)\2dx Jfi < / \u(x,i) - uk(x,t)\2dx
Jn
it follows that ju(x, 0) =
3.3
+ / \uk(x,t) -juk(x,
Jn
0)\2dx,
O
UQ(X).
Galerkin's Method
In this section another important method available to parabolic equations, called Galerkin's method, is introduced. This method is efficient in both theory and practical computation. The basic idea of the method is to choose a suitable basic space X and a standard orthogonal basis {u>i(x)} and then oo
to find a solution of the form Yjcj(i)a;i(x). i=l
As a typical example, we still consider problem (3.1.1), (3.1.2), (3.1.3). In order to apply Galerkin's method to prove the existence of weak solutions, we need the following Hilbert-Schmidt's Theoremfsee [Jiang and Sun (1994)]) Let H be a separable Hilbert space, A be a bounded and self-adjoint compact operator and {Aj} be all eigenvalues of A. Then there exists an orthonormal basis {WJ}, such that Auii = AjWj. The existence of weak solutions is proved in four steps: Step 1 Construct basis. Define operator A = ( - A ) " 1 : L2(n) - L2(Q),
f -> Af,
86
Elliptic and Parabolic
Equations
where Af is the unique solution of the problem —Au — / , x £ ( l , an
= 0.
From the L 2 theory of elliptic equations (see Theorem 2.1.1 and Remark 2.2.2 of Chapter 2) we see that Af e H2(Q.) D #o(fi) and \\Af\\HHQ) < C\\f\\LHn), i.e. A is a bounded operator. Integrating by parts leads to
J
/ g(Af)dx
=- [
JQ JQ
A(Ag)(Af)dx
JQ
= - [ A(Af)(Ag)dx
= f f(Ag)dx,
JQ
Vf,g £ L 2 (ft),
JQ
which means that the operator A is self-adjoint. Since HQ(£1) can be compactly embedded into L2(£l), the operator A is compact. Therefore, by Hilbert-Schmidt's theorem, there exists a standard orthogonal basis {ui}iZi, such that AuJi = AjWj. Since
Au = o, x € n, = 0 dQ
admits only a trivial solution, it is certainly \%^Q and hence - A u > j = -— w». Aj
Using Theorem 2.2.4 of Chapter 2 and the embedding theorem, we conclude Wj € C 2 (fi) provided dd is appropriately smooth. S t e p 2 Construct approximating solutions. Set t*o = / ^ QWi and let i=l
m
satisfy 'dUjn
, dt
u>k ) = (Aum,LJk) + {f,uk),
k = 1,2, ••• ,m,
(3.3.1)
L2 Theory of Linear Parabolic Equations
87
where (•, •) is the inner product in L2(Q). Since
x
'
i=\ m
1
(Au m ,w fe ) = ^c™(t)(Awi,u f e ) = ——cjp(i), (3.3.1) implies ±cT(t) = -j-cT(t)+fk(t),
(3.3.2)
where fk(t) = (/.wjt)- Hence # ( * ) = e-*/A* f cfc + y e^x"fk(r)dA
.
Step 3 Estimate approximating solutions. Multiplying (3.3.1) by c™(£) and then summing on k from 1 up to m yield
namely, 2^ll u m(-,i)ll|2(n) = -||V« m (-,t)lli,a(n) + (/(-»*).«m(-.*))Integrating over (0,£), we further obtain
2ll«i"(-.*)llia(n) - 2ll u '"( - ' 0 )lli 2 (n) = — II JJQt
\Vum\2dxdt+
//
fumdxdt.
JJQt
Using Poincaxe's inequality and Cauchy's inequality with e leads to sup | | w m ( - , i ) | | | 2 ( n ) + / / te[0,T]
\Vum\2dxdt
JJQT
(3.3.3)
88
Elliptic and Parabolic
Equations
Next we multiply both sides of (3.3.1) by —c™(£) and sum on k to obtain dum
dum\
( = Aum
-dr'-dr)
dum\
( + f
{ >-dr)
dun
{ >-d-t
Integrating over (0, T), integrating by parts with respect to x and using Cauchy's inequality we are led to \dum jf Jf^\ dxdt+\\Vum(;T)\\lHn) dt
fdxdt.
(3.3.4)
IQ JJQT
Combining (3.3.4) with (3.3.3) yields
I
QT
du, m |Um| + |Vu m | + dt \ 2
2
2
dxdt < C,
(3.3.5)
where C is a constant independent of m. S t e p 4 Complete the limiting process. The estimate (3.3.5) implies the existence of a subsequence of {um}, supposed to be {um} itself, and a function u S W2' {QT), such that um , dum ,_ ,, du , _ converges to u, and —-— and \um converge weakly to -— and vu respecat at L2(QT)-
tively, in
The function u is expected to be a weak solution of problem (3.1.1), (3.1.2), (3.1.3). 0
11
First we have u £W<2 {QT)- In fact, by Theorem 1.5.4 of Chapter 1, O i
i
O i
i
um GW2' (QT) and u is the weak limit of um in Wi (QT)Let h S C 2 (Q) and ip e C 2 [0, T] be arbitrarily given functions such that h
= 0 , ip(0) = tp(T) = 0. Choose a sequence 3 h x
j( )
'^2ajkUk(x),
= fe=l
1
converging to h in H ^). Multiply (3.3.1) by ip(i), integrate over (0,T) and integrate by parts with respect to x. Then let m —-> oo to obtain //
—Ukipdxdt = -
Vu • Vu)ki>dxdt + / /
fuikiidxdt.
L2 Theory of Linear Parabolic
89
Equations
Prom this it follows by multiplying by ajk and summing on k from 1 up to j , that du
hjtpdxdt
JJQ-
-II
II
Vu • Vhjipdxdt + / /
JJQT
fhjipdxdt
JJQT
Letting j —> co then leads to
JJQi
du hipdxdt dt
Vu-Vhipdxdt+ JJQ-
//
f hipdxdt.
JJQT
Because of the arbitrariness of h and ip, we may assert that for any ip £ C O ° ( O T ) (for instance, use Lemma 3.5.1 of §3.5.3) // JJQT
—
Vu • Wipdxdt + / /
JJQT
fipdxdt.
JJQT
This formula can be further proved without difficulty to hold for any o
namely, u is a weak solution of (3.1.1). It remains to verify 7u(:r,0) = UQ(X). We have €C°°(<9T)I
/ \u(x,t) — uo(x)\2dx Ja ; / \u(x,t) - um(x,t)\2dx Ja.
+
J
Jn
+ I Ja
£(c?(t)-*)<*(*)
dx
i=l
^2 CiU)i(x) dx l=7Tl + l
=h + h + h Evidently, I\ and I3 can be made arbitrarily small by choosing m large enough. Once m is fixed, Ii can be made arbitrarily small if t > 0 is small enough. Thus we have lim / \u(x,t) — uo(x)\2dx = 0.
3.4
Regularity of Weak Solutions
We first discuss the interior regularity.
90
Elliptic and Parabolic Equations
Theorem 3.4.1 Let f € L2(QT) and u of problem (3.1.1), (3.14). Denote
€W\X{QT)
Q5T = Slsx
fls = {xe Q; dist(x, 9fi) > 6}, Thenu£Wl'l{QsT)
be a weak solution (0,T).
and
\M\WI\QI.)
<
(3.4.1)
with constant C depending only on n and 5. Proof.
By the definition of weak solutions,
//
(ut
JJQT
fipdxdt,
V
(3.4.2)
JJQT
Choose a cut-off function r)(x) £ 0^(0.),
such that rj = 1 on fig, 0 < rj{x) <
1 and |V77(ar)| < —. Since for small h, tp = A{*(r]2Aiu)x[o,s} we can substitute it into (3.4.2) to obtain ff JJQs
= ff
L A ^ V A > ) + Vu • V ( A j / (v2Aiu)} L
€WI'°(QT),
dxdt J
fAi*(r,2A{u)dxdt,
(3.4.3)
JJQs
where i = 1,2, • • • , n, X[o,s](t) is the characteristic function of the segment [0, s] (0 < s < T). Using the properties of difference operators and Lemma 3.1.2, yields JJ
utAih\n2Aihu)dxdt=JJ
^^r,2Aiudxdt
-\j]Q}t^^hu)2)dxdt r,2(Aiu(x,t))2dxt=S t=o
=1 f * Jn =\ f
2 V
(Aiu(x,s))2dx.
J it
Thus, from (3.4.3), we obtain \ f n2{Aihu{x,s))2dx+ z
Jn
< ff
ff
Vu.V(Ai*(r,2Aiu)dxdt
JJQ3 2
fA\*{r] Aiu)dxdt,
0 < s < T,
(i = 1,2, • • • , n)
L2 Theory of Linear Parabolic
Equations
91
which implies i
f rj2{Aihu{x,s))2dx
sup
2o<s
< ff
Vu-V(Ai*(r)2Aiu)dxdt
+ ff JJQT
fA^^A^dxdt,
(i = l , 2 , - . - , n ) .
(3.4.4)
JJQT
Starting from this, we can proceed as we did in Chapter 2 for Poisson's equation to derive / rj2(Aihu{x,s))2dx
sup
\A]yu\2dxdt
+ ff
0<s
JJQ%.
(t = l , 2 , - . . , n )
(3.4.5)
which implies, by virtue of Proposition 2.2.2 of Chapter 2, that r?|Vu| S L°°((0,T);L 2 (f2)), D2u € L2(Q5T) and sup / r,2\Vu(x, s)\2dx < C K(||V«||i a ( Q T ) + | | / | | i 2 ( Q T ) ) , o<s
(3.4.6)
\D2u\2dxdt < C (||V«||i a ( Q T ) +
(3.4.7)
ff
||/|||2(QT))
•
JJQT
The difference is that here there is an additional nonnegative term on the left side of the inequality and the integrals are with respect to both the space variables and the time variable. In addition, as the starting point of our derivation, (3.4.4) is an inequality rather than an equality as we got for Poisson's equation. However this does not prevent us from deriving the desired estimate. To derive the estimate on Ut, we use the difference operator in t, which is denoted by A° for simplicity. Extend u to Q x (-co, +oo) by setting u = 0 outside jf
QT-
Take tp = r]2A°hu
(^rj1Alu
€WI'°(QT)
+ Vu-V{r}2A°hu))dxdt
to obtain = ff
fV2A°hudxdt
ff
A°h{r)2\Vu\2)dxdt
or // JJQT
rj2^A°hudxdt &t
= ff
r)2fA°hudxdt-
JJQT
JJQT
- 2 // JJQT
r)A°huVu • Vrjdxdt.
92
Elliptic and Parabolic
Equations
Using Holder inequality, Cauchy's inequality with e and (3.4.6) gives
I
r]2—A0hudxdt dt
QT
<-
rj2(A°hu)2dxdt+
(I
[f
4 JJQT
+ \ fl II
* JJQT
+
A°h(r)2\Vu\2)dxdt
+ l [[ ' JJQT
r)2(A°hu)2dxdt + 4 / /
4 JJQT
<\
rffdxdt
JJQT
|Vr/| 2 |Vu| 2 dxrft
JJQT
r]2(A°hu)2dxdt+
f[
fdxdt
+ 4 sup
JJQT
f
r}2\Vu(x,s)\2dx
O<S
(T)71|V*|2<M •r,HK")2dxdt + C (ll V«||l, ( „ ri + VHIHOT))
•
>QT
By Proposition 2.2.2, we get
// a /(t)«4// 0 /(§^ + C(||VU||2L2(QT) + | | / | | | 2 ( Q T ) ) ,
which implies
JJ
s
(^fdxdt
< C (||VU\\2LHQT) + \\f\\lHQT)) .
(3.4.8)
Combining (3.4.8) and (3.4.7), we obtain the desired estimate.
•
Since we can treat the integrals containing ut in deriving the estimate near the lateral boundary similar to the interior estimate, we can obtain the estimate near the boundary for equation (3.1.1), and hence combine it with the interior estimate to obtain the following result on the global regularity. Theorem 3.4.2 Let f £ L2(QT) and u €W\'X{QT) be a weak solution of problem (3.1.1), (3.14). If dQ e C2, then u £ Wl'l{QT) and HIV^
W T )
< C{\\U\\W1,O{QT)
+ ||/||L'WT)),
(3A9)
with constant C depending only on n and fi. Remark 3.4.1
Under the assumptions of Theorem 3.4-2, there holds \\u\\w^(QT)
(3.4.10)
L2 Theory of Linear Parabolic
93
Equations
with constant C depending only on n and Q.. Proof.
By the definition of weak solutions, ff
(uttp + Vu-Vtp)dxdt=
JJQT
//
ftpdxdt,
V
ewl'°(QT)•
JJQT
Choose (p = u and use Cauchy's inequality with e and Poincare's inequality on fi to obtain - / u2(x,t)dx 2jn
= //
\Vu\2dxdt
+ //
«=o
JJQT
fudxdt
JJQT
<— ff
u2dxdt + £ ff
<\ ff
\Vu\2dxdt+^ ff
2/u JJQT 2 JJQT IQT
fdxdt
2 JJQT JJQT *2 JJQi
fdxdt,
where \x > 0 is the constant in Poincare's inequality. Since u satisfies the zero initial value condition, this leads to
ff
\Vu\2dxdt < fi ff
JJQT
fdxdt,
(3.4.11)
JJQT
from which we obtain by using Poincare's inequality on D. ff
u2dxdt < \x2 ff
JJQT
fdxdt.
(3.4.12)
JJQT
Combining (3.4.11) with (3.4.12) gives
Finally we substitute it into (3.4.9) to obtain (3.4.10).
•
Furthermore, we have *I I
T h e o r e m 3.4.3 Let u GW2 (QT) be a weak solution of problem (3.1.1), (3.14). If dQ, e C2k+2 and f € W22fe,fe(Qr) for some nonnegative integer k, then u e W2k+2'k+1 (QT) and \\u\\wik+2,k+nQT)
< C ( | | t i | | W a i . o W T ) + \\f\\w^k(QT))
where C is a constant depending only on k, n and Q,.
>
94
Elliptic and Parabolic
Equations
Corollary 3.4.1 Let u &W\'1{QT) be a weak solution of problem (3.1.1), (3.1.4). IfdSl G C°° and f G C°°(QT), then u G C°°(QT). L2 Theory of General Parabolic Equations
3.5
Now we turn to the general parabolic equations Lu— — - Dj(aijDiu) where ay, bt, c G Lco(QT), parabolicity condition A|£|2 < dijix^Uj
+ biDiU + cu = f,
f G L2{QT)
< A|^| 2 ,
(3.5.1)
and oy = aji, satisfy the uniform
V£ € R n , (x,t) G QT,
where A, A are constants with 0 < A < A. Here, as before, repeated indices imply a summation from 1 up to n. As in the preceding sections, we consider the problem for (3.5.1) with the initial-boundary value conditions u(x,t) = 0,
( x , t ) 6 3 f i x (0,T),
(3.5.2)
u(x,0) = u0(x),
l£fl.
(3.5.3)
Existence of weak solutions of the problem will be treated by means of the methods which we have applied to the heat equation in the preceding sections. The weak solution u GW2' (QT) of problem (3.5.1), (3.5.2), (3.5.3) is defined by
I
{ut
QT
ftpdxdt,
V GC°°(QT).
(3.5.4)
QT
All of the methods will be described in a brief fashion. 3.5.1
Energy
method
As shown in §3.1, the application of the method to the existence of weak solutions is based on a modified Lax-Milgram's theorem. This theorem can also be used to equations of the form (3.5.1) whose coefficients ay depend only on the space variable x. To demonstrate this fact, as we did in §3.1,
L 2 Theory of Linear Parabolic
95
Equations
we need the following identity which is equivalent to (3.5.4): (utft + aijDiuDjifit + biDiU(pt + cu
//
II
fipte-etdxdt,
V^eC°°(Qr),
where 8 > 0 is a constant which can be chosen arbitrarily. As in §3.1, we consider only the case UQ = 0. Denote the bilinear form a(u, v) as (utvt + UijDiuDjVt + biDiUvt + cuvt)e~~etdxdt,
a(u, v) = / / JJQT
UQW\'1{QT),V£V{QT),
whose boundedness is obvious. To prove the coercivity of a(u, v), i.e. for some constant S > 0, a(v,v) > % | | ^ I . I W T ) ,
VveV(QT),
(3.5.5)
we need to estimate all terms in the expression of a(v, v). Since a^ are independent of t, we may use the parabohcity condition to derive, similar to the proof of Theorem 3.1.2, for v € V(QT), a,ijDivDjVte~~0tdxdt
11 JJQT
^-(aijDivDjv)e~etdxdt
= 2 //
TT (aaDivDiVe"61)
=x / / 2 JJIQT QT -8T
2
dt
K
>
I auDivDjV
Jn
Ae- e T
f \Vv
t=r
Jo.
2
2
dx + - / /
dx + t=T
aaDivDiVe~etdxdt
dxdt + -
2JJQT
°4 II 2
JJQT
aaDivDiVe~etdxdt
\Vv\2e~etdxdt
JJQT
0t
e~ dxdt.
In addition, using Cauchy's inequality with e, we get // JJQT
biDiVvte~6tdxdt
(3.5.6)
96
Elliptic and Parabolic
v2e~etdxdt
<e ff
+-
Equations
ff
\Wv\2e~8tdxdt,
(3.5.7)
v2e~etdxdt,
(3.5.8)
and cvvte~8tdxdt\
// JJQT
<e / /
I 0t
v?e- dxdt
JJQT
+£
ff JJQT
where the constant C depends only on the bound of \bi\ and \c\. Combining (3.5.6), (3.5.7), (3.5.8) and using Poincare's inequality, we are led to a(v,v) >(1 - 2e) / /
v2e~etdxdt
JJQT
+ (e-±-C) ff \Vv\2e-0tdxdt 4
+I
e ,
JJQT
V
S-?)1" "^
where /x > 0 is the constant in Poincare's inequality. From this, (3.5.5) follows immediately by choosing e > 0 small enough and 6 > 0 large enough. Remark 3.5.1 Recalling Theorem 2.3.1 of Chapter 2, we observe that to prove the existence of weak solutions for elliptic equations, the coefficient c is required to be less than some constant CQ. However, as stated above, for parabolic equations, no other conditions in addition to the boundedness of c are required. This is an essential difference between parabolic equations and elliptic equations. 3.5.2
Rothe's
method
In applying Rothe's method to the heat equation in §3.2, we transformed the problem into the one of solving elliptic equations and establishing some necessary estimates; the resulting elliptic equations are treated by means of variational method. All of these are available to more general parabolic equations in divergence form. However, since the variational method can not be applied to elliptic equations involving terms of first order derivatives, here we merely discuss equation (3.5.1) with bt = 0(i = 1,- • • ,n). We stress that, different from the case of elliptic equations, in treating par-
L2 Theory of Linear Parabolic
97
Equations
abolic equations, no other conditions in addition to the boundedness of the coefficient c, are required. In the present case, the elliptic equation obtained by discreticizing (3.5.1) with respect to t is ,,m,k-l
j . + c)v = fm'k + ——, whose corresponding functional is J{v) = - / (aijDivDjV
+[r+
.
,.m,fc-lN
cjv2 \
-2(/ m ' ,s + ^ — y)dx,
Wei^O).
Since in the expression of J(v), the coefficient of the term v2 is — + c h which can be made nonnegative if h > 0 is small enough, because of the boundedness of c. Thus the boundedness of J(v) from below is obvious. 3.5.3
Galerkin's
method
In applying Galerkin's method, the key step is to choose a suitable basic space and a standard orthogonal basis in it. For general parabolic equations (3.5.1) in divergence form, we choose L2(Q) as the basic space. The existence of the needed basis is proved in the following Lemma 3.5.1 Assume 80, 6 C2. Then there exists an orthonormal basis ( w i}iSi in L2(Q) satisfying the following conditions: i) ua e C2(f2), Ui = 0, (ui,Wj)Li(n) = <5y; oil
it) For any ip £ HQ (Q.) and e > 0, there exists a function of the form N
cteR,
i=l
such that \\V~
<£;
Hi) For any v € C2(QT) vanishing near the lateral boundary
DIQT
and
98
Elliptic and Parabolic Equations
e > 0, there exists a function of the form N
vN(x,t)
a(t) G C 2 ([0,T]),
= Y,Ci{t)^i{x), i=l
such that //
(\v-vN\2+
\Vv-VvN\2)dxdt
<e.
JJQT IQT
Proof. Since dCl € C 2 , by means of local flatting of the boundary, finite covering and partition of unity, we may assert the existence of a function C(x) G C2(f2), such that C = 0 and C(x) > 0 in fi. Let Q, C {x G R n ; 0 < ^ < Z, i = 1,2, • • • , n} and denote n/2
n
nSin(
Wfc(a;)
where A; = (fci, fc2,..., fcn) and fcj (j = 1, • • • , fcn) are positive integers. Given ip G Cg°(fi). By the definition of C, j G C$(fi). Hence for h > 0 small enough, ( ^ ) G Cg°(ft) and
<
#!(")
2'
where /^ denotes the mollification of / with radius h. Now we expand I — I in a Fourier series with respect to {u)fc}. Since this series and the series obtained by differentiating each term formally converge uniformly in Q, for any e > 0, there exists a function of the form V J Ck<*>k{x), such l
that
v s /
'»
^
l
i.e. f >»
£
l
£
c
(7) - E
CfcWfe
L 2 (fi)
<2' if!(fi)
L2
Theory of Linear Parabolic
+E
dX
i\<Jh
Equations
<
!
L2(Cl)
Thus
E
<£,
CkUJk 2
l
L (f2)
/V E a9^vcy
cfcDfc
^ ***
<e. 2
L (fi)
Setting wj(x) = ((x)wfc(:E) leads to
M7~ E l
CkGlk
L2(Q)
L (fi)
and
£
dw
^
v-^
9^t
i
^ L2(n)
i=l n
sE
-^ 7 - y " cfcwfe M ^ 1<7?
L2(n)
i=\
+E c
d ftp dxt \ C
5Z c* dxi
duk
Kk i
»=i
L2(n)
V? -
5 Z l
CkU}
*k m(n)
where the constant C depends only on C and is independent of
100
Elliptic and Parabolic
Equations
It remains to prove that {uJk} satisfies iii). To this end, we first expand /
\ .
T=T
. i
i .
2rrm
| , „ — < ) m = _ and ( c o s _ , | ^ v(x, t) in QT with respect to ^ ssin t ,
,.
v x
v-^
/ N . 2m7r
t) = Z-, am{X) m=l
(,
y-^ . , .
Sin —^-t
+ 2 ^ Pm(x) m=0
:
2m-K COS - ^ - * ,
where am,/3m € C 2 (fi) and a„ = /? m = 0. Since this series and the an Ian series obtained by differentiatii series obtained by differentiating each term formally converge uniformly in QT, for any e > 0, there exists a integer Ni, such that //
(\v - vNl | 2 + |Vu - VvNl | 2 ) ctecft < e,
where /
\
V"»
/ \ •
2m7T
^r—>. . .
2m7T
vNl (x, t) = 2 ^ am(aO sin -—-* + 2 ^ ftn(i) cos —jT*m=l
m—0
Prom ii), it follows that there exists a positive integer N, such that
am-
Yl
c'k,mUk
i
Hi(n) 2 c
<—,
fc,mwfc
l
m = 0,l,--- ,iVi.
tfi(n)
Hence //
{\v-vN\2
\Vv-VvN\2)dxdt
+
JJQT
where NI
VN
2TO7T
(x, t) = 2^ sin - — t m=l
2_,
V^l
1
)
l
cos—-t m=0 nN
•:^2ci(t)Wi(x). i=l
2^ \
Ck.mWkW
L2 Theory of Linear Parabolic
Equations
101
The proof is complete.
•
Now we use Galerkin's method to prove the following Theorem 3.5.1
Assume that dCl G C2, o y , bt, c, -^-
L 2 (0), a,ij = aji satisfy the parabolicity condition and
G L°°(fi), / G
G HQ(Q). Then °II problem (3.5.1), (3.5.2), (3.5.3) admits a weak solution in WY (QT)UQ
Proof. We merely prove the conclusion of the theorem for the case atj G Cl{QT), c,f£ C(QT). The conclusion under the assumptions of the theorem can be carried over by approximation. We proceed first to construct the approximating solutions. Since UQ G m
HQ(£1),
by Lemma 3.5.1 ii), there exists u™ — ^ J c ™ ^ converging to
UQ
in
i=i
Ho(fi). Approximating solutions to be found are of the form m
um{x, t) = Y2 9T(t)ui(x)
(m = 1,2, • • •),
i=l
satisfying /
(dum
[~ariJl<: +
dum
ai
duik , , dum
i~d~ ' ~Q~.
+
bi
~Q~UJk
, +
^rn^k
^,
~ J^kjdx
fc = 1 , 2 , •• • , m ,
n
= 0,
(m = l , 2 , •••)
or g™(t) (i = 1,2, • • • , m) satisfying equations of the form jt9r(t)=fi(t,9T(t),---,9Z(t))
(t = l > 2 , - . - , m ) ,
where fa are some linear functions of g™,g™, • • • ,g^initial value conditions 9T(0)=c?
(» = l , 2 > - . - > m )
(3.5.9) I*1 addition, the (3.5.10)
should be satisfied. By the theory of ordinary differential equations, problem (3.5.9), (3.5.10) admit solutions $"(*) G C^O.T]. For approximating solutions thus constructed, we can prove \\um\\w^{QT)
(m = l,2,---)
without difficulty and then take limit of a subsequence of {um} to obtain the desired weak solution. D
102
Elliptic and Parabolic
Equations
Exercises 1. Prove Theorem 3.4.2. 2. Establish the theory of regularity of weak solutions for general parabolic equations. 3. Define weak solutions of the Cauchy problem du — -Au
+ Xu = 0,
(x,t) e l " x (0,T),
Oil
u(x,0) = u0(x),
x £
and prove the existence and uniqueness, where A g R and UQ £ H 1 (R"). 4. Consider the first initial-boundary value problem du
- A u = 0,
(i,f)6Qr =
u{x,t) = 0,
(x,t)edQx
u(x,0) = u0(x),
fix(0,r), (0,T),
xeQ.,
where O c R n is a bounded domain and u0 £ L2(Q,). i) Define weak solutions of the above problem and prove the existence and uniqueness; ii) Prove that if u is a weak solution of the problem, then u £ C°°(QT) and if, in addition, dQ £ C°°, then u £ C°°(Ti x (0, T)). 5. Let u £ CQ(QT) be a weak solution of the equation du
du
Hi - Au + IVul + at
+ up = f,
fat)
G QT = SI x
(0,T),
where fi c R n is a bounded domain, / £ L2(QT), p > 0. Prove u £ 2 H (QT). 6. Define weak solutions of the second initial-boundary value problem r du dt du
-Au = f,
•u(x,0) = u0(x),
(i,t)GQr = nx(0,T), (x,t) e f f l x (0,T), x &Q,
and prove the uniqueness, where 0 C R™ is a bounded domain, / € g e L2{dCl x (0,T)) and u 0 £ L 2 (^).
L2(QT),
L2 Theory of Linear Parabolic Equations
103
7. Define weak solutions of the initial-boundary value problem ' du — + A2u = f, ( i , t ) e O r = n x ( 0 , r ) , du (x,t)£dQx (0,T), u = — =0, ov u(x,0) = 0, and prove the existence and uniqueness, where fl c Rn is a bounded domain, / G L2(QT) and v is the unit normal vector outward to 80..
Chapter 4
De Giorgi Iteration and Moser Iteration
This chapter is devoted to a discussion of properties of weak solutions. Two powerful techniques, the De Giorgi iteration and the Moser iteration, are introduced, which can be applied not only to linear elliptic and parabolic equations in divergence form, but also to quasilinear equations, not only to the estimate of maximum norm, but also to the study of other properties, such as regularity of weak solutions. In order to expose the basic idea and main points of these techniques in a limited space, we confine ourselves basically to Poisson's equation and the heat equation, and apply the techniques merely to the estimate of maximum norm of weak solutions.
4.1
Global Boundedness Estimates of Weak Solutions of Poisson's Equation
In this section we illustrate the De Giorgi iteration by applying it to the estimate of maximum norm of weak solutions for Poisson's equation. 4.1.1
Weak maximum equation
principle
for solutions
of
Laplace's
Definition 4.1.1 Let u G Hl{1H). The least upper bound and the greatest lower bound of u on 0, and dQ, are defined as supu = inf{/; (u — /)+ = 0, a.e. in Q,}, supu = inf {7; (u — l)+ S
HQ(Q)},
dU
inf u = — sup(—u), inf u = — sup(—u), n an n an 105
106
Elliptic and Parabolic
Equations
where s+ = max{s,0}. In case u is continuous on Q, the definition of supu, inf u and supu, n n an inf u coincides with the usual one. For sup u and inf u, this is obvious and an (f n for sup u and inf u, this follows from the discussion of the trace of functions an
9Q.
in Hl(Sl) (see §1.5). Let Cl c M.n be a bounded domain. Consider Laplace's equation - A u = 0,
xeCl.
(4.1.1)
Let u e iJ x (fi) be a weak solution of Laplace's equa-
Proposition 4.1.1 tion (4.1.1). Then
supu < supu. n an Proof.
By the definition of weak solutions, u satisfies
/ Vu-V
(4.1.2)
JQ
for any tp £
CQ°(Q)
and hence for any ip €
any k > I, (u — k)+ £
HQ(CI).
Set I = supu. Then for an From Proposition 1.3.10 of Chapter 1,
!
du dx~' 0,
HQ(Q).
., l
, >
' if tx < fc.
Choosing (p = (u — k)+ in (4.1.2) gives 2
/ |V(u-fc)+| da; = 0. /n Jo.
Thus from Poincare's inequality (Theorem 1.3.4 of Chapter 1), we obtain / \(u - k)+\2dx < n f |V(u - k)+\2dx = 0,
Jo.
Jn
where y. > 0 is the constant in Poincare's inequality, which implies (u — k)+ = 0, or u < k a.e. in Q.. Thus the conclusion of the proposition follows from the arbitrariness of k > I. • Corollary 4.1.1 (4.1.1). Then
Let u G if 1 (£2) be a weak solution of Laplace's equation inf u > inf u. n an
De Giorgi Iteration and Moser
Iteration
107
Choosing functions of the form (u — k) + as test functions to derive the estimate of maximum norm is an important technique in establishing a priori estimates. However, the argument as simple as above cannot be used to the same estimate for general equations, even Poisson's equation. For such equations, instead, one has to proceed by means of some iteration techniques, among them is the De Giorgi iteration introduced in the following. 4.1.2
Weak maximum equation
principle
for solutions
of
Poisson's
Lemma 4.1.1 Let ip(t) be a nonnegative and nonincreasing function on [fco,+oo), satisfying
ip(h) < ( ^ — J
[¥>(*)]", Vh>h>k0
(4.1.3)
for some constants M > 0, a > 0, (3 > 1. Then there exists d > 0 such that (fi(h) = 0 , Proof.
V/i > fc0 + d.
Set ks = k0 + d- —,
s = 0,1,2, •••
with constant d > 0 to be determined. Then from (4.1.3) we obtain the recursive formula V(ks+i)<
j
a
[
(s = 0 , 1 , 2 , . . . ) .
(4.1.4)
From this we can prove, by induction, ¥>(*.) <
^
(s = 0 , 1 , 2 , . . . )
(4.1.5)
with constant r > 1 to be chosen. Once this is proved, letting s —> oo then derives ip(ko + d) = 0 and the conclusion of the lemma by the nonincreasingness of
108
Elliptic and Parabolic
Now we choose r = 2a/^~1\
Equations
Then
¥>(fc.+i) < ^
^
Hh)f
[f{ko)f-1
< 1,
'-
Prom this, we see that if d > 0 satisfies Ma2a/9/(/J-l)
Ta
i.e.
then (4.1.5) is also valid for s replaced by s + 1.
•
Now we turn to Poisson's equation -Au = f(x),
xen.
(4.1.6)
Theorem 4.1.1 Let f € L°°(fi) and u £ i7 1 (fi) be a weak solution of Poisson's equation (4-1-6). Then s u p u < supu + C||/|| L oo(m, n an w/iere C is a constant depending only on n and CI. Proof.
By the definition of weak solutions, u satisfies / Vu • Vipdx = / fipdx Jn Jn
for any ip £ CQ^(Q) and hence for any
/ \tp\Pdx Jn /
Jn
\f
(4.1.7)
De Giorgi Iteration and Moser Iteration
109
where the constant C depends only on n and si and
{
+00,
n = l,2,
In other words 2/P
f
\
>A(k) jA(k)
' ~ f
/I
' IX, \f
where A(k) = {x e fi;u(x) > fc}. Prom this, using Holder's inequality
/
\
\f
./Ac*:)
\v,4(it)
y
If
|/|«dx)
\-/>i(fc)
,
y
where 9 is the conjugate exponent of p, i.e.
i + i-i. P
Q
we obtain M ' d x \)
(/"
V-^w
y
i/p
< C (/ /
l/l'efa]\
v^(fc)
1/9
.
(4.1.8)
y
Since h > k implies A(h) C A(k), and (p> h — k on A(h), we have /
Mpdi> /
|^|p^>(/i-fc)p|^(/i)|,
where, as before, \E\ denotes the measure of E. This combined with (4.1.8) gives (h-k)\A(h)\1'p
\A{h)\<(CU^~W)YlA{k)lv/<,.
110
Elliptic and Parabolic
Equations
Since p > 2 implies p > q, from Lemma 4.1.1 we obtain \A(l +
d)\=0,
where
^=C||/|| L ~ ( n)|^(0l (p_< ' )/(p9) 2 p/(p - 9)
« < i + c|n|^-«)/^2"/<"-«)||/|| L =o (n) .
n Corollary 4.1.2 (4.1.6). Then
Lei / € L°°(0) and u £ /f 1 (fi) be a weafc solution of
infu>infu-C||/||Loo(n), where C is a constant depending only on n and 0 . The De Giorgi iteration technique can also be applied to more general elliptic equations in divergence form. For instance, for the slightly general equation - A u + c(x)u = f(x) + divf(x),
ie(l,
(4.1.9)
we have Theorem 4.1.2 Assume that p > n > 3, 0 < c(x) < M, f € L p *(fi), / £ L p (fi;R") andu £ iJ1(f2) is a weak solution of equation (4-1.9). Then supu < s u p u + + C (||/||LP.(n) + ||/1lLP(n)) | f i | 1 / n " 1 / p , inf u >mf u_ - C (||/||LP. (n ) +
||/||L^))
|0|1/n"1/p,
where p* = np/(n + p), s_ = min{s,0} and C is a constant depending only on n, p, M and Q, but is independent of the lower bound of |fi|. For the proof we leave to the reader.
111
De Giorgi Iteration and Moser Iteration 4.2
G l o b a l B o u n d e d n e s s E s t i m a t e s for W e a k S o l u t i o n s of the Heat Equation
In this section we apply t h e De Giorgi iteration technique t o t h e estimates of maximum norm for weak solutions of t h e heat equation.
4.2.1
Weak maximum principle geneous heat equation
for
solutions
of the
homo-
Definition 4.2.1 L e t u e W2hl(QT), where QT = ftx(0,T) with ft C W1 being a bounded domain. Define t h e least upper b o u n d and t h e greatest lower b o u n d of u on QT and dpQr as s u p u = inf{Z; (u — l)+ = 0, a.e. in
QT},
QT
sup u = inf{/; (u - l)+
eW2'
(QT)},
dpQr inf u = — sup(—u), QT QT
inf u — — sup (—u). dpQT dpQT
First consider t h e homogeneous heat equation flit
^ - A u = 0,
(x,t)eQT-
(4.2.1)
P r o p o s i t i o n 4.2.1 Let u £ W2' (QT) be a weak solution neous heat equation (4-2.1). Then
of the
homoge-
s u p u < sup u. QT
Proof.
BVQT
By the definition of weak solutions, u satisfies //
(ut
(4.2.2)
JJQT
o for any ip &C°°(QT)
°1 1 and hence for any
(QT)-
Set I = sup u and 9VQT
choose (p = (u — k)+ with k > I. T h e n
CWl'^Qr).
112
Elliptic and Parabolic Equations
Substituting
(u - k)t(u - k)+dxdt +
JJQT
W(u-k)-V(u-k)+dxdt
= 0,
JJQT
i.e.
- if
—(u-k)%dxdt+
IQT
\V(u-k)+\2dxdt
[J
Ul
= Q.
JJQT
Thus, we obtain from Lemma 3.1.2 of Chapter 3, \
I (u(x, T) - k)\dx - \ \ (7((u(ar, 0) «/ 12
k)+))2dx
+ ff
2
\\7(u-k)+\ dxdt = 0.
JJQT 'QT
Since by Corollary 1.5.6 of Chapter 1, / ( 7 ( K a ; , 0 ) - f c ) + ) ) 2 d a ; = 0, we have \V(u-k)+\2dxdt<0.
// JJQT
Combining this with Poincare's inequality, we further obtain //
(u - k)\dxdt
|V(u - k)+\2dxdt < 0,
< M//
JJQT
JJQT
where /x > 0 is the constant in Poincare's inequality. Hence u(x,t) < k a.e. in QT and the conclusion of the proposition follows from the arbitrariness of k > I.
D
Corollary 4.2.1
Let u G W^iQr)
be a weak solution of (4-2.1). Then
inf u > inf u. QT
4.2.2
9PQT
Weak maximum principle for solutions mogeneous heat equation
of the
nonho-
Now we turn to the nonhomogeneous heat equation ^ - A u = f(x,t),
(x,t)€QT.
(4.2.3)
De Giorgi Iteration and Moser
Iteration
Theorem 4.2.1 Let f £ L°°(QT) and u £ W2' (QT) of the nonhomogeneous heat equation (4-2.3). Then
113
be a weak solution
s u p u < sup u + C||/|| L < »(Q r ) , QT
9PQT
where C is a constant depending only on n and il. Proof.
Denote sup u = 1. For k > I and 0 < t\ < t2 < T, we have
(u — k)+X[t1,t2] GWV (QT), where X[ti,t2]W 1S ^e characteristic function of the interval [ t i , ^ ] . Thus we may choose tp as a test function in the definition of weak solutions (see Remark 3.1.2 of Chapter 3) to obtain //
(u-k)t(u-k)+X[t1,t2]dxdt+
//
JJQT QT
JJc 'QT
X[ tl ,t 2 ]|V(u -
k)+\2dxdt
JJQT JJQT
f(u-k)+X[tut2\dxdt.
Hence 2o
/ H (u-k)2+dxdt+ Jti dt Jn
/ Jti
/ Jn
\X7(u-k)+\2dxdt
< 1^ I \f\(u-k)+dxdt, Jti
i.e.
Jn
\(ik(t2) - Mil)) + r i iv(u - k)+\2dxdt 1
Jti
Jn
< [* I \f\(u - k)+dxdt, Jti
Jn
where Ik(t)=
/(u-k)\dx. Jn
Assume that the absolutely continuous function Ik(t) attains its maximum at a e [0,T]. Since Jfe(0) = 0, h(t) > 0, we may suppose a > 0. Taking ti = a — e, t2 = o with e > 0 small enough so that a — e > 0 and noticing that
h(°) - h(°-~ s) > 0,
114
Elliptic and Parabolic Equations
we obtain f |V(« - k)+\2dxdt < f
f
J<7—£ JQ
f \f\{u - k)+dxdt.
J a—E JQ
Thus it follows from - / e
/ l v ( u - k)+\2dxdt <- f £
Jcr-e Jfl
[ \f\(u - k)+dxdt
J as
JQ,
by letting e —> 0 + that / \V{u(x,a)-k)+\2dx
< / \f(x,a)\(u(x,a)
JO.
- k)+dx.
JO.
This is an analog of (4.1.7) with (p = (u — k)+ obtained for Poisson's equation. Having this in hand, we may process as in the proof of Theorem 4.1.1 to establish the desired estimate. To this end, denote Ak(t) = {x;u(x,t)>
k},
\ik = sup \Ak(t)\. 0
Then, similar to the derivation of (4.1.8), we may deduce (u-k)p+dx)
(/ \JAk(cT)
[
\f\"dx)
\JAk(cr)
< C||/|| L ~ (QT)/ i fc 1/9 ,
)
where +oo, 2
2n
U-2
n = 1,2, n>3,
Q
P-1'
Applying Holder's inequality to Ik(a) and combining the result with the above estimate, we are led to h{°) <( [ (« - kf+dx J \JAk(a) J
\Ak(a)\^-^p
<(cn/ik~(Q T) ) 2 /4 3p - 4)/p . Hence, for any t € [0, T], /*(*) < hi*) < (C\\f\\L~(QT))2tfp-i)/p.
(4.2.4)
De Giorgi Iteration and Moser Iteration
115
Since for any h > k and t £ [0, T],
(u - k)2+dx >(h- k)2\Ah(t)\,
h(t) > f JAh(t) from (4.2.4) we obtain
(h-k)W<(C\\f\\L^QT))2^4)/p, i.e.
(C\\f\\ L^iQrA
(3p-4)/p
Using Lemma 4.1.1 and noticing that p > 2 implies P
P
we finally arrive at /xj+d = sup \Ai+d(t)\ = 0 , 0
^cifil 1 - 2 /^^- 4 )/^- 4 )!!/!!^^). This means, by the definition of A(k),
uKl + ClQf-V^-W^Wfh^Q^, Corollary 4.2.2 of (4.2.3). Then
Let f £ L°°(QT)
}?*"-/>"# QT
a.e. i n Q r .
Q
and u £ W^' ( Q T ) &e a weofc solution
U-C
1I/IU~(QT).
OPQT
where C is a constant depending only on n and 0 . The De Giorgi iteration technique can also be applied to more general parabolic equations in divergence form to establish the weak maximum principle. For instance, for the equation du -* — -Au + c(x,t)u = f(x,t)+divf(x,t), we have
(x,t)eQT,
(4.2.5)
116
Elliptic and Parabolic
Equations
Theorem 4.2.2 Assume p > n > 3, 0 < c(x,t) < M, f e L°°(0,T;LP'(D.)), fe Loo(0,T;IA'(Q,Rn)), and u € W\<X{QT) is a weak solution of equation (4-2.5). Then supu< supu+ + c f QT
dpQT
mfu> QT
inf u . - c ( OPQT
sup | | / | | i , . ( n ) + sup ||/|U P (n)) M 1 / n _ 1 / p , \0
0
/
sup | | / | | L , . ( n ) + sup ll/IU^n) V | 1 / n - 1 / p , \0
0
J
where p* = np/(n+p) and C is a constant depending only on n, p, M and Cl, but is independent of the lower bound of |fi|. The proof is left to the reader. Remark 4.2.1 Among the assumptions of Theorem 4-2.2, c(x,t) > 0 is not necessary. If we replace this condition by \c(x,t)\ < M for some constant M, then the same estimates hold, but the constant C depends on T in addition to n, p, M and Q. In fact, under such condition, we may introduce a new unknown function w = e _ A t u with A > 0 to be determined and change equation (4-2.5) into ^ - Aw + (A + c)w = e - A t ( / + div/). at If we choose A = ||c||i,°o(Qr), then A + c(x,t) > 0 and thus the conclusion of Theorem 4-2-2 can be applied to the new equation to obtain the desired estimate.
4.3
Local Boundedness Estimates for Weak Solutions of Poisson's Equation
Another important technique in estimating the maximum norm of solutions is the Moser iteration. In this section, the main points of this technique will be illustrated by applying it to Poisson's equation in establishing the local boundedness estimate of weak solutions. 4.3.1
Weak subsolutions
(supersolutions)
In order to further discuss the local boundedness of weak solutions of Poisson's equation, we introduce the concept of weak subsolutions (supersolutions) of equation (4.1.9).
De Giorgi Iteration
and Moser
Iteration
117
Definition 4.3.1 A function u € H1^) is said to be a weak subsolution (supersolution), if for any nonnegative function
Jn
g"(u)\Vu\2
< / c{g{u) Jn
ug'(u))ipdx.
Here we have used the condition g"(s) > 0. To prove that w = g(u) is a subsolution of (4.1.9), besides c > 0, cp > 0, it suffices to note that 5(0) = 0 and g"{s) > 0 imply g(s) - sg'(s) < 0,
Vs e R. D
118
Elliptic and Parabolic
Equations
Remark 4.3.1 Conditions g"(s) > 0 and g'(s) > 0 in Proposition 4-3-1 can be replaced by that g(s) is a nondecreasing convex Lipschitz function; a typical example is g(s) = sp+,
seR
(p>l).
We leave the proof to the reader. 4.3.2
Local boundedness estimate Laplace's equation
for
weak
solutions
of
Theorem 4.3.1 Let x° & Q, BR = BR(x°) Cfl andut H1 (SI) n L°°(0.) be a weak subsolution of Laplace's equation (4-1.1). Then 1/2
— uu2+,tdx sup u < C I — —/ R n BR,2 \ JBR
where C is a constant depending only on n. Proof. Prom Proposition 4.3.1, we see that u+ is also a weak subsolution of Laplace's equation (4.1.1). So we may assume u > 0. The proof will be proceeded in three steps. Step 1 Prove the inverse Poincare's inequality, namely, for any p > 2, r)2\Vup'2\2dx
f
up\Vn\2dx,
JBR
(4.3.1)
JBR
where rj(x) is the zero extension of a cut-off function on Bp>, relative to Bp (0 < p < pf < R), namely, rj € C$>(Bp,), 0 < n(x) < 1, n(x) = 1 on Bp, r](x) = 0 on Cl\Bp> and \S7n(x)\ < — 2 p_1
Choose 7y u Then
.
as a test function in the definition of weak subsolutions.
/ Jn
Vu-V(n2up-l)dx<0,
or (p-l)
f
n2up-2\Vu\2dx
riup-lVu
+2 J
J BR
J
• Vrjdx < 0.
BR
Hence ^ ^ P
/ JBR
ri2\Vup/2\2dx
+ f JBR
r)up'2Vup/2
• Vrjdx < 0.
De Giorgi Iteration and Moser Iteration
119
Since using Cauchy's inequality with e gives /
r,up'2^up'2
JBR
r)2\Vup/2\2dx
-S/r]dx<^ 2 [
+ 2e l-
JBR
up\Wr]\2dx,
[ JBR
(4.3.1) can be obtained by choosing e > 0 small enough. Step 2 Prove the inverse Holder's inequality, namely, for any p > 2,
i-zn
[
\R JBp
upqdx)
)
* n 2
r= f
,
updx),
2
~ \R - (p -p) JBl>,
J
(4.3.2) y
'
where +oo,
n = 1,2,
1
n . n-2'
n > 3.
Prom Remark 1.3.6 of Chapter 1, we have / , , \1/(2<j) / , _L/ rpundx]
\i/(2g) W'2)*>dx\ \V{riup/2)\2dx\
,
or
where C is a constant depending only on n. On the other hand, using the inverse Poincare's inequality gives / \V(r]up/2)\2dx J BR
= f \r)Vup'2 + J BR <2 [
r]2\Vup/2\2dx
JBR
Therefore (4.3.2) holds. Step 3 Iterate.
up/2VV\2dx +2 [ JBR
p
2
u |Vry|
up\Vr)\2dx
120
Elliptic and Parabolic
Equations
Denote
and choose p = 2qk, p = pk+i, p' = pk in (4.3.2). Then \
„
i/(2gfc+1)
\u\2"k+1dx
.
RnJB.
/
N V<2«*>
Iterating repeatedly leads to /
i
\1/{2qk+1)
/•
S i r
\ V2
where OO
-
OO
fc
2o '
^
^
fc + 2 2o f c '
Since 9 > 1 implies the convergence of these series, a, (3 are both finite numbers. Thus for some constant C depending only on n, v l/(29fc+1)
/
1/2
Prom this the desired conclusion follows by letting k —• oo. 4.3.3
Local boundedness equation
estimate
for solutions
of
• Poisson's
Now we turn to equation (4.1.9) with / = 0, namely, -Au + c(x)u = f(x),
xGfi.
(4.3.3)
Theorem 4.3.2 Let 0 < c(x) <M, f G L°°(ft) and u G if 1 (fi)nL°°(n) 6e a weafc subsolution of equation (4-3.3). Then there exists a constant Ro > 0 depending only on M, such that for any x° G Cl, there holds supu
V-"
JBR
u2dx)
+C||/||Loo(n), J
De Giorgi Iteration and Moser Iteration
provided 0 < R < R0 and BR = BR(X°) depending only on n, RQ and M. Proof.
121
C Q, where C is a constant
For any x° e £1, denote u(x) = u(x) + \x- z°| 2 ||/|| L =c(n),
x € ft.
Then, in the weak sense — Au + cu 2n||/|| L oo ( n ) +cu + c\x - x°| 2 ||/||L°°(fi)
= -Au-
in SI.
Then, in the weak sense,
-Au + cu<0,
mQr\BRo(x°),
namely, u is a weak subsolution of equation —Av + cv = 0 in Q D By Proposition 4.3.1, u+ also satisfies - A u + + cu+ < 0,
BR0(X°).
in Q, n Bfl0 (x°)
and hence mQ,nBRo(x0)
-Au+<0,
in the weak sense, namely, u+ is a weak subsolution of Laplace's equation in fi n jBfl0(a;0). Thus, from Theorem 4.3.1, we obtain \l/2
sup i p u+ u++ <
Ban ~ BR
WJBJ
z2 \u+\ dx ] + do
'
J
,
and hence the conclusion of the theorem follows.
0 < R < Ro •
R e m a r k 4.3.2 What we have adopted to establish the local boundedness estimates in Theorem 4-3.1 and Theorem 4-3.2 is the so-called Moser iteration technique, a technique of extreme importance. The method is based on the fact IMU°o = lim
\\U\\LP-
122
Elliptic and Parabolic
Equations
The basic idea in establishing the boundedness estimates is to choose suitably Pk and pk such that po = R, lim pk = R/2 and lim pk = +00, and then fe—»oo fe—»oo
try to prove that Ak =
\\u\\L'k(BPk)
satisfies the recursive formula < Ca"Ak
Ak+1 00
with otk > 0 such that the series V J ctk is convergent. fc=o
4.3.4
Estimate Poisson's
near the boundary equation
for
weak solutions
of
All estimates presented above are established in a neighborhood of the interior points of Q. In addition to these interior estimates, we need to establish estimates near the boundary points. As an example, we will do this for a domain of the form Q+ = {x € M.n; |XJ| < 1 (1 < i < n),xn > 0}. For general domains, we may transform the neighborhood of the boundary point to a domain of this special form by means of local flatting of the boundary. Of course, after a local flatting transformation, the shape of the equation will be changed. Theorem 4.3.3 Assume that 0 < c(x) < M, f £ L°°(Q+) and u € H1(Q+) n L°°(Q+) is a weak subsolution of equation (4-3.3) with trace vanishing on the bottom of Q+, i.e. ju(x\,--,a;„_i,0) = 0. Then there exists a constant Ro £ (0,1] depending only on M, such that for any point x° on the bottom of Q+,
sup u
R/2
u2dx\
I \
R
+C||/||L~(Q+), /
provided 0 < R < RQ and B^ = B+(x°) C Q+, where B^(x°) = BR(x°) n {x s Rn;xn > 0} and C is a constant depending only on n, Ro and M. The proof is similar as the one of Theorem 4.3.2. We leave it to the reader.
De Giorgi Iteration and Moser
4.4
Iteration
123
Local Boundedness Estimates for Weak Solutions of the Heat Equation
In this section, we use the Moser iteration technique to the local boundedness estimates for weak solutions of the nonhomogeneous heat equation. 4.4.1
Weak subsolutions
(supersolutions)
We first introduce weak subsolutions (supersolutions) of equation (4.2.5). Definition 4.4.1 A function u G W^iQr) is said to be a weak subsolution (supersolution) of (4.2.5), if for any nonnegative function
(ftp - f-
JJQT^
Vip)dxdt. '
Sometimes, a weak subsolution (supersolution) u of (4.2.5) is said to be a function satisfying OVL
~*
— - Au + c(x, t)u < ( > ) / ( ! , t) + div/(x, t) in the weak sense. Proposition 4.4.1 Assume that f = 0, / = 0, c(x,t) > 0 and u G W 2 1 ' 1 (gr)nL 0 0 (Qr) is a weak subsolution of (4.2.5). Ifg"(s) > 0, g'(s) > 0 and g(0) = 0, then g(u) is also a weak subsolution of (4-2.5). The proof is similar to the elliptic case (see Proposition 4.3.1). R e m a r k 4.4.1 Conditions g"{s) > 0 and g'(s) > 0 of Proposition 4-4-1 can be replaced by that g(s) is a nondecreasing convex Lipschitz function; one of the typical cases is g(s) = sp+, 4.4.2
seR
(p>l).
Local boundedness estimate for weak solutions homogeneous heat equation
of the
T h e o r e m 4.4.1 Let (x°,t0) G QT, QR = QR(x°,t0) = BR(x°) x {t0 R2,t0 + R2) C QT and u G Wl'1(QT) n L°°(QT) be a weak subsolution of
124
(4-2.1).
Elliptic and Parabolic
Equations
Then
ST~C(n^l/dxdt)"2 with C depending only on n. Proof. Similar to the discussion for Poisson's equation, we proceed in three steps. Step 1 Derive an estimate similar to the inverse Poincare's inequality. For any p, p' such that R/2 < p < p' < R, choose a cut-off function rj(x) on Bpi, relative to Bp, namely, n G CQ°(BP>), 0 < n(x) < 1, r](x) = 1 on Bp, and |V^(x)| < — 2
and extend 77 to be zero for x G Q\Bpi. For
2
any s G (to — P ,h + R ), choose £ G C°°(—00, s], such that £(t) = 1 on [t0 - p2,s], £(t) = 0 on (-00, t 0 - p'2] and 0 < C(t) < —
^ for t < s,
(P - P)
and extend it to be zero for t > s. We may assume that u > 0; otherwise we use u+ instead of u. Choose ip = £2n2u as a test function in the definition of weak subsolutions to obtain {ut£"n2u + Vu • V ( £ V u ) ) dxdt < 0,
// JJQT IQ
namely,
Wt{ev2u2)dxdt~U s ti'^dxdt
\jj. p'
p'
2
+ if
tW\Vu\ dxdt
+ 2 if
*
u^nVu
• Vrjdxdt < 0,
pi
where Qp = Bp x (to - p2,t).
Since f (t) = 0 at t = t 0 - p' , we have
[f ±L(Z2r,2u2)dxdt = f JJQ* or JB
ZWu2
t=s
dx.
Using Cauchy's inequality with e gives u£2rjVu • Vrjdxdt
2 // p'
<--If ~2
Z2r,2\Vu\2dxdt
+ 2 ff
Z2u2\Vr)\2dxdt.
125
De Giorgi Iteration and Moser Iteration
Hence \
[
£ W
<- ff tW\Vu\2dxdt 2 JJQp <\ 2
er?\Vu\2dxdt
dx+ [[
yjQ',
, (.P
JJQS^
C _
tt'r}2u2dxdt
ff
JJQ^
£2ri2\Vu\2dxdt+
ff
£2u2\Vr)\2dxdt+
+ 2 //
u 2 dxdi.
// P) .A/Q',
Therefore /•
sup
to-p2
C_ <, . ^ xo / / '{P'-PYJJQ JJQj
/
r)2u2(x,t)dx+
rto+P
r
/
/
Jto-p2
T72|Vu|2dxdt
JBp,
u2da:eft.
S t e p 2 Derive an estimate similar to the inverse Holder's inequality. Let x ( 0 be the characteristic function of the segment [to — p2,to + 2 p }. Then x{t)rlu G V^Q/j) and by the t-anisotropic embedding theorem (Theorem 1.4.2 of Chapter 1),
wt) )2w
(sw/ 0 „ '" 'r
sup
+ IJ =C{n)R7n(
JBR
\X(t)V(Vu)\2dxdt).
sup
rto+P
r)2u2(x,t)dx
/
^t0-p2
+ /
(x(t)7?u(a:,t))2dx
/
^t0-R2
JB-i r
/
.
\r]Vu +
where 5/3,
n = l,2,
1 + 2/ra,
n > 3.
uVr)\2dxdt),
126
Elliptic and Parabolic
Equations
Using the inequality obtained in Step 1, we further deduce 1/9
'Qt
j
Dn+2
JJ {X{t)rjufdxd^
sup V *o-p2
+ /
/
Jto-p2 n
r?u2{x,t)dx
/ JBp,
.
uVri\2dxdt)
\r]Vu +
JBp,
'
sup
rj2u2(x,t)dx
/
^t0-p2
JBp, 2
fto+p2
2
+ 2/ / n \Wu\ dxdt + 2 / Jto-p2 JB„,2 Jt0-p2
< „ _n , ^
— //
R {p'-P)2 JJQ P
U dl^.
/•
/ JB„,
.
u2\Vn\2dxdt) '
Proposition 4.4.1 shows that u 9 is also a weak subsolution of (4.2.1). So with uq in place of u, we obtain from the above inequality, x l/2qk+1
/
/
v
l/2qk
Step 3 Iterate. Similar to the case of Poisson's equation.
4.4.3
•
Local boundedness estimate for weak solutions nonhomogeneous heat equation
of the
Now we turn to equation (4.2.5) with / = 0, namely, Qu
— -Au
+ c(x,t)u = f(x,t),
(x,t)GQT.
(4.4.1)
Theorem 4.4.2 Let \c(x,t)\ < M, f £ L°°(QT) and u £ W 2 M (Qr) f~l L°°(QT) be a weak subsolution of (4-4-1)- Then for any (x°,to) € QT and
De Giorgi Iteration and Moser Iteration
127
R > 0 such that QR = QR{X°, t0) C QT, 1/2
sup u
U u22dxdt^j dxdt }
IJ/I
+C||/||LOOWT)
QR/2
with C depending only on n, M and T. Proof.
Let w(x,t) = e~Mtu(x,t),
(x,t)eQT-
Then, in the weak sense, w satisfies ^ - Aw + (M + c)w = e-Mt at
(^-Au \at
+ cu]< )
e~Mtf,
in QT.
So, without loss of generality, we may assume that c(x, t) in (4.4.1) satisfies 0
in QT-
Then, in the weak sense, u satisfies &&
A—
— at =
—
Au + cu
dit
~di ~Au + cu~ H/IU~(QT) _ ct\\f\\L^(QT) = / - II/IU»(QT) - CtWfh^iQr) <0,
in QT,
namely, u is a weak subsolution of (4.4.1) with / = 0. So s-i
—^- — Au+ + cu+ < 0,
in QT
(so
and hence — - Au+ < 0, dt
in QT,
namely, u+ is a weak subsolution of (4.2.1) in QT- Thus, from Theorem 4.4.1, we have
ss^-c(^LMdx)
128
Elliptic and Parabolic
Equations
Hence 1/2
sup u < C ( - L j f V- 0,
QR/2
u2dx)
JQR
+ C||/||L-(QT).
/
• Similar to the case of Poisson's equation, we can further establish the estimate near the boundary for the heat equation.
Exercises 1. Prove Theorem 4.1.2. 2. Let A > 0, n C R" be a bounded domain, / G L°°(fi) and u G HQ(Q) n L°°(fi) be a weak solution of - d i v ( ( u 2 + A)Vu) +u = f,
iefi.
i) Use the De Giorgi iteration technique to establish the maximum norm estimate for u; ii) Whether the maximum norm estimate holds in case A = 0? 3. Prove Theorem 4.2.2. 4. Let 0 C t n be a bounded domain, p > 2 and / G L°°(QT)Consider the first initial-boundary value problem for p-Laplace's equation — - divfl V u r 2 Vu) = / ,
(x, i ) G Q r = O x (0, T),
= 0. dPQx
i) Define weak solutions of this problem and prove the existence and uniqueness; ii) Use the De Giorgi iteration technique to establish the maximum norm estimate for weak solutions of the problem. 5. Prove Remark 4.3.1. 6. Prove Theorem 4.3.3. 7. Let Q C K n be a bounded domain and u G L°°(Q). Show that IMU~(n) = limJ|u|| L P (n).
De Giorgi Iteration and Moser
8. Let u\ and U2 € HQ(Q) ^>e a of Poisson's equation
wea
—Au = / ,
Iteration
129
k solution and a weak subsolution i£fi,
where Q C R" is a bounded domain and / € L2(Q,). Show that ui(x) > U2{x),
a.e. x € £1.
9. Prove that u £ W2' (QT) is a weak solution of the equation — -Au
+ c(x,t)u = f,
(x,t)eQT
=
flx(0,T)
if and only if u is both the weak supersolution and the weak subsolution of the equation, where fi C W1 is a bounded domain, c S L°°(QT) and 2 / G L (QT). 10. Prove Proposition 4.4.1 and Remark 4.4.1. 11. Let u G W2' (QT) n L°°(QT) be a weak solution of the equation
W " div ( ( ^ r l ) V u )
+
<X^U
= *> ( x '*) e gr = fi x (°'T)
where Q. C R" is a bounded domain, c £ L°°(QT) and / € L2(QT). Use the Moser iteration technique to derive the local boundedness estimate for u. 12. Use the Moser iteration technique to derive the local boundedness estimate near the boundary for weak solutions of the heat equation.
Chapter 5
Harnack's Inequalities
In this chapter, we continue our study of properties of weak solutions, we will concentrate our attention on Harnack's inequalities which reveal deeply the properties of solutions of elliptic and parabolic equations. Such kind of inequalities hold not only for general linear elliptic and parabolic equations in divergence form, but also for quasilinear equations (see Chapter 10). However, we merely illustrate the argument for the simplest equations such as Laplace's equation and the homogeneous heat equation, although the basic idea is available to general linear and quasilinear equations.
5.1
Harnack's Inequalities for Solutions of Laplace's Equation
In this section, we are concerned with Laplace's equation x£Rn.
- A u = 0, 5.1.1
Mean value
(5.1.1)
formula
Theorem 5.1.1 Let u € C2(Rn) ball BR = BR{y) C R n , u
(y) =~
be a solution of (5.1.1) Then for any
u x
^T=T / n
l
nunR
( )ds,
(5.1.2)
J9BR
(5.1.3) u(y) =—5^
/
u(x)dx,
where un is the measure of the unit ball in M.n. 131
132
Proof.
Elliptic and Parabolic Equations
Integrating (5.1.1) over the ball Bp = Bp(y) with p € (0, R) gives /
IB0 JB„
Audx = I/ Tr^ds — 0, JdB Jdi B du
where v denotes the unit normal vector outward to dBp. Introduce the polar coordinate p=\x — y\,z = x-y Then from the above formula, we have, with u = u(x) = u(y + pz), 0
dv
Jdi IdB,
-L
d_u(y + pz)ds aBp(o) dp d_u(y + u)ds (0) dp
7
JdBx d
„ n - i'— =p-
dp
/
u(y + u)ds
JdB^o)
p1
dp
n
u(y + pz)ds JdBp(0)
Hence pl~n
( u(y + pz)ds =R1~n I u{y + Rz)ds JdBp(o) JdBR(.o) =i?1-" / JdBR
u(x)ds.
Sending p —• 0 + and noting lim pl~n P-+0+
/
u(y + pz)ds =
nunu(y),
JdBp(0)
we then derive (5.1.2). Since (5.1.2) holds for any R > 0, we have ncjnpn~1u(y)
= / u(x)ds, JdBp
Integrating over (0,-R) with respect to p then gives (5.1.3).
•
Harnack's Inequalities
5.1.2
Classical
Harnack's
133
inequality
Theorem 5.1.2 Let u € C 2 (R n ) be a nonnegative solution of (5.1.1). Then for any ball BR = BR(y) C M.n, sup u < C inf u, B
R
BR
where the constant C depends only on n. (In fact, we may take C = 3n). Proof.
For any x1, x2 G BR(y), BR{xl)
c B2R(y) C
B3R(x2).
Using the mean value formula, we have ufx1) = — — - /
u(x)dx
<—— / u(x)dx unRn 7s 2R ( y ) <'—^-
/
u(x)dx
=3nu(x2). The desired conclusion then follows from the arbitrariness of x 1 and x2. O We have proved Harnack's inequality for classical solutions of Laplace's equation. In fact, such kind of inequality also holds for weak solutions. To prove, we proceed in several steps. 5.1.3
Estimate
of sup u Ben
Lemma 5.1.1 Let 0 < To < T\ and
for any t, s such that TQ < t < s < T\, where 9 < 1, A, B and a are nonnegative constants. Then
^-
C
((fl-p)«
+ g
)'
r0
where C is a constant depending only on a and 9.
134
Proof.
Elliptic and Parabolic
LetT0
Equations
Denote
ti+1=U
+ 0--T)Ti(R-P)
(t = 0 , l , " - )
with r G (0,1) to be specified. By assumptions of the lemma, we have
^)<Wti+i)+{{1_T)Ti{R_p))a+B,
i = 0,l,....
For any integer k > 1, iterating gives
rtto) < *
|>-.)«.
The desired conclusion follows by choosing r such that Qr~a < 1 in the above inequality and sending k —> oo. • Theorem 5.1.3 Let u G Hioc(M.n) be a bounded weak subsolution of (5.1.1). Then for any p > 0 and 0 < 6 < 1, 1/p
s u p u ^ c f - z j - r / (u+) p dx) Ben Vl-Dfll VB H / where C is a constant depending only on n, p and {1 — 0)- 1 Proof. From Proposition 4.3.1 of Chapter 4, u+ is also a weak subsolution of Laplace's equation (4.1.1). Hence, we may assume that u > 0. Using Theorem 4.3.1 of Chapter 4 (There we have treated the special case 6 = 1/2; the general case 0 < 9 < 1 can be treated similarly) we can derive the conclusion for p = 2. We can prove the conclusion for p > 2 in a similar way or by using Holder's inequality directly on the conclusion for p = 2. Now we consider the case 0 < p < 2. We first use the result for p = 2 to obtain supu
( f
Bert
\JBR
u2dx\ J
O
r BR
\ 1//2 vPdx) J
and then use Young's inequality with e to derive supw < \ supu + C((l BBR
l
BR
9)R)-n/"\\u\\LP{BR).
,
135
Harnack's Inequalities
Denote (f(s) = sup u and set s — 6R, t = R in the above inequality. Then B,
1 C V(») < gVW + ( t _ 8 )n/pll U H^(gi»)'
0 < S < i < i?.
Thus from Lemma 5.1.1, we deduce the desired conclusion
Checking this proof, we may get that Theorem 5.1.3 still holds if u G Hioc(Rn)
Remark 5.1.1
5.1.4
Estimate
is replaced by
of inf it Ben
Lemma 5.1.2 Let 3>(s) be a smooth function on R with $"(s) > 0 and u G Hloc(Rn) be a bounded weak solution of (5.1.1). Then v = $(u) G Hyoc(Rn) is a weak subsolution of (5.1.1), namely,
J
Proof.
Vu • V < 0,
VO < y> G
C^(Rn).
Ju In fact, for any nonnegative function ip G Co°(R n ), we have /
Vv • Vipdx
JRn
= /
$'(u)Vu • V(fdx
JR*
= [
Vu-V(&{u)
JRn
<; /
f
&'(v)\Vu\2
JRn
Vu • V{$'(u)
JR"
=0.
a
Remark 5.1.2 The smoothness condition for $(s) can be weakened to local Lipschitz continuity. Lemma 5.1.3
Assume that w G L2{B2) and satisfies
( f (v2\w\2h)"dx) \JB2
< C(2h)2h + Ch2 [ J
JB2
(|VT7|
+ r])2\w\2hdx
136
Elliptic
and Parabolic
Equations
for any h > 1 and any cut-off function r] on Bi, where q > 1 and the constant C is independent of h and r\. Then there exists a constant C > 0, such that for any integer m>2, \w\mdx)
< Cm.
Proof. Set hi = qi~1,
S0 = 2,
S i ^ S ^ - y ,
i = 1,2, ••• .
Choose T) to be a cut-off function on Bsi_1, relative to Bst, namely, 77 G C ^ C ^ i - i ) . 0 < r){x) < 1, »j(x) = 1 in BSi and |V??(x)| < 2*C. Then from the assumption of the lemma, we see that 1/9
(L
^dx
< Cq^-W-1
J
\
With It= [ I
+ C{2q)W-V
f
{w^dx.
l/(2
Zq
\w\ 'dx
, we further have
7.
1
)(29)(i-i)/9i-1/._1)
i =
i,
2 )
... .
Iterating then gives Ij < C J ] g*"1 + C Q ' (29)ft Jo,
j = 1,2, • • • ,
i=l J
J 1 ' i— 1 * = where Oj = 5 3 o »_i > &' 5 3 ; - i • Since 5 ^ ^
i=l
9
i=l
9
_ 1
- ^q^, the above
i=l
inequality implies Ij^Cqi+CIo,
j = 1,2, • • - .
From this, noting that for any fixed integer m > 2, there exists an integer j , such that 2qi~1 <m< 2qj, and using Holder's inequality, we have l/m
< CIj
Harnack's
Inequalities
137
Thus, with another constant C independent of m, we finally obtain \w\mdx B1 Lemma 5.1.4
< Cm. J
D n
Assume that w € H^c(R ),
/
w(x)dx = 0 and
JBi
Aw + \Vw\2 < 0,
x e W
in the sense of distributions, i. e. -f
Vw-Vipdx+
\Vw\2
J
V0<
Then there exists a constant p > 0 depending only on n, such that
r (P\w\y JBi
<2~m,
m = 2,3,---.
(5.1.4)
m\
Proof. We proceed in two steps: first prove the conclusion for m = 2, and then use the standard Moser iteration to reach the conclusion for m > 2. Choose ri G C$°(B3) such that 0 < r) < 1, 77 = 1 in B2 and |Vr7| < C and set
- /
\Vw\2n2dx
• Vwdx + /
< 0.
Integrating by parts and using Cauchy's inequality with e, we deduce /
\Vw\2r)2dx < /
JB3
V(T7 2 )
• Vwdx
JB3
=2 /
77V77 • Vwdx
JB3
<- I \Vw\2n2dx + 2 I \Vrj\2dx. 2 JB3 JB3 Thus /
\Vw\2dx
(5.1.5)
JBi
Prom this it follows by noting /
w(x)dx = 0 and using Poincare's inequal-
JB2
ity, / JB2
w2(x)dx
\Vw\2dx < C, JBo
138
Elliptic and Parabolic
Equations
where fi > 0 is the constant in Poincare's inequality. Choosing p < (2C)- 1 / 2 then gives (5.1.4) for m = 2. To prove the conclusion in the case m > 2, we take tp = rj2\w\2h with ft > 1 and 77 being a cut-off function on B^. Then ri2\w\2h\Vw\2dx
/ JB2
V{r?\w\2h) • Vwdx
< [ JB2
r)\w\2hVrj • Wwdx + 2 f t /
=2/ J1B2 B2
77 2 |w| 2/l_1 sgni(;|Viy| 2 dx.
J Bo B2
Using Cauchy's inequality and Young's inequality with e, 277IV7? • Ww\ < —r]2\Vw\2 + 4/i|V7y|2, An 2h\w\2h-1
2 h
< ^ ^ H
+
{2h)2h~\
we then derive ri2\w\2h\Vw\2dx
/ JB2
< \ [ r)2\w\2h\Vw\2dx 4ft JB2
JB2
2
2h
2
+ (2h)2h~1
: ^ L Z ! f r, \w\ \Vw\ dx 2ft JB2
+
=
\Vr)\2\w\2hdx
+ 4h f
(1~ih)
[
+ 4h f
rP\Vv,\2dx
f JB2
V2\w\2h\Vw\2dx + (2h)2h~l
f
2 V
\Vw\2dx
\Vr)\2\w\2hdx,
JB2
which combined with (5.1.5) leads to r)2\w\2h\Vw\2dx
/
< C(2h)2h + 16h2 f
JB2
|Vr7|2|u;|2^.
JB2
Thus \V(V2\w\2h)\dx
/ JB2
<2h f JB2
rt2\w\2h-1\Vw\dx
+2 f JB2
r]\Vri\\w\2hdx
Hamack's Inequalities
<[
2 V
\w\2h\Vw\2dx
+ h2 f
139
rfH^-^d
r]\Vr]\\w\2hdx
+ 2 1 JB2
< f
r,2\w\2h\Vw\2dx
+ h2 f
h—1, .o/. 1 , , 7 ? 2 f c w\2h + -)dx
2h + 2 [ r,\Vr,\\w\ dx
>B2 JBn i « 22 +, 16h
2h \2h
(\Vr,\+r,)2\w\2hdx.
f JB2
Prom the embedding theorem, we further obtain 1/9
f {ri2\w\2h)qdx)
\V(r)2\w\2h)\dx
J B2
/
J B2
(\Vr]\+r,)2\w\2hdx,
JBn. IB2
where +00,
1
n
. n-2'
n = 1,2,
n > 3.
Therefore, using Lemma 5.1.3 we infer /
\w\mdx < (Cm)m,
m = 2,3
Since m m < e m m!, we finally arrive at /
\w\mdx <
(Ce)mm\,
JBx
which is just (5.1.4), if we take p = (2Ce) _ 1 .
•
n
Theorem 5.1.4 Let u G Hloc(M. ) be a nonnegative and bounded weak solution of (5.1.1). Then for any 0 < 6 < 1, there exist constants po > 0 and C > 0 depending only on n and (1 — 6)"1, such that \ 1/Po
inf u > ^ f T^-T / Ben
C\\BR\JBR
up°dx) J
(5.1.6)
140
Elliptic and Parabolic
Proof.
Equations
Without loss of generality, we may assume that inf u > 0; other-
wise we may replace u by u + e (e > 0) and then let e —> 0 in the inequality for u + e. For simplicity, we assume R = 1; for the general case, we may use the rescaling technique. For any p > 0, set $(s) = —. Then $"(s) > 0 and Lemma 5.1.2 shows that $(u) is a weak subsolution of —An = 0. Thus from Theorem 5.1.3, we have sup
—dx,
—
Be U"
JB
UP
and hence L
1
(¥•)'**(/.
-l
u ~pdx
namely, inf u Be
1 X
~C IP
/
\-I/P
f
(V""")
=
/
po u-po dx dx Jf
JBi
upodx < C.
(5.1.7)
JB1
Set w = lnu — j3 with (3 = -—- /
\nudx. Then /
u;(a;)(ix = 0. It is
IB2
easy to see that (5.1.7) holds if e^i^ldx < C.
(5.1.8)
JB, IB! In fact, (5.1.8) implies that f
epo{0-lnu)dx
<
C j
/"
e po(lnu-/3) r f 2 .
<
C]
and hence (5.1.7) holds. We now use Lemma 5.1.4 to prove the existence of p0 > 0 such that (5.1.8) holds. To this end, we need to check the condition of Lemma 5.1.4,
Harnack's
141
Inequalities
namely, to prove that, in the sense of distributions, Aw + \Vw\2 < 0,
xeW1.
(5.1.9)
Since u is a bounded weak solution of (5.1.1) and inf u > 0, we have, for R"
any nonnegative function y> S Co°(R"), /
Vu • V ( - ) dx = 0,
namely, /
—y>|Vu|2d:r = 0
- V u • Vv?cta - /
or /
VwVipdx-
,/R"
j
|Vu/|Vfa: = 0,
V0 <
7R"
which shows that w satisfies (5.1.9) in the sense of distributions.
•
Checking this proof and the proofs of Lemmas 5.1.2-5.1.4, we may get that Remark 5.1.3 ueH\B3R).
Theorem 5.1.4 still holds if u G Hloc(Rn)
is replaced by
If we replace B\ and f?2 by B^g+iy2 and -B(0+3)/4, respectively, in the proof of Theorem 5.1.4, we may further get by a similar process of proof with some modifications that R e m a r k 5.1.4 u G H\BR). 5.1.5
Harnack's
Theorem 5.1-4 still holds if u G H*oc(M.n) is replaced by
inequality
T h e o r e m 5.1.5 Let u € H1{B3R) be a nonnegative and bounded weak solution of (5.1.1). Then for any 0 < 9 < 1 and R > 0, sup u < C inf u BBR
B
«H
with the constant C depending only on n and (1 — 0 ) - 1 .
142
Proof.
Elliptic and Parabolic
Equations
From Theorem 5.1.3 and Remark 5.1.1, for any p > 0, SUP U < C I -j-jr-r / UPdx ) B Ben \\ R\ JBR J
On the other hand, from Theorem 5.1.4 and Remark 5.1.3, there exists a constant po > 0, such that
W*h{ikL'f"*y Combining these two inequalities derives the desired conclusion.
•
Furthermore, the condition u G H1(B3R) in Theorem 5.1.5 may be replaced by u G H1{BR), according to Remark 5.1.4 or by the finite covering theorem. More general, we have T h e o r e m 5.1.6 Let fi C W1 he a domain and u G Hl{Q) he a nonnegative and bounded weak solution of (5.1.1). Then for any bounded suhdomain
Q' cc fi, supu
(5.1.10)
fi'
with the constant C depending only on n, fi' and fi. Proof.
Fix
0 < i ? < Jdist(fi',<9fi). For any x G fi , from Theorem 5.1.5, there exists a constant C depending only on n such that sup u < C inf u. BR(X)
Choose xx,x2
B
(5.1.11)
H W
G fi so that sup u = supu, BR(x*)
fi'
inf u = infu. BR(x2)
(5.1.12)
H'
Let r C fi be a closed arc joining a;1 and x2. By virtue of the finite covering theorem, T can be covered by a finite number N (depending only on fi' and fi) of balls of radius R. Applying the estimate (5.1.11) in each ball and
Harnack's Inequalities BR{X1)I
BR{X2),
143
respectively, and combining the resulting inequalities, we
obtain sup u
inf u.
Then, (5.1.10) follows from this and (5.1.12). 5.1.6
Holder's
•
estimate
The following auxiliary lemma is useful in proving the Holder continuity of solutions. Lemma 5.1.5 Let w(R) be a nonnegative and nondecreasing function on [0, RQ\. If there exist 6, n £ (0,1), 7 £ (0,1] and K > 0, such that UJ{6R)
< T]UJ(R) + KRt,
0
(5.1.13)
then there exist constants a € (0,7) and C > 0 depending only on 6, i), 7, such that u(R)
[u(Ro) + KI%],
0
(5.1.14)
Proof. We may take 77 so close to 1 that 0_7?7 > 1 and (5.1.13) still holds. Let Ro € (6Ro,Ro] and denote RS=6SR0,
s = 0,l,2,---.
Then, from (5.1.13) we have u(Rs+1)
+ KiFs,
s = 0,l,2,---.
Iterating gives, for s = 0,1,2, • • • , ? « A )
L
m=0
J2 (e^rj)7 m=0
^ui^
+
KRp^-^^^
144
Elliptic and Parabolic
Equations
#-7
Rs where C = — . Since s = log^ -=4, we have 0 JV ~ 1 Ro
w(Ra)<(5A
[w{Ro) + CKRl]
-C[lt)
M ^ + ^ o ] , s = 0,l,2,--.,
where a = r—- G (0,7). Let .ffo vary over (9R0,Ro\. Then fta (s = hit) 0,1,2, • • •) varies over (0, RQ\. Thus we can obtain (5.1.14) from the above inequality. • Theorem 5.1.7 Let ! ! c R " be a domain and u G i7 1 (fi) be a bounded weak solution of (5.1.1). Then for any bounded subdomain il' CC fl, there exists a constant a G (0,1) such that [u]a;{V < C, where C is a constant depending only on n, il' and Q. Proof.
For any fixed x° e ft' and 0 < R < -dist(fi', dfl), denote o m(R) = inf u, M(R) = supu, B
where BR = BR(x°).
«
BR
Let
v(x) = u{x) —m(R),
w(x) = M(R)
-u(x).
Then v,w G H1(B3R) are nonnegative and bounded and —Av — —Aw = 0 in B3R in the sense of distributions. Using Harnack's inequality to v and w gives sup v < C inf v, BR/2
B
«/2
sup w < C inf w, BR/2
B
«/2
namely, M{R/2)-m{R)
< C[m{R/2)-m{R)\,
M{R)-m(R/2)
<
C[M{R)-M{R/2)
We may assume C > 1; otherwise we replace C by C + 1. From the above two inequalities, we see that M(R/2)
- m(R/2) < ^±[M(R)
-
m{R)\.
Harnack's
Inequalities
145
C —1 Denote f(R) = M(i?) — m(R), 77 = ——-. Then f(R) is nonnegative and G+ 1 nondecreasing and satisfies /GR/2) < r,f(R). By the iteration lemma (Lemma 5.1.5), there exists a £ (0,1), such that f(R) < CRa, namely, Ma;BR(x°) < C. The conclusion of the theorem can then be completed by an easy covering argument. •
5.2
Harnack's Inequalities for Solutions of the Homogeneous Heat Equation
In §5.1, we have proved that for nonnegative and bounded weak solution u e H^c(Rn) of (5.1.1), there holds sup u < C(n, 9) inf u, Ben
B
V0 e (0,1).
<>R
A natural question is that for the homogeneous heat equation — - Au = 0,
(x, t ) £ f x R+
(5.2.1)
whether the analogous inequality sup u < C(n, 9) inf u, QeR
V6> € (0,1)
(5.2.2)
Go*
holds? The following example shows that the answer is negative. Example 5.2.1 The equation ut — uxx bounded solution on (—R,R) x [0,i? 2 ],
= 0 has a nonnegative and
146
Elliptic and Parabolic
Equations
where £ is a constant. Let 9 G (0,1). Then for fixed x G (-OR, 0) U (0,9R) and* G [0,.R2], we have u(0,t) u(cc,i)
f 2x£ + x2 I \4(t + R2))
= eXP
'
a-55 £ s S n x ^ - ° ° >
which shows that (5.2.2) does not hold. However, for equation (5.2.1), there holds another version of Harnack's inequality. We proceed to establish such kind of Harnack's inequality in several steps. Let (x°,t0) G Rn x R+, and R2
BR = BR(x°) QR = QR(x°,t0)
R2,t0),
x (t0 -
x (to - R2 - OR2, t0 -
QeR = BeR(x°) 5.2.1
= BR(x°)
Weak Harnack's
R2).
inequality
Lemma 5.2.1 Let 0 C R" be a bounded convex domain, H be a measurable subset o/fi and u G W1'P(Q) with 1 < p < +oo. Then \\u - u/s\\LP{n)
< C-^j-(diamn)" +1 ||Vu|| L p ( f2),
=
w\Lu{
where uyy = -rrTi / u(x)dx and C is a constant depending only on n. Proof. Since C°°(Q) is dense in W1,P(Q), it suffices to prove the conclusion for u G C°°(Ti). Let u G C°°(T7). Then for x,y&9,, x
u(x) - u(y) = - / ,0 Jo
v
' du(x + ruj) , dr. dr
OJ
—
y—x \x-y
Integrating over Af with respect to y, gives
\M\(u(x) - UM) = - J J
^
drdy.
Denote d = diamfi and V(x + rw) =
du(x + rw) dr 0,
when x + ru> G Cl, when x + rw € R n \ Q .
Harnack's Inequalities
147
Then \u(x) -utf\
I pn-lV{x
J\u\=iJo
dn n
W\
+ ru)dpdujdr
x + rw)dwdr JO J0
j J\ui\ \ u \=
l
^w\L]x-yll~n]Vu{y)ldyHence /
Jn
p \u(x ) — u/s\ dx
4^"L(ilx-yll"'Wy)]dv)'di-
(523)
--
If p = 1, then the conclusion of the lemma follows by exchanging the order of the integral in (5.2.3); if p > 1, n = 1, then the conclusion follows immediately by using Holder's inequality to the integral on the right side of (5.2.3). Now we discuss the case p > 1, n > 2. Choose /x € (0,1), such that 1
_ ^ _ < / p(n — 1)
x
<
^ - 1 ) p(n — 1)
or /i(l — n)
P
> — n, (1 — /i)(l — n)p > — n; p-1 the existence of /x is obvious. Using Holder's inequality, we have
= (_£ (\x - yr{1-n))
(\x - j/|<1-"><1-n>|Vu(j/)|)dj/)P
< ( [ \x - ylM-nWto-Vdy)
\Jn
f \x-
Jn
J 1
y\{l-^){1-n)p\Vu{y)\pdy
148
Elliptic and Parabolic
Equations
Integrating over fi with respect to x and exchanging the order of the integral, we further obtain
jf Qjjz-yMVuMldi/Yda;
f \x -
y\Q-lM-n)P\S7u(y)\Pdydx
JnJn = C - d M(l-n)lH-n(p-l)
f f f ^ _ y^-Ml-nfrfa}
|Vu(y)|"dj/
\Vu(y)\pdy
\Vu(y)fdy.
Substituting this into the right side of (5.2.3) then derives the desired conclusion. • Lemma 5.2.2 For any constant 7 > 0, there exists a nonnegative function j ( s ) s C 2 ( 0 , +00) with the following properties: i) For any s > 0, g"(s) > [g'(s)}2 - yg'(s), g'(s) < 0; ii) g(s) ~ —Ins as s —> 0 + ; Hi) g(s) = 0 for s > 1. Proof.
We first observe that if g(s) satisfies i), then /(s) = - e - s ( s \
s>0
satisfies f"{s) + jf'(s) = e-^[g"(s) - (g'(s))2 + 7 0, namely, h(s) = f'(s) + 7/(s) is nondecreasing on (0, +00). We try to consider the function (to(s)= ( - I n — — J
,
s>0.
It is easy to see that go{s) is nonincreasing and satisfies conditions ii) and iii). In addition, the function -7s
-
1
1_e-T>-
}>
S > 0
satisfies fS(s) + jf^(s) = e"*. W[0»( a ) - (g>0(s))* + ig'0(s)} = 0 for s ± 1. Roughly speaking, go(s) satisfies all conditions i), ii), iii) except at s = 1, where go(s) loses the smoothness.
Hamack's
Inequalities
149
The above analysis leads us to find a suitable smooth approximation of go(s)- A simple calculation shows that
M * ) = /o(*)+7/o(s) = <
~-—, 1 e - " —7,
for
7
s € [0,1), for s G (l,+oo)
and lim ho(s) < lim ho{s). Moreover s—+l~
a—>1 +
f
e^sho(s)ds = - e 2 7 .
JO
Now we construct a smooth approximation of ho(s), denoted by h(s), which satisfies the following conditions: h(s) G C°°[0, +00), h(s) < 0, h'{s) > 0 and h{s) = h0(s) for s G [0,1/2] U [2, +00),
/ Jo
ersh(s)d s =
Then we determine f(s) by solving the equation f'(s) + jf(s) /(0) = 0. f(s) can be expressed as / ( s ) = e - 7 * / e~<sh(s)ds, /o
-e2\ = h(s) with
s > 0.
A simple calculation shows that f(s) = - 1 ,
s > 2.
Finally, we define g(s) = - l n ( - / » ) ,
5>0.
It is easy to verify that g(s) is the required function. In fact, since
(f'(s)e^)'
= (f"(s) + ff'(s))eT
= h'W
> 0,
s > 0,
f'(s)e'ys is nondecreasing. For s > 2, we have f'(s)ey3 = 0, so for s > 0, /'(s)eT s < 0, namely, f'(s) < 0, and hence g'(s) = - ( / ' ( s ) / / ( s ) ) < 0. This together with ft'(s) > 0 shows that g(s) satisfies i). • R e m a r k 5.2.1 If g(s) satisfies i), then G(s) = g(as + (3) possesses the same property, where a > 1, 0 > 0 are constants.
150
Elliptic and Parabolic Equations
Let u £ H{oc(Rn x R + ) be a nonnegative weak solution of
Lemma 5.2.3 (5.2.1). If
mes{(x, t) eQR-,u(x,t)
>1} > fimesQR,
0 < / i < 1,
(5.2.4)
1—u n 2 then, for any a £ (0, (J,) and f3 £ (/i, 1) satisfying 0 = -, there 1 — cr 3 exists a constant h £ (0,1) depending only on n and /J,, such that to — o-R2
mes{a; £ BpR; u(x, t) >h} > -mesBpR,
Proof.
Let £ be a cut-off function on BR relative to BpR, namely, C £
CS°(BR),
0 < C(a?) < 1, C(z) = 1 on 5 ^ and |V<(a;)| <
2
u
as a
(1
°)R-
Take
es
yj = C X[tj,t 2 ]^'( ) t * function in the definition of weak solutions of s ne (5.2.1), where X[ti,t2] * * characteristic function of the segment [ii,^]) t0-R2 < h < t2 < t0, and G(s) £ C 2 (R) is a function satisfying G'(s) < 0, G"(s) - (G'(s))2 > 0. Then [
2
(C2G'(u)ut + V(C 2 G'(u)) • Vu)dxdt = 0
/
Jti
JBR
or f2 Jti
{C2G'(u)ut + {2G"(u)\Vu\2
j
+ G'{u)Vu-V(C2))dxdt
= 0.
JBR
With w = G(u), we have 2
wt + C2|Vu;|2 + Vu> • \/{C,2))dxdt
Jt!
= f2 Jt!
JBR
C 2 [(G'(u)) 2 - G"(u)]\Vu\2dxdt
I
< 0.
JBR
Using Cauchy's inequality with e to obtain |V«> • V(C 2 )| = 2C|VW • VC| < \
C2wtdxdt+-
\
2 Jtl
JBR 2
JBR n
\V<:\ dxdt < CR Jn
JBR
C2\Vw\2dxdt
/
< CmesBR.
(5.2.5)
Harnack's
Inequalities
151
Now we take w = G(u) = g(u + h) with g being the function constructed in Lemma 5.2.2 and h to be specified later. Denote jET(t) = mes{x G BR-,u(x,t) > 1},
JVt = {x e BpR-,u(x,t) > h}.
Then by the assumption of the lemma, we have /•to
-p{t)dt >/imesQfi = i?2jUmesB/j.
/
2 Jto-R >to-
On the other hand, obviously /•to
~p{t)dt < aR2mesBF
/ So 2
rto-
fi(t)dt > (/i — a)R Jto-
mesBR.
R2
Hence, by the mean value theorem, we see that there exists r G [to — R2, to — CTR2}, such that /X(T) >
n
[t0 - crR2,t0] in (5.2.5) and note that /3 G (fi, 1). Then
Take h =r,t2€
Thus
JT
'BR JB,
mes.BR. l — cr
(,2wtdxdt < CmesBjt < C(/x)mesJBMij <
C(fi)mesBpR.
(2(x)w(x,t2)dx BR
=
/
C,2wtdxdt + /
C,2{x)w(x,T)dx
C2{x)w(x,r)dx.
+ /
(5.2.6)
JB I RBR
Since w — g(u + h) and g'(s) < 0, we have / •IBR
C,2(x)w(x,t2)dx
> /
w(x,t2)dx JB0R\Nt2
>mes{B0R\Nt2)g(2h)-
(5-2.7)
152
Elliptic and Parabolic
Equations
Again note that g(s) = 0 for s > 1. We also have C,2(x)w{x,r)dx
/
< I
JBR
w(x,r)dx
JBR
—/
w(x,r)dx
< (mesBR — Jl(T))g(h)
J{X€BR;U(X,T)<1}
^ f1 ~ Y ~ ) 9(h)mesBR = -L=J~g(h)mesBm.
(5.2.8)
Combining (5.2.7), (5.2.8) with (5.2.6) and noting that — — / 3 ~ n = - , we 1—a 3 arrive at ,„ S.T s ^ 3C(n) + 2g(h) mesB mes^.jAAU < 3 (2M WSince g(s) ~ —Ins (s —> 0 + ), we may choose h so small that 3 mes(B /3/i \Af t2 ) < -mesB^fi. Therefore Vt2 € [<0 - o-R2,t0],
mes7Vt2 > -mesBpR, namely,
mes{x € Bj3R; u(x, t) > h} > -mesBpn,
to - eR2
a Remark 5.2.2
/ / condition (5.2.4) is replaced by
mes{(x,t) G QR;u(x,t)
> e} > /xmesQ/j,
0 < fj, < 1,
then, since — is still a nonnegative weak solution of (5.2.1), we can use e Lemma 5.2.3 to derive mes{z G BpR; u(x, t) > eh} > - m e s S ^ , 4
t0 - o-R2
n *j Ton Lemma 5.2.4 Let u G H^ x R + ) be a nonnegative weak solution of oc(R 'locV (5.2.1) satisfying
mes{z £ BpR; u(x, t) > h} > ismesBpR,
t0 - crR2
153
Harnack's Inequalities
where 0 < v < 1. Then for 6 = l/2min(/3, - 0 depending only on n, v, h and 9, such that u(x,t)>j,
(x,t)£QeR-
Proof. Denote w = G(u), where G(s) € C 2 (E) satisfies G'(s) < 0, G"(s) — (G'(s))2 > 0. Then, in the sense of distributions, w satisfies wt - Aw = G'(u)ut - G'(u)Au - G"(u)\Vu\2
= -G"{u)\Vu\2
< 0.
This means that w is a weak subsolution of (5.2.1). Thus we can use the local boundedness estimate for solutions of the homogeneous heat equation (see §4.4) to obtain SupW
2
<-^HH2(02eR),
(5-2.9)
where the constant C depends only on n and 6. Let £ be the function used in Lemma 5.2.3. Take t\ = to — CTR2, ti = to in (5.2.5). Then we obtain / / {(?w)tdxdt + \ [° [ <;2\Vw\2dxdt
(5.2.10)
(0
the function constructed in Lemma 5.2.2, then by the monotonicity of g(s), w < g (— J and hence
/
/
(C w)tdxdt
Jto-oR2 JBR £2(x)w(x,t)dx
t=t0 t=to-
>-
f
<:2(x)w{x,t0-aR2)dx>-CRng(^).
(5.2.11)
IBR
Combining (5.2.10) with (5.2.11) gives rto
[° f \S7w\2dxdt
(5.2.12)
By the assumption of the lemma and the fact that g(s) = 0 for s > 1, we have mes{a: 6 BpR;w(x,t)
= 0} > vmesBpn,
to — crR2
154
Elliptic and Parabolic Equations
Using Lemma 5.2.1 with Q = BpR and N = {x £ BpR; w(x, t) = 0}, we see that, for any to — aR2
w2(x,t)dx
JB0R
\Vw(x,t)\2dx
[ \M I
JB0R
\Vw{X:t)\2dx
JB0R
\Vw{x,t)\2dx.
f J BgR
Prom (5.2.12), we further obtain /•to
r
/
/ 2
Jto-aR
/-to
r
w2(x,t)dxdt
/ 2
JBgR
Jto-crR
\Vw\2dxdt
JB0R
< C ( l + g(±))
-£&\MIHQMR)
.
(5.2.13)
Choosing 7 so small that 27 < h and 2
we must have u > 7 on Qefi- Suppose, to the contrary, there exists (x, i) € QOR, such that u(x,i) < 7. Then from (5.2.13), we would have
which contradicts (5.2.14).
D
Prom Remark 5.2.2 and Lemma 5.2.4, we obtain Theorem 5.2.1 (Weak Harnack's Inequality) Assume that u £ 77[QC(R™ x R_|_) is a nonnegative weak solution of (5.2.1) and for some constants e > 0 and fi £ (0,1), mes{(a:,£) £ QR\u(x,t)
> e} > ^imesQR.
Harnack's
Inequalities
155
Then there exist a constant 6 £ (0,1/2) depending only on n and fi and a constant 7 > 0 depending only on n, /x, e and 0, such that u(x,£)>7, 5.2.2
Holder's
V(x,t)€Q6R.
estimate
Applying the weak Harnack's inequality, we can establish interior Holder's estimate for solutions of the homogeneous heat equation. Lemma 5.2.5 Let u e i? 1 (Qfl 0 ) be a bounded weak solution of (5.2.1) in QnQ. Then there exist constants 6 € (0,1/2) and a € (0,1) depending only on n, such that for any 0 < R < RQ/2, either i) osc u < CR QOR or ii) osc u < aoscu. QeR
Proof.
QR
Denote M = sup u. Without loss of generality, we may assume QR
that ui(R) = osc u = 2M; otherwise we consider v = u — — ( sup u + inf u 2 \
QR
Q R
QR
instead of u, which is a bounded weak solution of (5.2.1) with oscw = osc it = 2supv. QR
QR
QR
If M < R, then for 6 € (0,1), we have osc u < osc u — 2M < 2R, QSR
QR
which implies i). To prove ii) in case M > R, note that one of the following two cases must be valid: mes{(a;,t) € QR,U > 0} >
-mesQR
and mes{(a;,i) € QR, —U > 0} > -mesQj?.
156
Elliptic and Parabolic
Equations U
For defmiteness, we assume that the first one is valid. Let u = 1 H . _, M Then u > 0 and mes{(:r,£) G QR,U > 1} > - m e s Q ^ . From the weak Harnack's inequality (Theorem 5.2.1), it follows that there exist 6 G (0,1/2) and 0 < 7 < 1, such that u(x,t)>7>
(x,t)eQdR.
Thus - M ( l - 7) < u(x, t) < M,
(x, t) G Q8R.
Hence ui{6R) = supu - inf u < 2 M ( l - - ) = acj(R) 7 with a = 1 — —. 2 Using the iteration lemma (Lemma 5.1.5), we can obtain
n
Corollary 5.2.1 Let u G i/ 1 (Qfl 0 ) be a bounded weak solution of (5.2.1). Then there exist constants a G (0,1) and C > 0 depending only on n, such that osc u < G -=—
osc u + Ro , 9*o
\RO
QR
0
Furthermore, we have Theorem 5.2.2 Let u G H^^W1 x R + ) be a bounded weak solution of (5.2.1). Then there exists a constant a G (0,1), such that for any QR C QR C M.n x R+, b4a,QR < C, where the constant C depends only on n and QR. 5.2.3
Harnack's
inequality
Using the weak Harnack's inequality (Theorem 5.2.1) and Holder's estimate (Theorem 5.2.2). We can derive the following Harnack's inequality.
157
Harnack's Inequalities
Theorem 5.2.3 Let u G Hloc(Rn x R + ) be a nonnegative and bounded weak solution of (5.2.1). IfAR2 < to, then there exists a constant 6 € (0,1) depending only on n, such that sup u < C inf u, eR
QSR
where the constant C depends only on n and 6. Proof. We may assume that R = 1, since in the general case, we can transform the problem to the case R = 1 by rescaling. Suppose first supu = 1. Prom Theorem 5.2.2, there exist constants Oi
a G (0,1) and C > 0 depending only on n, such that [U] Q ,Q 2 < C. Since ©i C Q2, there must be constants e > 0 and \x G (0,1) depending only on n but independent of u, such that mes{(x,t) G Q2\u{x,t)
> e} > /xmes<52-
From Theorem 5.2.1, it follows that there exist a constant 9 G (0,1) depending only on n, // and a constant 7 depending only on n, /i, e and 8, such that u(x,t) > 7,
(x,t) G Qe-
Thus sup u = 1 < — inf u 0!
7 Qe
and the conclusion of the theorem follows with C = —. 7 The case supix = 0 is trivial. For the general case supu > 0, we consider 6i
w =
01
u, which is also a nonnegative and bounded weak solution of supw 0i
(5.2.1) with supw = 1. Thus, from what we have proved, ©i
supw — 1 < Cinf w. Qo
0i
Multiplying both sides by sup u leads to Qi
sup u < C inf u. ©i
Qe
n
158
Elliptic and Parabolic
Equations
Exercises 1. Assume that fi C KB is a bounded domain and u € C(Cl) satisfies the mean value equality, namely, for any y £ft and R > 0, u
(y) = „ , Pnn _ !l / u(x)ds, nwnR JdBR
provided equation
BR(J/)
C Bji(y) C CI. Prove that u € C 2 (fi) and satisfies Laplace's - A u = 0,
x e Q,
where Bn{y) is the ball in Rn of radius R, centered at y, wn is the measure of the unit ball in R n . 2. Prove Remarks 5.1.1, 5.1.3 and 5.1.4. 3. Assume that BR is a ball in R" of radius R, f € L°°(BR) and u G H1 (BR) is a nonnegative and bounded weak solution of -Au = / ,
x£
BR.
Set u = u + i? 2 ||/|| L oo (BK ). Prove that for any 0 < 6 < 1, sup u < C inf u, BeR
B
OR
where C > 0 is a constant depending only on n, (1 — 0 ) _ 1 and ||u||i,«>(BR). 4. Establish the weak Harnack's inequality for solutions of the nonhomogeneous heat equation.
Chapter 6
Schauder's Estimates for Linear Elliptic Equations
In this and next chapters, we introduce Schauder's estimates for linear elliptic equations and linear parabolic equations of second order respectively. These estimates will be applied to the existence theory of classical solutions in Chapter 8. In this chapter, Schauder's estimates will be established first for Poisson's equation. To establish Schauder's estimates for such a typical and simple in form equation, one can easily expound the basic idea of the method and catch the essence of the argument. Based on the results obtained for this equation, we finally complete the estimates for general linear elliptic equations. To establish Schauder's estimates, we will adopt the theory of Campanato spaces. By means of such approach, the derivation will be more succinct compared with those based on the potential theory or based on mollification of functions given by Trudinger. In addition, this approach is available not only to linear elliptic equations and systems of second order, but also to equations and systems of higher order.
6.1
Campanato Spaces
Schauder's estimates are a priori estimates on the Holder norms of the derivatives of solutions, which are certain kind of pointwise estimates. It is well-known that in many cases, it is quite difficult to derive pointwise estimates directly from the differential equation considered. However, to derive integral estimates is relatively easy. Thus it is reasonable to ask if there is some approach based on an integral description, instead of the above pointwise estimate. The answer is positive. In this section the Campanato spaces are introduced to describe the integral characteristic of the Holder continuous functions. 159
160
Elliptic and Parabolic
Equations
Definition 6.1.1 Let fi C 1 " be a bounded domain. If there exists a constant A, such that for any x &Q and 0 < p < diamfi, \Qp(x)\ > Apn, where Clp(x) = Q,nBp(x),
then fi is called a domain of (A)-type.
Definition 6.1.2 (Campanato Spaces) For p > 1, p, > 0, the subset of all functions u in LP(Q) satisfying l/V
Mp,wn =
SU
P
\P " \
•en \V 0
J£l„(x) ./oc-ri
\u(y) ~ uxJpdy J
< +00
i/
and endowed with the norm \\u\\Cv.» = \\u\\a>^(n) = [u]p^-ii + ||u||z,p(n) is called a Campanato space, denoted by £ p ' M (fi), where Ux p =
'
T?nvYT /
u dy
^ -
Note that [ix]p,M;n is a semi-norm rather than a norm, since [u]p>M;n = 0 dose not imply u = 0. It is easy to verify Proposition 6.1.1
Cp'^(Q)is a Banach space.
Proposition 6.1.2 (Property of the Mean Value) Let Q C Rn be a domain of (A)-type and u € £ p '^(fi). Then for any x € Q and p such that 0 < p < R < diamfi, there holds K * - «*,p| < c[u] p , M ; n / o-" / p i? M / p , where the constant C depends only on A and p. Proof.
For any y £ O p (x), we have \ux,R - uxJp
< 2p-\\ux,R
Integrating over Clp(x) C
$IR(X)
- u(y)\p + \ux
u(y)\p).
with respect to y leads to
\tiP{x)\\ux,R - uXyP\p <2p-x
( [ \ux,R - u(y)\pdy + [ \uXiP - u(y)\pdy) . \JnR(x) Jo.f,(x) J
Schauder's Estimates for Linear Elliptic Equations
161
Hence Apn\ux,R
- u XiP |" < C M ^
; n
( ^ + pP) <
C[u]l^nR»
or C[u}PMnp-n^R^
\UZ,R ~ «x,p| < with another constant C.
D
Theorem 6.1.1 (Integral Characteristic of Holder Continuous Functions) If Q is a domain of (A)-type and n < p, < n + p, then £p,M(f2) = Ca(U) and Ci[u]a-n < MP,M;n < C2[u)afr, ix — n where a =
P on n, A, p, p.
and C\, Ci are some positive constants depending only
The precise meaning of £"'"(0) = C Q (fi) is that C a (H) c £ ^ ( 0 ) and for any u G £ p , , i (Q), there exists a function u G Ca(Q.) such that u = u a.e. in fi. Proof. Let u G C a (fi). y G fip(x), we have
Then for any a; G fi, 0 < /o < diamfi and
1
(u{y) -
*(y) - u^l = " \np(x)
u(z))dz
Qp(x)
<- ——Y, /
\u(y) -
< Ma-,n
Iw-z^dz
<M^i /
/
u(z)\dz
krdz
-^jf"*"*""** where w„ is the measure of the unit ball in R™. Hence
p-" /
|«(y) - Ul,„|*dy < c'K a;fi r-^,(^)l < o w < . n
162
Elliptic and Parabolic
Equations
Mp,M-,n < C[u]a.}n
(6.1.1)
with another constant C depending only on n, A and p. This, together with ||u||z,P(n) < C|u|o ; n implies u € £ ^ ( 0 ) and Ilull£p.c(n) < C|u| a ; n. Conversely, assume u € Cp'^(il). We will prove that there exists a function u € C a (fi) such that u = u a.e. in CI. S t e p 1 Construct u. For any fixed x € Ti and 0 < R < diamfi, let Ri = R/2i (i = 0,1,2, • • •). Then by Proposition 6.1.2, < C[u] P l M i nie ( "- n ) / p 2 i < n -">/* + "/'\
\ux,Ri - uXtRi+1\
Hence for any integer j such that 0 < j < i, we have i-l
r,fl, - « * , * ! < C [ U ] p , w n ^ - n ) / p ^ 2 , : ( " " M ) / p + " / p i-i —C,1nl'p\t\\
-oz
-T>(v--n)/Pni(n-n)/p
iujp,M;n/t
\
z
1
_
/ 2(n_M)/p
or K i t , - «*,«, I < C[u)p^nR^'n)/p
(6.1.2)
with another constant C depending only on n, A, p and /i. This implies that for any x £ fl and 0 < i? < diamfi, {ux,iJi}i^o ' s a Cauchy sequence and hence UR(X) = lim u I | f i i ,
a; G fi.
For any 0 < r < R, let r< = r/2* (i = 0,1,2, • • •). Then by Proposition 6.1.2 K A -Ux.rJ
(7 ) p\
n
/p
^-" )/p -
Schauder's
Estimates for Linear Elliptic Equations
163
Since \i > n, we have lim |ux,Ri - Ux,n | = 0. i~+00
Hence UR(X) = ur{x), which means that UR(X) is independent of R. Denote u(x)
Step 2
= UR{X),
I£(1.
Prove u = u a.e. in Cl.
Take j = 0 in (6.1.2) and let i —> 00. Then we obtain K H - u(x)\ < C[u]p^R{tl-n)/p.
(6.1.3)
Hence u(x) = lim u x A,
a; £ fi.
ij—0+
On the other hand, by Lebesgue's theorem, u(x) = lim ux
a.e. x € fi.
R,
fi->0+
Therefore u = u a.e. in Q. Step 3 Prove fi G C a ( 0 ) . For any x,y £fl, x ^ y, denote R = \x — y\. Then |u(:r) - u(y)| < \u{x) -
UX,2R\
+
\UX,2R
- uyaR\
+ \uy,2R - u{y)\.
From (6.1.3), we have \u(x) - ux,2R\ + \uy,2R - fife)I < C[u] p , M i nii ( M - n ) / p . Denote G = f22fi(x)nQ2ij(y). /
\Ux,2R -
Then
Uy,2R\dz
JG
<
\ux,2R - u(z)\dz+ v/f!2R(x)
/
|u„ i 2 fl-u(z)|dz
Jtl2n.(y) i/p 1
<|^2JR(^)| -
1/P
( /
|u I l 2 R -u(z)|P(fc
\JCl2R(x)
j
+ |02fl(y)|1-1/p ( f \Jn2R(y) 1 1 p
<(2i?)^|n2fl(x)| - / NP,wfi +
\uyt2R -
u(z)\pdz) )
{2RylP\£l2R{y)\l-l'P\u]p^n
164
Elliptic and Parabolic
Since SlR(x) C G, we have ARn < inequality,
\SIR(X)\
Equations
< \G\. Thus from the above
C[u}p^nR(,i-n)/p
\ux,2R - uyt2R\ < with another constant C. Therefore
\u{x) - u(y)\ < C[«]PlMin|a: - y\^-n)/v,
(6.1.4)
which implies u e Ca (Q) and [«]0in < C[«] Pi/i . n
(6.1.5)
with constant C depending only on n, A and p. Summing up, we have shown £ p , / i (0) = C"*(0) and completed the proof of the theorem. • Remark 6.1.1
For any u e £ ^ ( 0 ) = Ca(U) \u\0;Cl < C||li||£p,n(fi).
In fact, by the continuity of u, there exists z £ Q, such that
Thus, for any x € fi, using (6.1.5), we have \u{x)\ <\u{x) — u(z)\ + \u(z)\
1 I f =\u(x) - u{z)\ + ]QT / u(y)dv
\a\-1/p\\u\\LP{n)
\ft\-1/p\\u\\LPin)
For any 0 < A < 1, we may define 1/p
p 0
(p x
M
/
p
l«(y)-«*,pl dy 1 '
which is also a semi-norm. From the proof of Theorem 6.1.1, we see that, II
if a =
JT
e (0,1], then this semi-norm is equivalent to the Holder
Schauder's
Estimates for Linear Elliptic
165
Equations
semi-norm [u]a-n, i.e. CiMain
(6.1.6)
where C\,C2 are positive constants depending only on n,A,p,/i
and A.
In fact, the second part of (6.1.6) follows from (6.1.1) and the obvious inequality
MiJU < ("]p,wn where the constant Ci can be chosen independent of A. On the other hand, similar to the proof of (6.1.4), we may obtain \u]W
=
SUD
Hx)-U(y)\
i,»en 0<|x—y|
I J>
(A)
i/|
which in combination with (A) [u]«;n
implies that the first part of (6.1.6) also holds, but the constant C\ depends on A. Proposition 6.1.3 Let O be a domain of (A)-type. Then for fi> n + p, all elements of £ P ' M (Q) are constants. Proof.
Prom (6.1.4) in the proof of Theorem 6.1.1, for any x, y £ Cl, \u(x)-u(y)\
_ , , . ^—n , , d u , . , n . By the assumption, fx > n + p or > 1, thus —— n = 1,2, • • • , n) p
dXi
exist and equal zero. 6.2 6.2.1
•
Schauder's Estimates for Poisson's Equation Estimates
to be
established
From now on, we are devoted to Schauder's estimates for solutions of linear elliptic equations. To obtain the estimate for solutions on a bounded domain Q, we first establish the local interior estimate, i.e. the estimate on any ball contained in fl and the local estimate near the boundary dCl, i.e. the estimate on the small neighborhood of any point of dCl, and then use
166
Elliptic and Parabolic
Equations
the finite covering technique. Since the smooth boundary can be locally transformed to a superplane by flatting technique, to obtain the local estimate near the boundary, it suffices first to establish the estimate on any small semiball. We begin our discussion with Poisson's equation -Au(x)
= f(x),
x £ Rl,
(6.2.1)
and hope to establish the following estimates: i) Interior estimate. If u £ C2'a(BR)(0 < a < 1) is a solution of equation (6.2.1) in BR = BR(X°), then [D2u]a;BR/2 < C ( - ^
\U\0,BR
+ - ^ | / | 0 ; B R + [/] Q;Bfl ) ;
(6.2.2)
ii) Near boundary estimate. If u £ C2'a{BR) (0 < a < 1) is a solution of equation (6.2.1) in J5^ = B^(x°) = {x £ BR{XQ);XU > 0} satisfying 0,
(6.2.3)
x„=0
then
P a < B j „ * C ( ^ > I O ; B + + ^I/IO;B+ + I / U j ) •
(6-2-4)
In (6.2.2) and (6.2.4), C is a constant depending only on n. R e m a r k 6.2.1 From the interpolation inequality (Theorem 1.2.2 of Chapter 1), we see that in (6.2.2) and (6.2.4), [D2u]a can be replaced by
M2,aR e m a r k 6.2.2 u
/ / instead of (6.2.3), the boundary value condition is
=
xn=0 u — ip.
R e m a r k 6.2.3 In the proof of (6.2.2), (6.2.4) (and their preparatory propositions) stated below, we always assume that the solution u considered is sufficiently smooth. This is reasonable, because we have the following proposition. P r o p o s i t i o n 6.2.1 / / the estimate (6.2.2) ((6.2.4)) holds for any R>0 and any sufficiently smooth solution u of (6.2.1) on BR = BR(X°) (BR = ~B+R{xQ) with (6.2.3)), then (6.2.2) ((6.2.4)) also holds for any R>0 and any solution u £ C2>a{BR) (C2^(BR)) of (6.2.1).
Schauder's Estimates for Linear Elliptic
Equations
167
Proof. Given R > 0. Suppose that u £ C2'a(BR) is a solution of (6.2.1) in BR. Denote Re = R-e with 0 < e < R. Let £ e Cfi°(BR) be a cut-off function on BR relative to BRC and v = £u. Then v £ C2'a(BR) satisfies -Av
= g,
x£ BR,
(6.2.5)
= 0,
(6.2.6)
dBR
where g = £f - uA£ - V£ • Vtt. Since u € C2'a(BR), we have g € Now we choose a sequence {ffm} C C°°(BR), converging to g in as m —• co and consider the approximating problem
Ca(BR). Ca(BR)
XGBR,
Prom the L 2 theory (see Theorem 2.2.5 of Chapter 2), this problem has a solution vm £ C°°(BR) \\Vm -Vl\\H*(BR)
and < C\\9m-gi\\L*(BR),
( " I , / = 1, 2, • • • )
where the constant C depends only on n and B« (see Remark 2.2.2 of Chapter 2). This implies that {vm} converges in H2(BR) as m —> oo, whose limit function is obviously a solution of (6.2.5), (6.2.6) and hence it is equal to v almost everywhere in BR by the uniqueness of the solution. Since vm € C°°(BR), by the assumption of the proposition, we have [D2Vm}a]BR/2
< C ( ^2+S>m|0;B* + -j^\9mW,BR + [9m]a;BR J ,
(6.2.7)
where the constant C depends only on n. According to the maximum principle for Poisson's equation, it is easily seen that {vm} is uniformly bounded and uniformly converges on BR as m —• oo, whose limit function is just v. Since the right side of (6.2.7) is bounded, using Arzela-Ascoli's theorem, we see that there exist a subsequence {vmk} of {vm} and a function w € C2'a(BR/2), such that as k —> oo we have Vmk{x)->w(x), DiVmk(x)->Diw(x), Dijvmk(x)-^Dijw(x), uniformly on BR/2-
1 < i < n, l
168
Elliptic and Parabolic
Equations
Now we take m = rrik in [D Vm]a;BRc/2
< C (
R2+a\vm\o;BRe
+ -j^\9m\o;BRs
+ [9m]a;BRc
J ,
which holds by the assumption of the proposition, and let k —> oo to obtain \D2wUBRc/2
< C l^^\v\0.BRc
Since clearly w = v on BRe/2 just \D2uUBRc/2
+ -j^\g\0;BRc + [g]a;BReJ ,
and £ = 1 on BRC, the above inequality is
< C(-jJ2+^W\0;BRc + - ^ I / I O J B K , +
[f]a;BRt),
from which (6.2.2) follows by letting e —> 0 + . Similarly we can prove the second part of the proposition. 6.2.2
Caccioppoli's
•
inequalities
we first prove Caccioppoli's inequalities for solution of Poisson's equation. Theorem 6.2.1 Let u be a solution of (6.2.1) in BR. 0 < p < R and A € i , there hold [ \Du\2dx
L
\Dw\2dx
where w = Diu{l
Then for any
[ f2dx], J JBR
(6.2.8)
^ f (w-X)2dx+[ (f-fR)2dx\, (R - PYJBR JBR J
(6.2.9)
< i < n), fR =
\BR\
depending only on n.
/
f(x)dx
and C is a constant
JBR
Proof. Let 77 be a cut-off function on BR relative to Bp, i.e. 77 6 and satisfies 0 < T)(x) < 1,
r)(x) = 1 in Bp,
\Dr](x)\ <
CQ°(BR)
C R-
p
2
To prove (6.2.8), we multiply both sides of (6.2.1) by rj (u — A), integrate over BR and integrate by parts to derive / JBR
T]2\Du\2dx =-2
j JBR
r]2(u-\)fdx.
rj{u - \)Drj • Dudx + JBR
Schauder's
Estimates for Linear Elliptic Equations
169
Using Cauchy's inequality with e to all terms on the right side we are led to r]2\Du\2dx
/ JBR
<\ I r)2\Du\2dx + 2 [ 2 J BR J BR
(u-X)2\Dr,\2dx
which implies (6.2.8). To prove (6.2.9), we multiply both sides of the equation for w, i.e. - A t » ( i ) = Dif(x)
= Di(f(x)
- fR),
xeBR
by T]2(U — A), integrate over BR and integrate by parts to derive ri2\Dw\2dx
/ JBR
= - 2 /
n(w-
\)Dr) • Dwdx - f
JBR
n2(f -
fR)Diwdx
JBR
- 2 /
v(w-\)(f-
fR)Dir1dx.
JBR
Then we use Cauchy's inequality with e to all terms on the right side to obtain r)2\Dw\2dx
/ JBR
<\ 2
I
r)2\Dw\2dx + C f
(w - X)2\Dr]\2dx
JBR
JBR
+ C f V2(f~fR)2dx JBR
C
(/ - fRYdx. / (f-M*
IBR JBR
170
Elliptic and Parabolic Equations
and hence (6.2.9) follows. Corollary 6.2.1
Let u be a solution of (6.2.1) in BR. Then
\D2u\2dx
JB
•
(±
j
B
u2dx + Rn\f\20,BR + Rn+2a[f}l,B^
,
where C is a constant depending only on n. Proof.
Taking p and R to be — and -R and A = 0 in (6.2.9) gives \Dw\2dx < jL J
f JBR?R/2 I2
w2dx + C f
JJB30/4
fR)2dx;
(f-
JD •'B3R/4 3R/4
&3R/4
3 taking p and R to be -R and R respectively and A = 0 in (6.2.8) gives w2dx < - ^ /
/ JB3R/4
R
u2dx + CR2 f
JBR
fdx.
JBn
A combination of these two inequalities leads to
\D2u\2dx
f <£z R
f
<|_ f K
u2dx + C f
JBR
f2dx
u2dx
JB3R/i
n
+ CR
{f-fR)2dx
+C f
JBR
\f\l,BR
+
CR^2a[f}2a,BR. •
JBR
Corollary 6.2.2 holds
If f = 0 in B\, then for any positive integer k, there
IMItf*(B1/2) <
C||W||L J (BI)»
where C is a constant depending only on n and k. Proof. The conclusion for fc = 1 follows immediately from Caccioppoli's inequality (6.2.8). Now we consider the case fc = 2. Prom (6.2.8), we have [ \Du\2dx < — ^ / u2dx. JBP {R-PY JBR
(6.2.10)
Applying (6.2.8) to DjU (j = 1,2, • • • , n) leads to
/ JBD
\DDjU\2dx < .J3 {
R
,2 f ~ P)
JBR
\DjU\2dx.
(6.2.11)
Schauder 's Estimates for Linear Elliptic Equations
171
3 1 3 Taking p = -, R = 1 in (6.2.10) and p = - , R = - in (6.2.11), we obtain \Du\2dx
J J B3/4
u2dx,
J B\
\DDju\2dx
j JBx/2
\Dju\2dx. JB3/I
Thus IMIff*(B1/a) < C||u|| L 2 ( B l ) . For the case k > 2, we may prove by analogy. Corollary 6.2.3
•
lff = 0in BR, then sup \u\ < C ( — / BR/2
\
K
u2dx I
JBR
J
where C is a constant depending only on n. Proof. Assume R = 1 for the moment. Choose k > n/2 in Corollary 6.2.2 and use the Sobolev embedding theorem. Then we obtain SUp \u\ < C | | u | | f f f c ( B l / 2 ) < C | | u | | L 2 ( B l ) . B1/2
For the general case R > 0, the desired conclusion can be obtained by rescaling. • T h e o r e m 6.2.2 Let u be a solution of problem (6.2.1), (6.2.3) in B^. Then for any 0 < p < R, there hold j
\Du\2dx
JB+
l so
1
(R-P)
f JB+
l / \Dw\2dx
where w — Diu(l
u2dx + (R-p)2
f2dx\,
[ JB+
w2dx+
< i < n), / » = -—xr /
[ (f-fR)2dx\, JB+ f(x)dx
J
(6.2.12) (6.2.13)
J
and C is a constant
depending only on n. Proof. We merely prove (6.2.13). The proof is similar to that of Theorem 6.2.1, the only difference is that here we multiply the equation for w - A » ( i ) = Dif{x) = Di(f(x)
- /„),
x&B+
172
Elliptic and Parabolic
Equations
by r)2w and then integrate over JB^ to obtain T]2wAwdx = /
- / JB+
Since n €
C^(BR)
n2wDi(f
-
fR)dx.
JB+
and w
0, the integral over the boundary dB^ xn=0
resulting from integrating by parts is equal to zero.
•
Remark 6.2.4 In Theorem 6.2.2, we use fa to denote the average of f over the half ball B^, while in Theorem 6.2.1, $R denotes the average of f over the ball BR. In order to abbreviate the notations, here we use the same notation to denote slightly different things. But no confusion will be caused. Remark 6.2.5
Using (6.2.13) and the equation n-l L'nn'U'
= ~ ^2 DkkU ~ f>
xeB
R
fc=l
we can derive
L
\D2u\2dx
dx+
(R
/
(f-fR)2dx
f fdx JBt
• l ^ j / \Du\2dx+ (R - P)2 JBt
[ JBt
(6.2.14)
fdx
where C is a constant depending only on n. Remark 6.2.6 We cannot apply the method of the proof of Theorem 6.2.2 = 0 dose not imply w to w = Dnu, since u 0. x„=0
xn=0
In the proof of Theorem 6.2.2, we did not use n2{w — A) = -Xrf = 0 if and only if as a multiplier, because r)2(w — A)
Remark 6.2.7 A = 0.
xn=0
Combining (6.2.12) with (6.2.14) gives
xn=0
Schauder's Estimates for Linear Elliptic Equations
Corollary 6.2.4 j
173
Let u be a solution of problem (6.2.1), (6.2.3). Then
\D2u\2dx < C (^
u2dx + R«KB+R
J
+ R^[fl,B+)
,
where C is a constant depending only on n. Corollary 6.2.5 holds
If f = 0 in B*, then for any positive integer k, there
\\U\\H"(B+/2)
^
C
'II U IIL 2 (B+)>
where C is a constant depending only on n and k. Proof. The conclusion for fc = 1 follows from Caccioppoli's inequality (6.2.12) immediately. Using (6.2.12) for u and DiU (i = 1,2, • • • , n — 1) and combining with the equation, one obtains the desired conclusion forfc= 2. The case k > 2 can be discussed by analogy. • Similar to the proof of Corollary 6.2.3, we can use Corollary 6.2.5 and the embedding theorem to obtain Corollary 6.2.6
If f = 0 in B^, then
(i
1/2
V
r u2dx
sup \u\ < C I —- /
,
where the constant C depends only on n. 6.2.3
Interior
estimate
for Laplace's
equation
Theorem 6.2.3 Let u be a solution of equation (6.2.1) with f = 0 in BR. Then for any 0 < p < R, there hold I JBP
u2dx
v
#
y
I (u-up)2dx
. /
\BP\ JBD
u2dx,
I
(6.2.15)
JBR
I (u-uR)2dx,
(6.2.16)
JBR
u(x)dx and C is a constant depending only on n.
174
Elliptic and Parabolic Equations
Proof. R/2,
We first prove (6.2.15). By Corollary 6.2.3, we have, for 0 < p <
u2dx < \BP\ supu 2 < Cpn sup u2 < C (^Y
/ JB„
B„
For R/2 < p
BR/2
u2dx.
[
Vit/
JBR
obviously /
u2dx < f
JB„
JBR
u2dx < 2" (~Y y J R
u2dx.
/ JBR
A combination of these inequalities leads to (6.2.15) for 0 < p < R. Now we prove (6.2.16). Since Dju(j = 1,2, ••• ,n) satisfy Laplace's equation in BR, from (6.2.15), we have (DjU)2dx
j
(DjU)2dx,
< C (^)" J
j = 1,2, • • • ,n.
From this and Poincare's inequality ~2 / (u — up)2dx
\Du\2dx, JB„
we obtain, for 0 < p < R/2, [ (u - up)2dx < Cp2 [ JB„
\Du\2dx < Cp2 ( • § ) " / XR)
JB„
\Du\2dx. JBRn
On the other hand, if we choose p = R/2 and A = UR in (6.2.8), then we have \Du\2dx <%
/
K
JBR/2
f
(u -
uR)2dx.
JBa
Hence, for 0 < p < R/2, f {u-up)2dx
(u-uR)2dx.
f
JBp
JBR
Noticing that g(X) = /
(u — X)2dx (A £ R) attains its minimum at A = up,
JBP
for R/2 < p < R, we have / JBp
(u — up)2dx <
(u — UR)2dx < j
JBP
<2"+2(|)n+2y
{u — UR)2dx
JBR
(u-uR)2dx.
Schauder's
Estimates for Linear Elliptic
175
Equations
A combination of these inequalities shows that for any 0 < p < R, (6.2.16) holds. • 6.2.4
Near boundary
estimate
for Laplace's
equation
Theorem 6.2.4 Let u be a solution of problem (6.2.1), (6.2.3) with / = 0 in Bft. Then for any nonnegative integer i and any 0 < p < R, there holds
f ID^dx^cl^-Y
f iD^dx,
\R' JB+ where C is a constant depending only on n. JB+
Proof. We proceed to prove the theorem in five cases. i) The case i = 0. The conclusion can be obtained similar to the interior estimate ((6.2.15) in Theorem 6.2.3). ii) The case i = 1. Choose k > n/2 + 1. For 0 < p < R/2, from the embedding theorem and Corollary 6.2.5, we can obtain
/
.B
i
\Du\2dx
.
\Dju\2dx
'< Since u
— 0, we have xn=0
u2dx < CR2 f
f J +
B
(Dnu)2dx
< CR2 [
JB+
R
\Du\2dx.
JB+
Thus for 0 < p < R/2, I JB+
\Du\2dx
^ttJ
f
\Du\2dx.
JB+ J R
When R/2 < p < R, it suffices to take C > 2n. hi) The case i = 2. Since for j = 1,2,..., n — 1, DjU xn=0
use the conclusion for i = 1 to assert
f \DDjU\2dx
f
\DDjU\2dx.
= 0, we may
176
Elliptic and Parabolic
Equations
n-1
By virtue of the equation Dnnu = — V^ DkkU, we further obtain fe=i n-\
/
{Dnnufdx
J
V
KRJ
Bt
\DDku\2dx
/ JB+
£^I
R
Thus |D 2 u| 2 dx < C (4)"
/
|£> 2 u| 2 dx.
/
iv) The case i = 3. We first use the conclusion for i = 2 to assert that for j = 1 , 2 , . . . , n - l , |I>2I>7-u|2da; < C f ^ "
/
|£>2£>,u|2cfc.
/
n-l
By virtue of the equation Dnnnu
= — Y J DkknU, we further obtain fc=i
\Dnnnu\2dx
/ JB+
\D3u\2dx.
[ ^RJ
JB+
Thus f
+
\D3u\2dx < C ( | )
n
f
\D3u\2dx.
+
v)The case i > 3. We may prove by analogy.
•
Theorem 6.2.5 Let u be a solution of problem (6.2.1), (6.2.3) with / = 0 in B'R . Then for any 0 < p < R, there holds f
u2dx
u2dx,
f
XRJ
JB+
(6.2.17)
JB+
where C is a constant depending only on n. Proof.
0, from Theorem 6.2.4 we see that for 0 < p <
Since u x=0
R/2, f JB+
u2dx < Cp2 f JB+
(Dnu)2dx
< Cp2 ( • § ) " / V
^
\Du\2dx. JB+/2
Schauder's
Estimates for Linear Elliptic
Equations
177
Using Caccioppoli's inequality (6.2.12), we have \Du\2dx <-^[
/
R
JB+,„ J R/2
u2dx. JB+
Thus for 0 < p < R/2, (6.2.17) holds. For R/2 < p < R, (6.2.17) is obvious; it suffices to take C > 2"+ 2 . • 6.2.5
Iteration
lemma
Lemma 6.2.1 Assume that
4>(p)
0
where a, (3 are constants with 0 < 8 < a. Then there exists a constant C depending only on A, a and 0, such that + BR13},
0
Let v = -(a + 0) and choose r € (0,1) such that
(J){TR)
< 1.
BR0
=ATa-/T»4>{R) + BR0 4>{T2R)
AT01'"
< T"4>{R) +
BR0,
BT0R0
+ B{TV + + B(T2V
T0)R0,
+ BT20R0
+ TU+0 +
T20)R0,
<j>{rk+1R)
+ ••• + rk0)
=r(fc+1>"0(fl) + Brk0
(,><"-« + T^-^v~^
=r^>4>{R) + BT
(1
~r
'
V
R0
+ • • • + l ) R0
178
Elliptic and Parabolic
Equations
where C\ > 1 is a constant independent of k. Thus cj){TkR) < CITV-WMR)
+ BRf3},
Vfc > 0.
For any fixed 0 < p < R < Ro, choose a nonnegative integer k, such that rk+1R
6.2.6
Interior
estimate
for Poisson's
equation
Theorem 6.2.6 Let u be a solution of equation (6.2.1) in £?R0 and w = DiU (i = 1,2,..., n). Then for any 0 < p < R< RQ, there holds - L ;
/
\Dw- (Dw)p\2dx
\Dw - (Dw)R\2dx
<- g ^ /
+ C[f]%BR,
where C is a constant depending only on n. Proof.
Decompose w as follows: w = wi + u>2 with wi and w^, satisfying -Atoi = 0, WI
dBE
in BR,
= W,
and -Aw2 W2
= Dif = Di(f - f R ) ,
dBR
in BR,
0.
Now we apply (6.2.16) to Dw\ to obtain /
\DWl-{DWl)p\2dx
JBP
f XK/
Thus for any 0 < p < R < Ro, we have / JBD
\Dw -
(Dw)p\2dx
JBR
\DWl-{DWl)R\2dx.
Schauder's
Estimates
for Linear Elliptic Equations
\Dwi - (Dwi)p\2dx
<2 /
+2 /
JBP
179
(Dw2)p\2dx
\Dw2 -
JBP
+2
f
\DWl-(DWl)R\2dx
JBR
\Dw - (Dw)R\2dx
+C f
JBR
f
KRJ
\Dw2-(Dw2)R\2dx
+2 f
JBR
(Dw2)R\2dx
\Dw2 -
JBR
\Dw-(Dw)R\2dx
\Dw2\2dx.
+C f
JBR
J
BR
Multiply the equation for w2 by w2 and integrate over BR and note that w2 = 0 . Then we deduce dBR
\Dw2\2dx — — I
I JBR
w2Aw2dx
JBR
= I
w2Di(f -
fR)dx
{f -
fR)DiW2dx
JBR
= ~ J
BR
\Dw2\2dx + \f
<\f J BR
"
(f-fR)2dx. JBR
Thus \Dw2\2dx < / " ( / - fR)2dx
f
< CRn+2a[f}2a.BR.
(6.2.18)
>JBR BR
JBR
Hence /
\Dw -
(Dw)p\2dx
JBP
~C
(RT+2
jBR
lDW
- ^W^\2dX
+
CRn+2a
lft,BR-
Using the iteration lemma (Lemma 6.2.1) we finally obtain /
\Dw-{Dw)p\2dx
JBP
~C ( | ) " + 2 a (JBR \DW - (Dw)x\2d* + C*n+2alf}l;BR) •
a
180
Elliptic and Parabolic Equations
Theorem 6.2.7
Let u be a solution of equation (6.2.1) in BR and w =
DiU (i = 1 , 2 , . . . , n ) . Then for any 0 < p < —, there holds f \Dw-(Dw)p\2dx
(6.2.19)
JBP
where C is a constant depending only on n and
Proof.
According to Theorem 6.2.6 and Corollary 6.2.1, we have /
\Dw -
(Dw)p\2dx
JBP
^Pn+2a
( ^ k JBR/2
2
\DW ~ (Dw)R/2\
dx + lf}i]BR/)j
^+2"(^W/BR/2I^I^+[/IU/2) ^Pn+2a
{ j ^ JBR u2dx + ^ I / I U + \f]l,BR) ,
from which the conclusion of Theorem 6.2.7 follows. Theorem 6.2.8 [D2u]a;BR/2
•
Let u be a solution of equation (6.2.1) in BR. Then < C (^^\U\0;BR
+ ^\f\o-,BR
+ [/UBK) ,
(6.2.20)
where C is a constant depending only on n. Proof.
According to Theorem 6.2.7, for x G BR/2, 0 < p < —, we have \D2u(y)-(D2u)Bp{x)nBR/2\2dy
/ JBp(x)f\BR/2
\D2u(y) - (D2u)XtP\2dy
< f JBp{x)
a
f Ri+2a
\U\o;BR/2(x) + - ^ l/lo;BH/2(x) + U\a;BR/2(x)
( j ^ \ < B
R
+ ^\f\l,BR
+ [ft,BR)
,
Schauder's Estimates for Linear Elliptic Equations
181
where (D2u)B{x)r)B
K/i
=
} - / D2u(y)dy \Bp{x)r\BR/2\ JBp(x)nBR/2
Hence 1/2 D
[
U
U
]2,n+2a;BR/2
^ ( p 4 + 2 a \ \o;BR+^2^\f\o;BR+[f]a;BR
)
and (6.2.20) follows by using Remark 6.1.2. 6.2.7
Near boundary
estimate
for Poisson's
D equation
Theorem 6.2.9 Let u be a solution of problem (6.2.1), (6.2.3) in B^, w = DiU (i = 1,2,..., n — 1). Then for any 0 < p < R < Ro, there holds
• ^ jB+ ( E l^'H 2 + \Dnw - (Dnw)p\2 J dx fg+ f E \Diwf + \D"W ~ (A^)fil 2 dx
^J^
+ c[f)lBt, where vp = Proof.
, \BP
/ I
(6.2.21)
v(x)dx and C is a constant depending only on n.
JB+
Decompose w as follows: w = W1+W2 with w\ and u>2 satisfying ' -Awi 9Bt
= 0,
in B^,
w
and ' -Aw2 = Dif = Di(f - f R ) , V)2
dBl
0.
in B+,
182
Elliptic and Parabolic Equations
For j' — 1,2,..., n — 1, from Theorem 6.2.5, we have / IDjWifdxKCf^Y v y JB+ -ft
/ IDjWxfdx, JB+
0
Hence, for any 0 < p < R < Ro, \Djw\2dx
/ JB+
\DjWi\2dx + 2
<2 J Bp
\DjW2\2dx J Bp
< C (v 4 ) " -ft'
+2
2
/
\DjWl\ dx
JB+
+2 f JB+
\DjW\2dx + C f
f
+
\DjW\2dx
\DjW2\2dx \DjW2\2dx
+ CRn+2a[f}2a;B+R,
(6.2.22)
where we have used the estimate \DjW2\2dx<
/ JB+
\Dw2\2dx
f JB+
whose proof is similar to (6.2.18). Thus for 0 < p < R/2, we obtain by using Poincare's inequality and Theorem 6.2.4, /
(Dnw)p\2dx
\Dnw -
JB+
\Dnwi - (Dnwi)p\2dx
<2 /
+2
JB+ 2
\Dnw2 -
{Dnw2)p\2dx
JB+ 2
/
\DDnwx\ dx
JB+
JB+
\D2Wl\2dx
f
^R'
\Dnw2\2dx
+C /
JB+„ J
R/2
+C f
\Dw2\2dx.
JB+
n-\ Using the equation DnnW\ = )> DjjW\ and Caccioppoli's inequality, we further obtain / JB+,„
\D2wi?dx
Schauder's Estimates for Linear Elliptic Equations
\Dnnwi\2dx
< /
+ 2V
D /2 JB+ R/2
= 1l j, =
n—l n-1
183
\DDjWi\2dx
/ D JB+„ R/1
.
\DDjWi?dx /R+ 71-1
| |2 +
1 i 2|2da:
417 B . ^ ^ ^l 4 ^
-
Thus, for 0 < p < R/2, we have (Dnw)p\2dx
\Dnw /. ,71-1
+ 2
^ ( 4 ) " XK/
l ^ ^ | 2 ^ + Ci?" +2a [/]^ + .
E /
(6.2.23)
JB
J=I R
Combining (6.2.23) with (6.2.22) we see that for 0 < p < R/2,
V] \DjW\2 + \Dnw - (Dnw)p\2 dx
I JB
*
\i=i
) ,2
p
C
I n—l
\
w 2 + £>nU
- (R)
I + I S l^ l +
C i r
l
£> u,
'~ ( " )«l
2 dx
,+2a[/]2;Bi-
Therefore (6.2.21) follows for 0 < p < R/2 by using Lemma 6.2.1. For — < p < R, (6.2.21) obviously holds. The proof is complete. Theorem 6.2.10
•
Let u be a solution of problem (6.2.1), (6.2.3) in B^
and w = DiU (i = 1,2,..., n). Then for any 0 < p < —, there holds
L
\Dw - (Dw)p\*dx < Cpn+^aMR,
B+
where C is a constant depending only on n and
(6.2.24)
184
Elliptic and Parabolic
Proof.
Equations
From Theorem 6.2.9 and Corollary 6.2.4, we have ~
/
71— 1
\DjW\2 + \Dnw -
(^
J
B+
{Dnw)p\2\dx
j=1
JB
R/2
j= l
^n+2a(^lBJD^d^KBR/2) ^f>n+2a (
jB+ «2*° + j^Mrt
^
+
K*)
which implies, in particular, that for j = 1,2, • • • , n — 1, / \Djw\2dx < J Bt
CPn+2aMR.
Hence
(DjW)2p = - J j - ( /
IZ^-Hdz J < 7^+7 /
+
l-D^I 2 ^ <
Thus, for j = 1,2, • • • , n — 1, /
\DjW —
(Djw)p\2dx
JB+
<2 / \D1w\2dx + 2 / |(£>,u))pl2da; is+ JB+
[
<
Cpn+2aMR.
Thus / JB+
\Dw - {Dw)p\2dx <
Cpn+2aMR
Schauder's Estimates for Linear Elliptic Equations
185
and we have proved (6.2.24) for w — DiU (i — 1,2, • • • , n — 1). Again using n-l
the equation Dnnu •= — V ^ Duu — / , we derive (6.2.24) for w = Dnu. i=l
Theorem 6.2.11 Then
Let u be a solution of problem (6.2.1), (6.2.3) in B^.
[£2<;B+/2(x°) ^C
( fi2+^ Ho;B+(xO) + ^
l/lo;B+(xO) + [/]a;B+(x<>) J ,
(6.2.26)
where Cis a constant depending only on n. Proof.
According to Theorem 6.2.10, for x € dB^/2(x°)
n
BR/2(x°),
0 < p < - j i we have
\DMy)-(D2u)BUx)fdy
/
\j^^-\u\liB+^x)
+ - ^ | / l o ; B + / 2 ( x ) + [/]a i B + / 2 (x)J
(6.2.27)
where 1 . ., . 1 .-,2 #4+2a N();B+(x°) + # 2 a l'lo;B+(i°) + V ia;B+(x°
^
Let x e B^^2(x°).
Denote x = (xi,- •• , x n _ i , 0 ) . If 0 < xn < —, xn < p <
—, then Bp(x)nB+/2(x°)
C B%p{x). Thus from (6.2.27), we have
|DVI/)-(^2«)B+(
2
dt/
(6.2.28)
If 0 < xn < —, 0 < p < xn, then by Theorem 6.2.6 and (6.2.28) for p = xn, we derive / JB„(x)
\D2u(y)-(D2u)Bf>{x)\2dy
186
Elliptic and Parabolic
f
s
Xn
Equations
r\Dw-(Dw)Bxn{x)fdX+[f}lM))
JBXn(x)
'
/. T>
D
On the other hand, H — <xn B+R/4(x°)
C B^(x°)
D
< —,Q < p < —, then Bp(x) C BR/4(x)
C
and hence, by Theorem 6.2.7,
/ <[
l- D 2 u (v)-( £ , a «)B p (*)nB+ a (*o ) l 2 dy \D2u(y)-(D2u)Bp{x)\2dy
JBJx) lB„{x)
CM1/2
- ^ (^^2+^l U lo;B+(xO) + ^ l / l o ; B + ( x O ) +
and (6.2.26) follows by using Remark 6.1.2. Remark 6.2.8 / / superplane xn = 0, same near boundary w = Diu(i = 1,2,- • the superplane.
[/]a;B+(x°)J
•
the boundary of the domain considered is not the but a superplane of other form, we still have the estimate. Of course, in proving we need to replace • ,n — 1) by the tangential derivatives with respect to
Remark 6.2.9 Using the interior estimate and the near boundary estimate after local flatting of the boundary, we can obtain the global Schauder's estimate on Q,. However, it is to be noted that, after local flatting of the boundary, Poisson's equation will be changed into another elliptic equation. We will discuss how to establish the global estimates for general linear elliptic equations in the next section.
Schauder's
6.3
Estimates for Linear Elliptic
187
Equations
Schauder's Estimates for General Linear Elliptic Equations
Consider the general linear elliptic equation Lu = —a,ij(x)DijU + bi(x)DiU + c(x)u = f(x),
x € CI,
(6.3.1)
where fi c R™ is a bounded domain. We merely study the Dirichlet problem, namely, the problem with boundary value condition =
(6.3.2)
oil
The purpose of this section is to establish Schauder's estimates for solutions of problem (6.3.1), (6.3.2) under certain conditions. We have the following theorem. Theorem 6.3.1 Assume that 0 < a < 1, 9 0 6 C2>a, a^^c £ Ca(Tl), aij = aji, and equation (6.3.1) satisfies the uniform ellipticity condition, namely, there exist constants A, A with 0 < A < A, such that A|£|2 < aijWtej
< A|£| 2 ,
a
2a
V£ e l
In addition, assume f £ C (fi),
n
, i £ 0.
Ifu£
C2'a(fl)
M2,a;ft ^ C G / k f i + M2,a;fi + Mo;fj),
is a solution
(6.3.3)
where C is a constant depending only on n, a, A, A, 0 and the Ca(Cl) norms of'ay, bi, c. Remark 6.3.1 It is to be noted that in Theorem 6.3.1, in order (6.3.3) holds, we need not require u £ C2'a(fl). In fact, u £ C2'a(Cl) n C(Cl) is enough. Under such condition, we have |u|2,a;n e < C(\f\a-n
+ M2,a;fi + W\o;n),
where Cl£ = {x £ Cl; dist(x, dd) > e} with e > 0 small enough. From this we finally obtain (6.3.3) by letting e->0. The proof of Theorem 6.3.1 will be completed by means of simplifying the problem and applying Schauder's estimates for solutions of Poisson's equation and the finite covering argument.
188
Elliptic and Parabolic
6.3.1
Simplification
of the
Equations
problem
First of all, we observe that in establishing the a priori estimate (6.3.3) for equation (6.3.1), without loss of generality, we may assume
= f(x) - Lip(x) € Ca(Q).
If we have established the estimate (6.3.3) for the special case (p = 0, then applying it to the above problem gives \w\2,a;Q < C(\f - L(p\a.Q + Mo;fi), from which (6.3.3) follows immediately. Next, we point out that it suffices to prove (6.3.3) for the equation without terms of lower order, namely, the equation of the special form —a,ij(x)DijU — f(x),
x £
fi.
(6.3.4)
In fact, if we can prove (6.3.3) for equation (6.3.4) with I ^ E O , then applying it to equation (6.3.1) gives |«|2, ai n < C(\f - biDiU - cu\a-a + \u\0.n) < C ( | / | a i n + |u|i, a ; n), and by the interpolation inequality, we obtain \u\2,a;U < C(\f\a-Q
+ |u|o ; fi)-
(6.3.5)
The above discussion shows that we need merely to prove estimate (6.3.3) for the special equation (6.3.4) with the special boundary condition u 6.3.2
= 0 , namely, to prove the estimate (6.3.5). Interior
estimate
We will prove the estimate (6.3.5) by means of the so-called method of solidifying coefficients. The basic idea is to fix a point x° € fl and treat (6.3.4) as an equation with constant coefficients —a,ij(x°)DijU = h(x),
(6.3.6)
h(x) = f(x) + g(x),
(6.3.7)
where
Schauder's Estimates for Linear Elliptic Equations
g(x) = (a,ij(x) -
189
aij(x°))DijU.
In order to estimate the solutions of (6.3.6) for a given smooth function h(x), we will change the variables to further simplify (6.3.6) to the form of Poisson's equation so that we can apply the results obtained in §6.2. Since A = (ciij(x0)) is a positive definite matrix, there exists a nonsingular matrix P , such that PTAP — In, where In is an n x n unit matrix. Let y = PTx =
(P^x,
where we regard the variables x, y as column vectors. Then du dxi d2u dxidxj
du dyk
dyk du D D A . dxi dyk d2u dyi = Pkia.. *.. ' ~^r = PkiPljDklU, dykdyi dx
d - d 2 where Dk = — , Dkl = ^ - ^ - , u{y) = u{{PT)-ly). -a,ij(x0)DijU Since PTAP
=
Hence
-aij{x°)PkiPijbkiu.
= In, we have a y (a; 0 )P fc< Py = 6kl =
Thus equation (6.3.6) becomes - A n = h(y),
where A
+ +
(6.3.8)
+ h{y)=K{pT) ly)
=!* M '""' 5r
~-
We may assert A"1/V - A
< \PTxl
- PTx2\
< \~l'2\xl
- x2\,
(6.3.9)
namely, the distance in the x-space is equivalent to that in y-space. In fact \y\ = {xTPPTx)x'2
=
(xTA~1x)^2.
190
Elliptic and Parabolic
Equations
If we denote by A and A the minimal and maximal eigenvalues of A then A and A are the minimal and maximal eigenvalues of A~1. Hence A~ I / 2 |z| < \y\ < A _ 1 / 2 |x|-
(6.3.10)
From the ellipticity condition we have A < A < A < A. Thus (6.3.10) implies (6.3.9). Now we proceed to apply the interior estimate obtained in §6.2 (Theorem 6.2.8) to equation (6.3.8). Thus, we obtain the following estimate
where BR denotes the ball in the y-space of radius R centered at such that BR C Cl = {y = PTx; x € 9,}. We assume 0 < R < 1. Now we return to (6.3.6) with h{x) given by (6.3.7). Since [g]a,BR
PTx°
+ [fiyUsJAifilo;^
2
+
\u\2;BR),
\9\Q]BR
+
^2\u\o-,BR)
+
^\u\0.,BR),
+
+
^\o-,BR) ^\u\0,BR).
This combined with (6.3.11) leads to
[D2u\a,BR/2
~]^\f\o;BR +
J^\U\0;BR lf}a;BR)-
Schauder's
Estimates for Linear Elliptic
191
Equations
Returning to the original variable x, we derive
where B'1 ensures
= {x = (PT)-1y;
No;B„ < C\U\0;B-1,
y € BR}. Here it should be noted that (6.3.9)
|/| 0 ;B f l <
C|/|0.B-I,
[f]a.6R
< C[/] a .B-i
with the constant C independent of x° € fi. In particular, we have ID2<-,B-}2
< c{Ra[D2u]a.n
+^ M o
;
n + ^ | / | o ; n + [/] a i n).
C BR1. Thus for 0 < R < 1 small enough, we
It is easy to check Bxi/2R have
[D2u]a.B2R
+ ^>|o;n
+ ^ l / | o : n + [/]„;n)
(6.3.12)
with another constant C. 6.3.3
Near boundary
estimate
To establish the near boundary estimate we adopt the local platting technique of the boundary. Let x° e dfl. Since dCl £ C 2 , a , there exist a neighborhood U of x° and a C 2 , a invertible mapping \I> : U —> Bi(0), such that V(U nfi) = B+ = {|/e Bi(0); yn > 0}, ¥(CA n dQ) = dB+ n {y £ R"; yn = 0}, where Bi(0) denotes the unit ball in the y-space. Denote p.
=d% 13
BXj'
k ij
d^k dXidXj'
192
Elliptic and Parabolic
Equations
where * = (* X) * 2 , • • • , *«)• Then
d2u dxidxj
du du dyk D dxi dyk axi dyk dyi du d2yk dxi dxj dyk dxidxj
d2u dykdyi
.
A
-
fe
°
" y
'
'
d d2 where Dk = -=—, Dki = -—5—, u(y) = u ( * ^j/)). Hence -aij(x)Diju
= -aij(y)PkiPijDkiu
aij(y)PfcjDku,
-
where a>ij(y) = aij('^~1(y)). Thus, with the transformation y = equation (6.3.4) turns out to be -ai^PkiPijDuu
- aij(y)PlcjDkU = f{y),
^(x),
(6.3.13)
where f(y) = / ( * - x ( » ) ) . Let G(y) = * ' ( * - 1 ( y ) ) ( * , ( * - 1 ( t f ) ) ) r . Since y = $(x) is a C2'a invertible mapping, \P'(\I>-1(y)) is nonsingular and hence G(y) is positive definite and continuous. Let m{y) and M{y) be the minimal and maximal eigenvalues of G(y) and denote m = min m(y),
M = max M{y).
yeB+
y€B+
Then 0 < m < M. Since (a^PkiPj)
= *'(*-1(2/))i(2/)(^(*-1(2/)))T,
where A(y) = (a,ij(y)), for any £ € M n , we have &ij{y)PuPijtei
= ^*'(*-1(y))i(y)(*'(*-1(y)))Te
Let 77(2/) = ( ^ ( t f - 1 ^ ) ) ) ^ . Then AM 2 < cuMPkiPiM
<
A|T?| 2 .
Since |?y|2 = rjTr] = £,TG(y)£, we have
m|£|2 < H2 < M|£|2. Hence Am|£|2 < cuMPkiPiMi
< AM|e| 2 .
Schauder 's Estimates for Linear Elliptic
Equations
193
This means that equation (6.3.13) is uniformly elliptic. Since * is an invertible C2'a mapping, there exist /xi,/X2 with 0 < /zi < fi2, such that for any xl,x2 £ ft fl U, H^x1 - x2\ < |^(x x ) - * ( x 2 ) | < n2\xl - x2\, which shows that the distance in the x-space is equivalent to that in the y-space. To establish the estimate near the boundary for solutions of equation (6.3.13), as we did for the interior estimate, we consider the equation without terms of lower order, treat it by means of the method of solidifying coefficients and change it to Poisson's equation by a transformation of variables followed by using the estimate near the boundary for this special equation. Here it should be noted that after changing variables, the boundary yn = 0 of B± = { ) / £ Bi(0);yn = 0} becomes a superplane of another shape. However as indicated in Remark 6.2.8, in this case, the near boundary estimate for Poisson's equation stated in Theorem 6.2.11 still holds. Using this result, coming back to the variable y, returning to the original equation and coming back to the variable x, we finally obtain [D2u)a,oR < c(Ra\D2u]a,u
+ ^ | « | o ; n + ^ l / k n + [/]«.;«),
where 0 < R < 1 and OR C fi is such a domain which depends on R and for some constant a > 0 independent of R such that Q fl BaR(x0) C OR. Thus for 0 < R < 1 small enough, there holds [D2uUn2R
+ ^^|«|o;n
+ ^l/lo;n + [/Un), with another constant C, where 6.3.4
Global
CIR
= ft n
(6-3.14)
BR(X°).
estimate
Now we proceed to combine the interior estimate (6.3.12) and the near boundary estimate (6.3.14) and use the finite covering argument to establish the global Schauder's estimate. Combining the interior estimate (6.3.12) and the estimate near the boundary (6.3.14), we see that for any x° £ ft, there exists 0 < R(x°) < 1,
194
Elliptic and Parabolic
Equations
such that for any 0 < R < R(x°), [D2u]a.,n2R < Co (Ra[D2u]a.t(l
+^|«|
0 i
n + ^ l / k n + [/]« ; n) ,
where CIR = fi n BR(X°), CQ is a constant independent of x°, R and R(x°) and either BR(X°) is included in Q or x° e dfl. We will assume that for any x° € fi, i?(x°) < -Ro < 1 with .R0 to be specified later. By the finite covering theorem, there exist a finite number of such open balls BR1(X1), 2 m J BR2 (X ), •••, BRm (x ) with x ' £ CI (j = 1,2, • • • , m), covering fi, and for any 0 < R < R3 < Ro, [D2u]a.U2R{xJ)
+ ^\u\0-n
+ ^|/|0in +
(j = l , 2 , - - - , m ) . Let x',x" € fi and assume x' G BRJO{X^°). two cases must occur: i) \x' -x"\ >Rjo; n)x"en2RJ0(xt°). If i) occurs, then \D2u(x') - D2u(x")\
IfUn) (6.3.15)
Then one of the following
^ 2 < -S5-I^«| 0 ,n. •""jo
If ii) occurs, then from (6.3.15) we obtain \D2u{x') -
D2u(x")\
<\D2u\a-,n2Rji
,.,„<**>)
+ - ^ > | o ; n + ^rl/lojn + [/kn
In either case, we have \D2u(x') -
D2u{x")\
\x' — x"
+ ~\D2u\0.tQ U
3o
+ 4 ^ > | o ; n + -j§-|/|o ; n + C 0 [/] Qi n. H
j0
Hence \D2u}a,n
S\D2u\o-,a R0
K
Jo
195
Schauder's Estimates for Linear Elliptic Equations
+ W ^ M o s n + S / l c n + CotfUci.
(6.3.16)
where Ro = min{.R1, R2, • • • , i? m }. Using the interpolation inequality gives •§-\D2u\o-,n < \{D2u]a.,ci + C\u\0,n.
(6.3.17)
Now we choose Ro < ( 3 C 0 ) _ 1 / a . Then CQRt[D2u\a,Q
< ^[D2u]a.,n,
(6.3.18)
Combining (6.3.17), (6.3.18) with (6.3.16) we derive [D2u]a,n < C(|u| 0 ; n + l / U n ) with some constant C. Using the interpolation inequality again then leads to (6.3.5).
Exercises 1. Prove that £ p ' M (fi) is a Banach space, where Cl C Rn is a bounded domain, p > 1, /x > 0. 2. Let fi C K™ be a bounded domain and u G L 2 (fi). Prove that the function g{\)
= f {u(x) - X)2dx, Jn
X£
attains its minimum at
x=
u
u{x)da ^W\i \n\
3. Let u £ C2'a(B) be a solution of the boundary value problem ' -Au + \Vu\ = / , u
x € B,
= 0, dB
where 0 < a < 1 and B is the unit ball in W1. Prove that there exists a constant C > 0 depending only on n, such that \u\2,a;B
< C{\f\a-,B
+ |w|o;fi)-
196
Elliptic and Parabolic
Equations
4. Establish Schauder's estimates for solutions of the following boundary value problem for the bihaxmonic equation
r A 2 U = /,
x € Q, X6ffl.
Chapter 7
Schauder's Estimates for Linear Parabolic Equations In this chapter, we introduce Schauder's estimates for solutions of linear parabolic equations of second order. We first consider the heat equation, establishing Schauder's estimates for this equation, and then applying to general linear parabolic equations. To obtain Schauder's estimates for solutions of the heat equation, we also adopt the theory of Campanato spaces.
7.1
t-Anisotropic
Campanato Spaces
In Chapter 6, we have introduced the Campanato spaces, and described the integral characteristic of the Holder continuous functions in spatial domains. In this section, we introduce the ^-anisotropic Campanato spaces, and describe the integral characteristic of the Holder continuous functions in parabolic domains. Let Q C R" be a bounded domain, T > 0. Denote Q = £1 x (0,T), Ip = IP{to) = {to-p2,t0
+ p2),
Bp = Bp(x°),
Qp{x°,t0)
=
BpxIp.
Definition 7.1.1 (Campanato Spaces) Let p > 1, n > 0. The subspace of all functions u in LP(Q) satisfying MP,M =[U1P,M;Q
=
sup (x,t)£Q
\P~ti \
\u(y,s)-uXit,P\pdyds) JJQC\QJx,t)
< + 00 197
I
198
Elliptic and Parabolic
Equations
endowed with the norm ||w|U».e =
IMUP.M(Q)
=
MPIM;Q
is called a Campanato space, denoted by £ "*•*•' = \nnn
P,M
(n- *\\ II
+
||U||LP(Q)
(Q), where "^'
s d
) Vds-
Comparing with Definition 6.1.2, here instead of fi n Bp(x), QnQp(x,t). R e m a r k 7.1.1 [u]p,/i;Q is a semi-norm, [U\P,IJ,;Q — 0 does not imply u = 0.
we use
rather than a norm,
since
It is easy to check £ P ' M (Q) is a Banach space.
P r o p o s i t i o n 7.1.1
T h e o r e m 7.1.1 (Integral Characteristic of Holder Continuous Functions) Let fi be an (A)-type domain, n + 2 < /x < n + 2 + p. Then £ p ."(Q) = Ca'a/2(Q) and Cl[u]ata/2,Q
where a = only on
< [u]p,n,Q < C 2 M a , a / 2 , Q )
and C\, C2 are some positive constants depending
P n,A,p,fi.
The proof is similar to that of Theorem 6.1.1 in Chapter 6. R e m a r k 7.1.2
MJ2;Q=
For 0 < A < 1, define the new semi-norm SU
P
(P~* II
(x.oeq V 0
l«(y. s ) ~ ux,t,P\pdyds I
JJQC\Qp(x,t)
Similar to the case of spatial domains, ifa=
.
) '
€ (0,1], then these
semi-norms are equivalent to the Holder semi-norms [u] a , a /2 ; Q, that is ClMa,a/2;Q < Mp^;Q ^ where C\,C2
C2{u)a^/2-Q,
are positive constants depending only on n,A,p,fi
P r o p o s i t i o n 7.1.2 constants.
and X.
If fi> n + 2+p, then all the elements in Cp,tl(Q) are
The proof is similar to that of Proposition 6.1.3 in Chapter 6.
Schauder's Estimates for Linear Parabolic Equations
7.2
199
Schauder's Estimates for the Heat Equation
7.2.1
Estimates
to be
established
Now we proceed to establish Schauder's estimates for solutions of the first boundary value problem for linear parabolic equations. Similar to the case of elliptic equations, we first establish the local interior estimates and local estimates near the boundary. Since for parabolic equations, the boundary condition is prescribed on the parabolic boundary, we need to establish the estimates near the bottom, near the lateral and near the lateral-bottom. We begin our discussion with the heat equation
QR
^ - A u = f(x,t).
(7.2.1)
i) Interior estimate. If u G C2+a'1+a/2(QR) = QR{XQM), then
is a solution of (7.2.1) in
\D2u]aia/2.QR/2
< C (jl^\u\0.QK
+ ^ l / l o i Q * + [/]a,«/2;QB) I
(7-2.2)
ii) Near bottom estimate. If u € C2+a<1+a/2(cfR) is a solution of (7.2.1) in Q°R = Q°R(x°,0) = BR(x°) xI°R = BR{x°) x (0,R2) satisfying = 0,
(7.2.3)
t=o
then [D2u}a,a/2]Q0R/2
< C ( ^ M o ; Q ° R + ^l/l0;QO R + [f\a,a/2&^
! (7-2.4)
iii) Near lateral estimate. If u e C 2 + a ' 1 + Q / 2 ( Q ^ ) is a solution of (7.2.1) in Q+ = Q + ( x V o ) = B+(x°) x IR(t0) = {x € BR(x°);xn > 0} x (t0 R2,t0 + R2) satisfying = 0,
(7.2.5)
x„=0
then [D2u]a,a/2.iQ+R/2
+ ^ 1 / l o i Q j + t / U / a , g j ) 5 (7-2-6)
iv) Near lateral-bottom estimate. If u e C2+a'1+a/2(QR ) is a solution of (7.2.1) in Q°+ = Q°R+(x°,0) = B+(x°) x 7° = {x e BR(x°);xn >
200
Elliptic and Parabolic
Equations
0} x (0,i? 2 ) satisfying (7.2.3), (7.2.5), then ^ ] a , a / 2 ^ 2 < ^ ( ^ k l o ; Q O R + + ^ | / | 0 l Q 0 B + + [/] a , Q/2;Q 0 R+ ) . (7.2.7) R e m a r k 7.2.1 By the interpolation inequality (Theorem 1.2.1 in Chapter 1) and equation (7.2.1), we see that in (7.2.2), (7.2.4), (7.2.6) and (7.2.7), [D2u]a^/2 can be replaced by |u| 2 +a,i+a/2R e m a r k 7.2.2
If the boundary value condition _
IS 111
—
{Q CLTld
dPQ
ip € C2+a'1+a/2(Q), then instead of equation (7.2.1), we may consider the equation for u — tp. R e m a r k 7.2.3 Similar to the case of elliptic equations, in the following arguments, we may always assume that the solution is sufficiently smooth. 7.2.2
Interior
estimate
Similar to the case of elliptic equations, we need to establish Caccioppoli's inequalities, which are slightly different in form. T h e o r e m 7.2.1 Let u be a solution of equation (7.2.1) in QR, W = DiU (1 < i < n). Then for any 0 < p < R and A £ R, we have (u — X)2dx + //
sup / h
JB„
(j
IP JBf,
*
where / H = on n.
f2dxdt\,
(7.2.8)
(w - Xfdxdt + Jj (f - fR)2dxdt],
(7.2.9)
(u- Xfdxdt
(w-X)2dx+
sup/
\Du\2dxdt
JJQP
\Dw\2dxdt JjQp
jj //
\QR\
+ (R- p)2 ff
f(x, t)dxdt, C is a positive constant depending only
JJQR
Proof. We only show (7.2.9); the proof of (7.2.8) is similar. Let r](x) be a cut-off function defined on BR related to Bp, that is, 77 £ CQ°(BR), 0 < T](x) < 1, r) = 1 in Bp and \Dr)(x)\ <
n
. Let f £ C°°(R), 0 < £(t) < 1, R—p (j
£ = 0 for t < t0 - R2, £ = 1 for t > t0 - p2 and 0 < £'(£) < — r?. (H - py
Schauder's Estimates for Linear Parabolic Equations
201
Multiplying the equation for w ^
- A W = DJ(x,t)
= Di{f(x,t)
by r]2t,2(w - A) and integrating over Q3R = BRx have
-
fR)
(t0 - R2, s) (s € IR), we
#,<'« , <"-*>!?*•* V2^2(w - X)Awdxdt + ff
//
r)2Z2(w - X)Di(f -
fR)dxdt.
Integrating by parts yields
itfe{w-\)2)dxdt-
\ll
ff
2 JJQR at
T]2£2\Dw\2dxdt-2
= - ff
rftfiw-Xfdxdt
JJQR
r)Z2(w-\)Dr)-Dwdxdt
ff
JJQ'R
JJQ'R
- ff
r?eU - fR)DiWdxdt
JJ
Q%
r)Z2(w - X)(f - fR)DiT]dxdt,
-2 ff JJQ%,
that is, 7]2^2{w-X)2dx
\f
= -2 ff
r)2£2\Dw\2dxdt
+ ff
2 JBR
S
JJQR
2
ri£ (w-X)Dri-Dwdxdt-
JjQR
r)2£2(f -
ff
fR)Diwdxdt
JjQn
tf2(w
- 2 ff
- A)(/ - fR)Dir]dxdt
+ ff
JJQR
r]2^'(w -
X)2dxdt.
JJQR
Applying Cauchy's inequality with e to the first three terms on the right side of the above formula, we obtain V2Z2(w - A)2dx
\ f 2 JBR
<\ ff
S
2 2
2 JJQR
+ ff
2
r) £ \Dw\ dxdt + C ff
JJQH
+ C ff JJQ%,
2 V
T]2t2\Dw\2dxdt
JJQR
e(f-fR)2dxdt+
£,2{w-X)2\Dr)\2dxdt ff JJQ'v
r,2£\e\(w - Xfdxdt
202
Elliptic and Parabolic
4 I.'Ok ^M"** + ^
Equations
Jf (» - A)'**
cJJa(f-fR)2dxdt.
+ Hence
r,2e(w - \)2dx
f JBR
S
C
<:
2
+ ff
V
e\Dw\'- dxdt
JJQR
ff (w- Xfdxdt + C ff (/ - fRfdxdt.
(R-P)2JJQR
Therefore, for any s e Ip, we have f (w-X)2dx
r,2£2(w-X)2dx
< f
JB„
S
JBR
S
-JiSw II .{w ~ x)2dxdt + c II .(/ " fRfdxdt >Q'n
C
<,R _
2
)2
JJ (w-X) dxdt + C Jf
(f-fR)2dxdt
and //
\Dw\2dxdt < ff
rj2(,2\Dw\: dxdt
C ff <-: (R-P)2JJQR JJQR
(w - Xfdxdt + C II (/ - fR)2dxdt, JJQR
which implies (7.2.9).
•
Corollary 7.2.1 Let u be a solution of equation (7.2.1) in QR, w = Diu{l
J BR/2
~W HQR
w2dx + //
\Dw\2dxdt
JJQR/2
UHXdt + CRn+2
W^«
+ CRn+2+2a
where C is a constant depending only on n.
Vta/2;QR'
203
Schauder's Estimates for Linear Parabolic Equations 7?
Proof.
"i
In (7.2.9), choosing p and R as — and -R respectively and setting
A = 0, we obtain w2dx + / /
sup / IR/2
•> BR/2
<•£ / /
\Dw\2dxdt JJQR/2
2
(/ - hR/i)2dxdt,
w dxdt + C ff
'Q3H/4
JJQsR/i
3 and in (7.2.8), choosing p and R as -R and i? respectively and setting A = 0, we obtain \w\2dxdt < - ^ / /
//
R
JJQ3R/4
u 2 dxdt + CR2 ff
JJQR
fdxdt.
JJQR
Combining the above two inequalities, we see that w2dx + / /
sup / IR/2
JBR/2
R
< £ n
JJQR/2
u2dxdt + C [[
<•£ / /
\Dw\2dxdt
JJQR
fdxdt
+ C ff
JJQR
u2dxdt + CR«+2\f\2;QR +
ff
(f -
hn/ifdxdt
JJQ3R/4
CR^2^[f}2a,QR.
JJQR
•
Repeated use of the inequality (7.2.8) leads to Corollary 7.2.2 have sup/
If f = 0 in Q\, then for any nonnegative integer k, we
\Dku\2dx+
h/2 JB1/2
ff
\Dk+1u\2dxdt
JJQI/2
u2dxdt,
ff
(7.2.10)
JJQI
where C is a positive constant depending only on n and k. Corollary 7.2.3
If f = 0 in QR, then
izM-c(^ILu2dxdt)
'
where C is a positive constant depending only on n. n +1 Proof. We first assume that R = 1. Take a natural number k > —-—. By the equation Dtu = Au
204
Elliptic and Parabolic
Equations
and Corollary 7.2.2 we see that, for i, j = 0,1, • • • , k, there hold \DjD\u\2dxdt
ff
\Dj+2iu\2dxdt
< ff
JJQI/2
u2dxdt.
JjQl/2
JJQl
Thus ll«llff*(Q1/a) < C||u|| L a W l ) . Applying the embedding theorem, we obtain sup u2 < C\\U\\HH{QI/2)
< C||u|| L 2 W l ) .
Ql/2
For the general case R > 0, we may use the rescaling technique to obtain the desired conclusion. • Theorem 7.2.2 Let u be a solution of equation (7.2.1) in QR and f = 0 in QR. Then for any 0 < p < R, we have ff ff
u2dxdt
(u-Up)2dxdt
where UR = -r-pr—r / / \QR\
ff
u2dxdt,
(7.2.11)
(u-uR)2dxdt,
(7.2.12)
u(x, t)dxdt, C is a positive constant depending only
JJQR
on n. Proof. We first prove (7.2.11). By Corollary 7.2.3 we see that, if 0 < p < R/2, then u2dxdt <\QP\ supu 2 < Cpn+2 sup u2 < C (^Y
// JJQ,,
Qp
QR/2
K
R
/
ff
u2dxdt.
JJQR
So, if 0 < p < R/2, then (7.2.11) is valid, while if R/2 < p < R, (7.2.11) is obvious with C > 2n+2. Next we prove (7.2.12). By the ^-anisotropic Poincare's inequality (Theorem 1.4.3 of Chapter 1), we see that //
(u - Upfdxdt
ff
\Du\2dxdt + p 4 / /
\Dtu\2dxdt
Using equation (7.2.1) and noticing that / = 0 in QR, we see that ff
(u - up)2dxdt
\Du\2dxdt + p4 ff
\Au\2dxdt)
).
Schauder's
Estimates
for Linear Parabolic
ff
\
205
Equations
\Du\2dxdt + p4 ff \D2u\ 2dxdt
JJQf,
.
JJQf,
In the above inequality, applying (7.2.11) and Caccioppoli's inequality (7.2.8) to Du and applying (7.2.11), Caccioppoli's inequality (7.2.9) and (7.2.8) to D2u, we see that if 0 < p < R/2, then up)2dxdt JJQP
2
dxdt + p4 II
\D2u\2dxdt
QR/2
+2
J
2+
^(r [(i) (i)li<«->^ and (7.2.12) follows by the choice of A = uR. If R/2 < p < R, then (7.2.12) is obvious with C > 2"+ 4 . • Similar to the treatment for Poisson's equation, we may apply the interior estimate (7.2.12) for the homogeneous equation and the iteration lemma (Lemma 6.2.1 of Chapter 6) to obtain the interior estimate for solutions of the nonhomogeneous heat equation. T h e o r e m 7.2.3 Let u be a solution of equation (7.2.1) in QR0, W = DiU (1 < i < n). Then for any 0 < p < R < RQ, we have
T ^ i i \Dw-(Dw)p\2dxdt JJQP
<-:
C Rn+2+2a
\Dw - (Dw)R\2dxdt + C\f\2a>a/2iQR,
ff JJQR
where C is a positive constant depending only on n. Proof.
Decompose w as w = w\ + W2 with w\ and w2 satisfying
dt
- Awi = 0,
W\ dpQR
= w
in QR,
206
Elliptic and Parabolic
Equations
and -^1-Aw2 W2
= DJ = Dt(f - f R ) ,
in QR,
= 0. dpQn
Applying (7.2.12) to w\, we obtain //
\Dwx -
(Dwi)p\2dxdt
JJQP
(DWl)R\2dxdt.
\Dw! -
So, for any 0 < p < R < Ro, //
{Dw)p\2dxdt
\Dw -
JJQP
\Dwi - {Dw{)p\2dxdt
<2 / /
+ 2 if
JJQP
(Dw2)p\2dxdt
\Dw2 -
JJQ„
-C{^Y+
\Dw! - (DWl)R\2dxdt
ff \Dw-(Dw)n\2dxdt
\Dw2\2dxdt
+ C ff
+ c ff \Dw2\2dxdt.
Multiplying the equation for w2 by w2, integrating over QR, integrating by parts and noticing that w2 0, we see that 9VQR
\jjJ-{wl)dxdt
+
•• II
fR)dxdt
w2Di{f -
jjjDw.
dxdt
JJQR
= - JJQR II (/ - /*)AtiW dxdt 2
<\ 1
jj
\Dw2\2dxdt + \ ff l
JJQR
(f-fR)2dxdt,
JJQR
from which, noticing // JJQR
—{wl)dxdt= ot
I JBR
wl(x,t)
dx>0, t=t0 + R2
Schauder's
Estimates for Linear Parabolic Equations
207
it follows
\Dw2\2dxdt < ff
ff JJQR
(/ - fRfdxdt
<
CRn+2+2a[f}la/2]QR
JJQR
and hence //
(Dw)p\2dxdt
\Dw -
JJQP
\Dw - (Dw)R\2dxdt + CRn+2+2a[f}la/2.
Using the iteration lemma (Lemma 6.2.1 of Chapter 6), we finally obtain //
{Dw)p\2dxdt
\Dw-
JJQP
-C(R)n+2+2a(JI Theorem 7.2.4
\Dw-(Dw)R\2dxdt +
R^2+2a[f]la/2.iQR).
Let u be a solution of equation (7.2.1) in QR, W =
DiU (1 < i < n). Then for any 0 < p < —-, we have ff
\Dw - (Dw)p\2dxdt
<
Cpn+2+2aMR,
JJQP
where C is a positive constant depending only on n, and MR
Proof. //
=
#4+2a MO;QH + ^2^l/lo ; Q R + [f]a,a/2;QR-
According to Theorem 7.2.3 and Corollary 7.2.1, we see that (Dw)p\2dxdt
\Dw -
JJQP
( ^ ^
\Dw - (Dw)R/2\2dxdt +
/
[f)l,a/2;QR/2)
\Dw\2dxdt + [f]l>a/2.iQR/2) JJQR/2
JJQR
u2dxdt + ^\f\20]QR + Wl,a/*QR).
from which we get the conclusion of the theorem.
•
208
Elliptic and Parabolic
Theorem 7.2.5
Equations
Let u be a solution of equation (7.2.1) in QR.
Then
[D2u}a,a,2;QR/2 ^C
( # T ^ M o ; Q « + Jfi\f\o;Qn + I/]<W2;Q*) ,
(7-2.13)
where C is a positive constant depending only on n. Proof.
By Theorem 7.2.4, for (x,t) e QR/2, 0 < p < —, we have \D2u{v,«) - (D2u)Qp{Xtt)nQ
//
\2dyds
JJQp(x,t)f]QR/2
\D2u(y, s) -
< ff
(D2u)XttJ2dyds
JJQp(x,t) 1
+
jR2Sn/'0iQ R / 2(i,t) + l / J a , a / 2 ; Q f i / 2 ( x , t ) J +
" ( ^ 4 + 2 Q I U I O ; Q B + ^ ^ l / l o ; Q H + [f]a,a/2;QRJ >
where
Thus 1/2 [- 0
U
] 2 , n + 2 + 2 a ; Q R / 2 ^ ^ ( p 4 + 2 c « l U lo;Q R + £ 2 ^ l / l o ; Q H +
- ^ ( ^2+^l"l°^«
+
^I^I°;QR
[/]a,a/2;QR
+ [f\a,a/2;QR ) •
Prom Remark 7.1.2, we obtain (7.2.13) and the proof is complete. 7.2.3
Near bottom
•
estimate
Since all the spatial derivatives on the bottom are tangential derivatives, the establishment of near bottom estimate is quite similar to the interior estimate. Firstly, we establish Caccioppoli's inequalities.
Schauder's Estimates for Linear Parabolic Equations
209
Theorem 7.2.6 Let u be a solution of equation (7.2.1) in QR satisfying (7.2.3), w = Dtu (1 < i < n). Then for any 0 < p < R, we have u2dx + //
sup / i°
JBP
\Du\2dxdt
JJQO
(7.2.14) w2dx + //
sup / i°P JB„ JBD 1
,p
s2 / / {R - pY JJQOR
where fn =
0
\QR\
\Dw\2dxdt
JJQ°
//
w2dxdt+ [[ (f - fR)2dxdt\, JJQ0R
(7.2.15) J
f(x, t)dxdt, and C is a positive constant depending
JJQI
only on n. Proof. We only present the proof of (7.2.15), which is similar to that of Theorem 7.2.1, the only difference is that here we multiply the equation for w dw - Aw = Dif(x,t) ~dt
= Di(f(x,t)
- fR),
by r)2w and then integrate over Q^8 = BR X (0, s) (s € IR) to obtain // ri2w-pr- dxdt = II ri2wAwdxdt+ ot JJQo,s iQr >QT
JJQ°-°
//
JJQO,S
tfwDAf
— fn)dxdt.
Since rj e CQ°(BR), the boundary integral vanishes when integrating by parts with respect to the spatial variables. On the other hand, by w = 0, t=o we have
UQR, f^dxdt
=\ j ^ rj2(x)w2(x,t)
4/./w
t=s
dx
t=o
w2(x, s)dx. D
Remark 7.2.4 We use the notation f& to denote the average of f over QR when treating the near bottom estimate, while use fR to denote the average of f over QR when treating the interior estimate. In what follows, we will use the same notation fR to denote the average of f over Q^ and
210
Elliptic and Parabolic
Equations
QR when treating the near lateral estimate and the near lateral-bottom estimate. No confusion will be caused from the context. Combining the inequality (7.2.14) with (7.2.15) we have Corollary 7.2.4 Letu be a solution of equation (7.2.1) in Q°R satisfying (7.2.3), w = Diu(l
w2dx + //
i°R/2 JBR/2
\Dw\2dxdt
JJQ°R/2
~W II o uHxdt
+ CSr 2
* W*«k
+ CRn+2+2a
Wl«,2;Q°R>
where C is a positive constant depending only on n. Repeated use of the inequality (7.2.14) leads to Corollary 7.2.5 have sup/ n/2
If f = 0 in Q\, then for any nonnegative integer k, we
\Dku\2dx+
JBU2
ff
\Dk+1u\2dxdt
< C ff
JJQ°1/2
u2dxdt,
(7.2.16)
JJQ\
where C is a positive constant depending only on n and k. By using (7.2.16), similar to the proof of Corollary 7.2.3, it follows that Corollary 7.2.6
If f = 0 in Q°R, then ( \1/2 2 sup H < C I -^-To / / u dxdt 1 Q%, \ R + JJQ°n J
where C is a positive constant depending only on n. Applying Corollary 7.2.6, similar to the proof of (7.2.11), we obtain the following Theorem 7.2.7 Let u be a solution of equation (7.2.1) in QR satisfying (7.2.3) and f = 0 in Q°R. Then for any 0 < p < R, we have ff JJQO
u2dxdt
ff
u2dxdt,
JJQOR
where C is a positive constant depending only on n.
(7.2.17)
Schauder 's Estimates for Linear Parabolic Equations
211
Furthermore, we have the following Theorem 7.2.8 Let u be a solution of equation (7.2.1) in Q°R satisfying (7.2.3) and f = 0 in QR. Then for any 0 < p < R, we have II
u2dxdt
I]
u2dxdt,
(7.2.18)
where C is a positive constant depending only on n. Proof. Obviously, Au satisfies the same equation and initial value condition as u, which implies that (7.2.17) is valid for Aw too, that is \Au\2dxdt
// JJQO
Noticing u
t=o
\Au\2dxdt,
ff
\RS
0 < p < R.
JJQOR
= 0, we see that if 0 < p < R/2, then //
u2dxdt
JJQ°
\Dtu\2dxdt
JJQ , i
\Au\2dxdt
=Cp ff JJQ°P
Using (7.2.15) and (7.2.14), we obtain \D2u\2dxdt <—r
// JJ
Q%,
R
ff
\Du\2dxdt < —
JJ
R
Q%R/i
f
u2dxdt.
JJ
Q°R
Therefore,
ff
u2dxdt
ff
u2dxdt
ff
u2dxdt,
that is, for 0 < p < R/2, (7.2.18) is valid. If R/2 < p < R, then (7.2.18) holds obviously with C > 2"+ 4 . • Similar to the treatment for the interior estimate, using the near bottom estimate (7.2.18) for the homogeneous equation and the iteration lemma (Lemma 6.2.1), we may obtain the near bottom estimate for the nonhomogeneous heat equation.
212
Elliptic and Parabolic
Equations
Theorem 7.2.9 Let u be a solution of equation (7.2.1) in QR satisfying (7.2.3), w = DiU (1 < i < n). Then for any 0 < p < R < RQ, we have
- ^
Ij^ \Dw\2dxdt < ^
Jl^ \DW\2dxdt + C[/]2Q,a/2;QoR,
^
where C is a positive constant depending only on n. Proof.
Decompose w into w = w\ + W2 with w\ and w2 satisfying - B - - Awi = 0, Wl
Wh
in Q°R,
w
and f
^
- Aw2 = DJ = A ( / - / « ) ,
™Q%
0.
W2 9PQ°R
Using (7.2.18) for Dw\, we obtain
/ / \DWl\2dxdt
JJQO
\R/
ff \DWl\2dxdt.
JJQ
So, for any 0 < p < R < Ro, we have // <2 / /
\Dw\2dxdt \Dw1\2dxdt + 2 / /
JJQ°
\Dw2\2dxdt
JJQ°P
jl \Dw\2dxdt + C ff JJQR JJQl
\Dw2\2dxdt.
Similar to the proof of Theorem 7.2.3, multiplying the equation for w2 by w2, and integrating over QR, we see that
\Dw2\2dxdt < ff
ff
(/ - fR)2dxdt
<
CRn+2+2a[f}2aiOc/2.iQ0R.
>QH
Therefore II
o
\Dw\2dxdt < C ( £ ) " + 4 IIQ
\Dw\2dxdt +
CRn+2+2a[f}2ata/2.iQoR.
Schauder's Estimates for Linear Parabolic Equations
213
Finally, using the iteration lemma (Lemma 6.2.1) we immediately get the conclusion of the theorem. D Theorem 7.2.10
Let u be a solution of equation (7.2.1) in Q°R satisfying
(7.2.3), w = DiU (1 < i < n). Then for any 0 < p < —, we have
\Dw - (Dw)p\2dxdt < Cpn+2+2aMR,
II JJQ°
where C is a positive constant depending only on n, and M
R =
Proof.
flS+^Ho;^
+ ^ l / l o ; Q ^ + U\l,a/2;QR-
By virtue of Theorem 7.2.9 and Corollary 7.2.4, we see that //
\Dw\2dxdt
Hs p
^pn+2+2a ( ^ W L u 2 d x d t + ^ ^ + [ / 1 - / ^ 0
{D
< - w? [L |D «"H £ m IL D|""2
and hence //
\Dw -
{Dw)p\2dxdt
JJQ°P
<2 II \Dw\2dxdt + 2 II JJQ° JJQ°P
\(Dw)p\2dxdt
Let u be a solution of equation (7.2.1) in QR satisfying
< C ^-^2T^Mo;Q°„ + -jp\f\o;Q°R + lf}a,a/2;QR) >
214
Elliptic and Parabolic
Equations
where C is a positive constant depending only on n. Proof. Similar to the case of elliptic equations (Theorem 6.2.11), we need only to use Theorem 7.2.10 and the corresponding result about the interior estimate (Theorem 7.2.4). • 7.2.4
Near lateral
estimate
Similar to the case of Poisson's equation, in establishing the near lateral estimate, we can estimate the tangential derivatives directly, while for the normal derivatives, we need to apply the equation and the results for tangential derivatives. Since in equation (7.2.1) there is a term of time derivative Dtu, we need to estimate it too. First, we establish the following Caccioppoli's inequalities. Theorem 7.2.12 Letu be a solution of equation (7.2.1) in Q\ satisfying (7.2.5), w = DiU (1 < i < n). Then for any 0 < p < R and any A e E", we have sup / u2dx + // \Du\2dxdt iP JB+ JJQt
l
u2dxdt + (R- p)2 [[
„ //
L(R-P)2JJQ+
w2dx + //
sup / IP JB+
1
\Dw\2dxdt
,2 II w2dxdt + [I {f- fR)dxdt
ID JB+
(7.2.19)
JJQp-
(7.2.20)
|£>,~)tu\ dxdt
\Du - X\2dx + / /
sup up /
fdxdt
JJQ+
JJQt
^—-r II \Du-\\2dxdt+ II fdxdt
(R - P) JJQt
where fR - -—q— //
f{x,t)dxdt
(7.2.21)
JjQ„ and C is a positive constant depending
JJQ+R
\QR\
only on n.
Proof. Since the proofs of (7.2.19) and (7.2.20) are similar, we merely show (7.2.20), which is similar to that of Theorem 7.2.1, the only difference is that here we multiply the equation for w ^
- Aw = DJ(x,t)
= Di{f(x,t)
- fR),
Schauder's
Estimates for Linear Parabolic
Equations
215
by r72£2w and integrate over Q^'s = B^ x (t 0 - R2, s) (s € IR). Then dw rfe^dxdt ~dt
[[ J 'QlJot-3
r)2Z2wAwdxdt+
if
r)2Z2wDi(f -
if
fR)dxdt.
JJQ+R'
JJQ+-*
Since r\ e C^{BR)
= 0, the boundary term resulting from
and w xn=0
integrating by parts with respect to the spatial variables is equal to zero. Now we show (7.2.21). Rewrite equation (7.2.1) as ^-div(£>u-A) = /(M). Let 77 and £ be the cut-off functions in the proof of Theorem 7.2.1. Multiply both side of the above equation by r)2£2Dtu and integrate over Q~^'s = BR x (£0 — R2,s) (s S IR). Then integrating by parts and noticing that = 0, we obtain
Dtu xn=0
r]2i2\Dtu\2dxdt
II JJQ+"
:- //
(Du - A) • D{r)2Z2Dtu)dxdt
rj2Z2Dtufdxdt
+ If JJ
JJQV
S
QR
2 2
-
r] (, (Du-X)-DDtudxdt JjQf - 2 // a
r)£,2Dtu(Du - A) • Drjdxdt + / /
JjQ+:
a
JJQ+
d_(r)2Z2\Du - X\2)dxdt + ~ 2 Hot dt QR
II
iftfl Du -
r)Z2Dtu{Du - A) • Drjdxdt + / /
2 II
r)2£2Dtufdxdt
JJQR-»
r)2£2Dtufdxdt.
JJQ+R->
Utilizing Cauchy's inequality with e, we further have // r?Z2\Dtu\2dxdt+\ 3 JjQi
*iL*
2
J 77 2 £ 2 |£>u-A| : JB+ dx
?\Du-\\* t=t0-R2
\\2dxdt
dx
216
Elliptic and Parabolic
Equations
+ 11 s V2tt'\Du-\\2dxdt+l JjQt-
[[ 2 JJQ+,s
r)2Z2\Dtu\2dxdt
+ 2 [[
r]2^2f2dxdt
i2\Dr1\2\Du-X\2dxdt
+ 4 // JJQR>
JJQ+°
2 2
<\ II
2
r, Z \Dtu\ dxdt
2 JJQ+--
+ TZ
\Du - M2dxdt + 2 / /
^r / / 2
(R - p) JJQ+-°
fdxdt,
JJQ+.>
from which (7.2.21) follows at once.
•
Combining the inequalities (7.2.19), (7.2.20), (7.2.21) with equation (7.2.1) we obtain Corollary 7.2.7 Let u be a solution of equation (7.2.1) in Q^ satisfying (7.2.5), w = Diu(l
sup/ IRIIJB+,„ VJBj/2
<—
II
(\Dtu\2 + \Dw\2)dxdt JJ ^QR/2
QR/2
2 u^^^+A.niin+2\t\2
+2+2 _1_/^.pn+2+2ar /l2
dxdt + CR^ \f\lQl
+ ^
«[/£,a/2;Q+,
where C is a positive constant depending only on n Corollary 7.2.8 If f = 0 in Q~^, any 0 < p < R, we have
\Dku\2dx+
sup/ h
then for any nonnegative integer k and
JBi
^TB^W
II
\Dk+1u\2dxdt
JJQ+
llQt}Dku?dxdt'
(7 2 22)
-'
where C is a positive constant depending only on n. Proof. We present the proof by considering the following four cases. i) The case k = 0. In this case, the conclusion can be obtained by using Caccioppoli's inequality (7.2.19) directly. ii) The case k = 1. For 1 < j < n, DjU still satisfies the homogeneous equation and the zero lateral boundary value condition. It follows from the conclusion for the case k = 0 that / / \DDjU\2dxdt JjQt
<— ^ // \ R - P) JJQR
{Djufdxdt
Schauder's
Estimates for Linear Parabolic
217
Equations
^wh? L ^ d x i t
(T2 23)
'-
Caccioppoli's inequality (7.2.21) then implies \Du\2dx+
sup /
//
iP JB+
\Dtu\2dxdt
JJQ+
tfQ+JDu?dxdt
-whr
<7-2-24)
n-l
Using the equation Dnnu = Dtu — \~] DjjU and the inequalities (7.2.23) and (7.2.24), we obtain \Dnnu\2dxdt
//
JjQt
< —
\Du\2dxdt,
rx / /
R
( ~ P) JJQ+
which together with (7.2.23) and (7.2.24) implies (7.2.22) for the case of k = l. iii) The case k = 2. For 1 < j < n, DjU still satisfies the homogeneous equation and zero lateral boundary value condition. It follows from the conclusion for the case k = 1 that \DDju\2dx+
sup / iP JB+
<,n^
C
//
\D2Dju\2dxdt
JJQ+
„ //
\DDju\2dxdt
2
(R-P) JJQ+
c_
< , „ ^ xo / / •{R-pyjjQ+R
\D2u\2dxdt.
Since Dtu still satisfies the homogeneous equation and zero lateral boundary value condition, it follows from the conclusion for the case k = 0 and the equation Dtu = Au that sup / Ip JB+
|D t u\ 2 dx + / / JJQ +
C_ <,„~ 2 ,„ / / \Dtu\2dxdt (R-P) JJQiJJQ+R C_ / / \Au\2dxdt \R-P)2JJQ+R JJQt
\DDtu\2dxdt
218
Elliptic and Parabolic
-{R-PYJIQI
Equations
\D2u\2dxdt.
Combining the above two inequalities and using the equation Dnnu n—1
=
n-1
Dtu - 2 J DJJU and the equality Dnnnu
= DnDtu
— J ^ Djjnu,
j=i
we get
j=i
the conclusion (7.2.22) for the case k = 2. iv) The case A; > 2 can be deduced accordingly.
•
Prom Corollary 7.2.8, we obtain Corollary 7.2.9 i, we have
If f = 0 in Q~l, then for any nonnegative integer k and
sup / \Dk+iu\2dx A/a JB+/2
+ ff \Dk+i+1u\2dxdt JJQt/2
{D^dxdt,
where C is a positive constant depending only on n and k. Using Corollary 7.2.9 and the embedding theorem, similar to the proof of Corollary 7.2.3, we obtain Corollary 7.2.10 we have
If f = 0 in Q^, then for any nonnegative integer i, 1/2 2
sup |D'u| < C ( - ^ - 2 / /
|Z3*«| dardt
where C is a positive constant depending only on n. By virtue of Corollary 7.2.10, similar to the proof of Theorem 7.2.2, we obtain Theorem 7.2.13 Let u be a solution of equation (7.2.1) in Q^ satisfying (7.2.5) and f = 0 in Q\. Then for any nonnegative integer i and any 0 < p < R, we have ff
ID^dxdt
JJQt
< C (^)n+2 y R /
[
ID^dxdt,
J +
QR
where C is a positive constant depending only on n. Furthermore, we have
(7.2.25)
Schauder's
Estimates for Linear Parabolic
Equations
219
Theorem 7.2.14 Letu be a solution of equation (7.2.1) in Q^ satisfying (7.2.5) and f = 0 in Q\. Then for any 0 < p < R, we have ff
u2dxdt
ff
u2dxdt,
(7.2.26)
where C is a positive constant depending only on n. Proof.
Noticing that u
= 0 and using (7.2.25), we see that if 0 < x =0
p < R/2, then u2dxdt
//
\Dnu\2dxdt
\Du\2dxdt
ff
\Du\2dxdt.
In addition, using Caccioppoli's inequality (7.2.19), we obtain // \Du\2dxdt < TPT / / u2dxdt. R JJ JjQln Qt Therefore II
u2dxdt
f
JJQp
u2dxdt,
0 < p < R/2,
JQR
that is, (7.2.25) is valid for 0 < p < R/2. For R/2 < p < R, (7.2.25) holds obviously with C > 2n+A. • Applying the near lateral boundary estimate (7.2.26) for the homogeneous heat equation and the iteration lemma (Lemma 6.2.1), we may obtain the near lateral boundary estimate for the nonhomogeneous heat equation. Theorem 7.2.15 (7.2.5), w = Diu(l
Let u be a solution of equation (7.2.1) in Q^ o satisfying
Fp(x,t) = \Dtu\ + Y^
\DJW\2
+ \Dnw -
Then for any 0 < p < R < Ro, we have pn+2+2a
JJ
Fp(x,t)dxdt
(Dnw)p\2.
220
Elliptic and Parabolic Equations
If F (x,t)dxdt <- c Rn+2+2oc JJQj R
+ C[f}la/2.Q+R,
where C is a positive constant depending only on n. Proof. We divide the proof into four steps. Step 1 Estimate Dtu. Decompose u into u = u\ + u2 with u\ and u2 satisfying -^r - Aui = fR, dt
in <2£,
dpQt
and - ^ - Au2 = f - fR, u2
in
Q-R-,
. =0. 9PQ+R
It is obvious that Dtui satisfies the homogeneous equation and zero lateral boundary value condition. So, (7.2.26) is valid for Dtui, that is
ff
\DtUl\2dxdt < C ( 4 ) " + / /
Wtu^dxdt.
Hence, for any 0 < p < R < RQ, we have //
\Dtu\2dxdt<2
\DtUi\2dxdt + 2
\Dtu2\2dxdt
ff \Dtu1\2dxdt JjQt ff
\Dtu2\2dxdt
+ 2 ff JJ
QR
\Dtu\2dxdt + C ff
\Dtu2\2dxdt.
Multiplying the equation for u2 by Dtu2 and integrating over Q^, we obtain // JJQ+
\Dtu2\2dxdt-
ff JJQ+
Au2Dtu2dxdt=
ff
(/ -
fR)Dtu2dxdt.
JJQ+
Integrating by parts for the left hand side of the above equality with respect to spatial variables, and using Cauchy's inequality to the right hand side,
Schauder's Estimates for Linear Parabolic
Equations
we see that dxdt <\ ft 1
Since u2
9
pQt
\Dtu2\2dxdt+±
JJQt
l
(f-fR)2dxdt.
[[ JJQt
= 0, it follows that ^\Du2\2dxdt
[[
= I
JJQ+ ot
JB+
\Du2?
dx > 0 . t=t0+R2
Therefore ff
+
\Dtu2\2dxdt < ff
(/ - fR)2dxdt
CRn+2+2a[f}2aa/
<
So, for any 0 < p < R < Ro, we have / / \Dtu\2dxdt JJQt
-c (l)" +4 IIQt
WtU?dxdt+CRn+2+2a
Ka/*Qf
Step 2 Estimate DJW(1 < j
.
„
— - Aw! = 0,
.
„+
m Q+,
W\
dvQt and ^--Aw2 w2
Wt
= DJ = A ( / - /«),
in Q+,
0.
When j = 1,2, • • • , n - 1, from (7.2.26), we have //
JJQt
| D j U ; i | 2 d a ; d t < cKKJ f4)n+4 /
JQ+ 'Qt
\DjWl\2dxdt.
221
222
Elliptic and Parabolic Equations
So, for any 0 < p < R < RQ, //
\Djw\2dxdt<2
JjQi
\Djw1\2dxdt + 2 JJQi
\Djw2\2dxdt JJQ+
-C(RT
II
\DJwi\2(ixdt
+ 2 (I
\DjW2\2dxdt
-C'(fl)n+
//
\DM2dxdt
+ C II
\DjW2\2dxdt.
Similar to the proof of Theorem 7.2.3, we multiply the equation for w2 by u>2 and integrate over Q^ to obtain //
\Dw2\2dxdt <
JJQ+
{f-fR)2dxdt
ft JJQ$
(7.2.27)
Then, for j = 1,2, • • • , n — 1 and any 0 < p < R < RQ, we have //
\Djw\2dxdt
^C ( l ) n + 4 UQ+R \DM2dxdt
+
CR^2^[f]2aa/2iQ+R
Step 3 Estimate Dnw. From the ^-anisotropic Poincare's inequality (Theorem 1.3.4), we have / / \Dnwi JJQt
I
(Dnwi)p\2dxdt
\DDnwi\2dxdt + p4 / /
\DtDnwi\2dxdt\
By virtue of the equality DtDnw\ = ADnw\, we see that // \Dnwi JjQt
(Dnwi)p\2dxdt
\DDnwi\2dxdt + p* / /
\D2Wl\2dxdt + p4 II
|ADnWi\2dxdt) \D3Wl\2dxdt\
.
.
Schauder's
Estimates for Linear Parabolic Equations
223
In addition from (7.2.22), we have < -z \\ \D2wi\2dxdt. 2 P JJQt
/ / ID^^dxdt JjQt So, for any 0 < p < R0/2, //
\Dnwi -
(Dnwi)p\2dxdt
JJQt
ff
\D2wi\2dxdt + p2 II
\D2wi\2dxdt)
\D2Wl\2dxdt.
Prom this, it follows by using (7.2.25) for wi, we derive for 0 < p < 1p < R < ilo, // \Dnw\ JJQt
(Dnwi)p\2dxdt
\D2Wl\2dxdt
[[\D2Wl\2dxdt.
So, for 0 < p < 2p < R/2 < R0/2, we have // \Dnw - (Dnw)p\2dxdt JJQt \Dnwi - {Dnwi)p\2dxdt + 2
<2 If JJQt 2
JJQt +2
\Dnw2 - {Dnw2)p\2dxdt
2
2
ff \D wA dxdt + C ff \Dw2\2dxdt. JJ JJQt,* Qt/2
By virtue of the equation Dtw\ = Au>i, it follows that // Qt„
JJ
\D2Wl\2dxdt
< ff \Dnnwi\2dxdt + 2y" ff \DDjWi\2dxdt JJ JJQt,* j^i Qt,2
(7.2.28)
224
Elliptic and Parabolic
IDtW^dxdt + cY]
JJ
Q+'R/2 R/2
Equations
if
IDDjW^dxdt.
(7.2.29)
JJ
J7^1 = l
Qh2 ^R/2
Using Caccioppoli's inequality (7.2.21) for w\ with X = (0,---
,0,{Dnw)R),
we obtain
If
lA^i \2dxdt
JJ
Qi/2
/n-1
- ^
// , I Y
\DJWl\2 + IA»U>1 - (DnW)R\2 | dxdt
^jp Jf +\Y' n - 1 \Diw? + \Dn™ - i.Dnw)R\z I dxdt + JJ2 II +[Y' n - 1 \D0W1? + \DnW2? ) dxdt
=IP IJ A
2
H
\Dw2\ dxdt,
dxdt
(7.2.30)
R -tR and using Caccioppoli's --inequality (7.2.20) for DjWi (1 < j < n), we obtain m o m m l i i - i r ^7 O 9fl\
Ik
fi-.r
7,.,
n-1 j= l
^
Y
\DDjWi\2dxdt
J
•' QR/2
n-1
II , |£j«>il2<*«ft
2 n~1 rr -"R2 Y
//
2
+(\Djw\
+
\DjW2\2)dxdt
<4oY If \Djw\2dxdt+-E2 " ~Z[JJQR
H
If \Dw2\2dxdt. JJQt
(7.2.31)
Schauder's Estimates for Linear Parabolic Equations
225
Combining (7.2.29), (7.2.30) and (7.2.31), we have \D2wi\2dxdt
// JJ +
Q
R/2
^§2
If
+
+
[Yl \Diwf
+ \D"W ~ (Dnw)R\2 J dxdt
w llQJDw^2dxdt>
which together with (7.2.27), (7.2.28) implies that for 0 < p < 2p < R/2 < Ro/2, there holds / / \Dnw - (Dnw)p\2dxdt JjQt
I 5Z \Diw\2 + \D"W ~ (Dnw)R\2 J dxdt
+, L/Kf-ipn+2+2alf]2
VL,a/2;Q+-
The above inequality is obviously valid for R/A < p < R < Ro (with C > 4 n + 4 ). Step 4 Estimate Dtu and Dw. Combining the estimates obtained from the above three steps, we see that for any 0 < p < R < Ro, // Fp(x,t)dxdt JJQt
~C ( ^ r 4 HQt Ffi(X't)dXdt
+
CRn+ +2a
* Vt«/2;Qf
Finally, we use the iteration lemma (Lemma 6.2.1 of Chapter 6) and immediately obtain the conclusion of the theorem. • Theorem 7.2.16
Letu be a solution of equation (7.2.1) in Q^ satisfying
(7.2.5), w = DiU (1
\Dw - (Dw)p\2dxdt < Cpn+2+2aMR,
(7.2.32)
226
Elliptic and Parabolic
Equations
where C is a positive constant depending only on n, and 1
• ,2
.
1
Proof. Let i = 1,2,••• ,n — 1. According to Theorem 7.2.15 and Corollary 7.2.7, we have jj
+
| \Dtu\2 + Y^
\DJW\2
+ \DnW - (Dnw)p\2
+ |D„io - (Dnw)R/2f)dx4t +
iCeM2°
J dxdt
lf?v/2lQiJ
JIQi » 2 " ' + j ^ l / H a t + W„/™s)
(s^s
(7.2.33)
In particular, the above inequality implies that for j = 1,2, • • • , n — 1,
[[ JjQt
\DjW\2dxdt < Cpn+2+2aMR.
Thus
{D
><=m\lUDMdxd)
-\QJ Qt\ JjQt
\Djw\'dxdt
<
Cp'aMR.
So, for j = 1,2, • • • , n — 1, we have II <2 / /
\DjW —
(Djw)p\2dxdt
\Djw\2dxdt + 2 / /
\{DjW)p\2dxdt
227
Schauder's Estimates for Linear Parabolic Equations
In addition, (7.2.33) implies / / \Dnw - (Dnw)p\2dxdt JjQt
<
Cpn+2+2aMR.
Therefore, / / \Dw - (Dw)p\2dxdt JjQt
<
Cpn+2+2aMR.
To sum up, we have proved (7.2.32) for w = Diu(i Similarly, using (7.2.33), we have / / \Dtu JjQt
= 1,2, • • • ,n — 1).
(Dtu)p\2dxdt
<2 / / \Dtu\2dxdt + 2 I JjQt JJQt n+2+2a
\(Dtu)p\2dxdt
71-1
Therefore, using the equation Dnnu
= Dtu — \] D^u — f again, we see i=i
that (7.2.32) holds for w = Dnu. Theorem 7.2.17 (7.2.5). Then
•
Let u be a solution of equation (7.2.1) in Q\
satisfying
[£> 2
Near lateral-bottom
estimate
Since on the bottom, all the spatial derivatives are tangential derivatives, we may establish the estimate near the lateral-bottom just as we did in establishing the near lateral estimate. Here, we only list the conclusions of such kind of estimate, whose proof are similar to the corresponding parts in the proof of the near lateral estimate.
228
Elliptic and Parabolic
Equations
Theorem 7.2.18 Letu be a solution of equation (7.2.1) inQR+ satisfying (7.2.3), (7.2.5), w = Diu(l
sup / i°
JB+
\Du\2dxdt
// JJQ°+
'.mh? IL°+u2dxdt+{R~p? IL>Q7- fdxdt w2dx + / /
sup /
\Dw\2dxdt
dxdt \Du\2dx+
sup / i°
JB+
,(Rp
\Dtu\2dxdt
JJQ°+
l
, //
- p)22 JJQo+
where / # = -—g^— / / \QR.\
//
\Du\2dxdt+ JJ[fO+ f2dxdt], 1 Q
f(x, t)dxdt and C is a positive constant depending
JJQI+
only on n. Corollary 7.2.11 Let u be a solution of equation (7.2.1) in Q°R satisfying (7.2.3), (7.2.5), w = Diu(l
^
w2dx+
fLu2dxdt
JJQR
(\Dtu\2 + \Dw\2)dxdt
//
+ CRn+2
KQ°R+
+CRn+2+2aKa/2-,Q^
where C is a positive constant depending only on n. Corollary 7.2.12 If f = 0 in Q°R , then for any nonnegative integer k and any 0 < p < R, we have sup / \Dku\2dx + (I \Dk+1u\2dxdt 0 79 JBt JJQ +
< lJ3
. , II (R-prJjQ0+
\Dku\2dxdt,
where C is a positive constant depending only on n. Proof. The proof is similar to that of Corollary 7.2.8. Here, we need to use the following fact: if u is appropriately smooth in QR and satisfies the homogeneous heat equation and the conditions (7.2.3), (7.2.5), then Dtu satisfies the same equation and boundary value condition. In fact, it
Schauder's Estimates for Linear Parabolic
229
Equations
is obvious that Dtu satisfies the equation and the lateral boundary value condition. In addition, using the smoothness of u and the equation, we have Dtu(x,0)
= lim Dtu(x,t) t->0+
= lim Au(x,t)
= Au(x, 0) = 0,
t->0+
that is Dtu satisfies the bottom boundary value condition. Corollary 7.2.13 and i, we have
If f = 0 in Q? , then for any nonnegative integers k
\Dk+iu\2dx
sup /
•
+
\Dk+i+1u\2dxdt
+ [[
< C [[
[D^dxdt,
where C is a positive constant depending only on n and k. Corollary 7.2.14 we have
If f = 0 in Q°R , then for any nonnegative integer i, I
\ 1/2
sup | ^ |
,
where C is a positive constant depending only on n. Theorem 7.2.19 Letu be a solution of equation (7.2.1) inQ°^~ satisfying (7.2.3), (7.2.5), f = 0 in Q°R~• Then for any nonnegative integer i and any 0 < p < R, we have [[ {D'uPdxdt < C (£-Y+ JJQ°+ \R/
f JQ°+
iD^dxdt,
where C is a positive constant depending only on n. Theorem 7.2.20 Let u be a solution of equation (7.2.1) in Q^ + satisfying (7.2.3), (7.2.5), f = 0 in Q0^ • Then for any nonnegative integer i and any 0 < p < R, we have If
{D^dxdt
< C ("|)"+
ft
^ufdxdt,
(7.2.34)
where C is a positive constant depending only on n. Proof.
Noticing that D%u
= 0 and using equation (7.2.1) and Theo-
rem 7.2.19, we see that if 0 < p < R/2, then ff
JJQ°P+
ID^dxdt
JJQ°+
WtD^dxdt
230
Elliptic and Parabolic Equations
=CpA (I
lADVdaaft
JJQV-
+
\Di+2u\2dxdt
JJQI
\Di+2u\2dxdt.
JJQV,*
In addition, using Corollary 7.2.12, we obtain //
\Di+2u\2dxdt <%r II
\Di+1u\2dxdt < £
2
JJQV/2
II R4
R JJQI+/4
l&ufdxdt. +
JJQ°R
Therefore II {D^dxdt^cl^-Y^yRj J Jot
I
ID^dxdt
JQ+
I
{D^dxdt,
0
JQ+R
that is, (7.2.34) is valid for 0 < p < R/2. For R/2 < p < R, (7.2.34) holds obviously with C > 2 n + 4 . • Theorem 7.2.21 Letu be a solution of equation (7.2.1) inCf^ satisfying (7.2.3), (7.2.5), w = Dtu (1 < i < n). Then for any 0 < p < R < R0, we have
^nJ^
/L^i 2 +M)***+^[/]^ /2iQ o R+ ,
JJQR
where C is a positive constant depending only on n. Proof. Similar to the proof of Theorem 7.2.15, we decompose u and w in the same manner. The estimates on DfU and DjW (1 < j < n) are quite similar to the corresponding estimates in Theorem 7.2.15. To estimate Dtu\, we need the fact used in the proof of Corollary 7.2.12. The estimate on Dnw is even easier than the corresponding estimate in the proof of Theorem 7.2.15. In fact, according to (7.2.34), we may estimate Dnu in the same way as what we do on DjW (1 < j < n). •
Schauder 's Estimates for Linear Parabolic
231
Equations
Let u be a solution of equation (7.2.1) in Q0^ satisfying
Theorem 7.2.22
(7.2.3), (7.2.5), w = Dtu (1 < i < n). Then for any 0 < p<—, we have \Dw - (Dw)p\2dxdt
[[
<
Cpn+2+2aMR,
+
JJQ°P
where C is a positive constant depending only on n, and MR
= ^4+2^lUlo;Q°R+ + ;R2^lo;Q 0 R + +
^l,a/2;Q°R+-
Theorem 7.2.23 Let u be a solution of equation (7.2.1) in Q°R+ satisfying (7.2.3), (7.2.5). Then lD2u]a,a/2;Q°+/2
<
C
( ] ^ M o ; Q R + + Jp\f\o;Q°+ + [/]a,a/2;Q°+J ,
where C is a positive constant depending only on n. 7.2.6
Schauder's equations
estimates
for general
linear
parabolic
Now we turn to the general linear parabolic equation and consider the corresponding first boundary value problem dii
— - aij(x,t)Diju
+ bi(x,t)DiU + c(x,t)u =f(x,t),
(x,t) € QT, (7.2.35)
u
=tfi(x,t),
(7.2.36)
dpQr
where QT = O x (0, T), Q c M™ is a bounded domain, T > 0. Similar to the case of elliptic equations, we may use the interior estimates and the near boundary estimates (including near bottom estimates, near lateral boundary estimates and near lateral-bottom estimates) for the heat equations, to establish the corresponding estimates for (7.2.35), and then use the finite covering technique to derive the global Schauder's estimates. Exactly speaking, we have the following theorem. Theorem 7.2.24 Let 0 < a < 1, dfl 6 C2'a, aihbuc £ Ca>a/2(QT), aij = aji and equation (7.2.35) satisfies the uniform parabolicity conditions, that is, for some constants 0 < A < A, A|e|2 < a«(a,t)&& < A|£| 2 ,
V£ € R", (x,t) € QT-
232
Elliptic and Parabolic
Equations
In addition, assume that f £ Ca'a/2(QT),
+ \p\2+a,l+a/2;QT
+ \u\o-,QT)>
(7.2.37)
where C is a positive constant depending only on n, a, A, A, f2, T and the C 2+a,l+a/2(Q T ) norm 0f a^^ c, Remark 7.2.5 Note that in order the estimate holds, it suffices to require u £ C2+a>l+a'2{QT) n C(QT) instead ofu£ C2+a'1+a'2(JQT).
Exercises 1. Prove Theorem 7.1.1 and Remark 7.1.2. 2. Prove Remark 7.2.3. 3. Establish the near lateral-bottom estimate for the heat equation. 4. Prove Theorem 7.2.24 and Remark 7.2.6. 5. Let u £ C2+a>1+a/2{BT) be a solution of the following initialboundary value problem ~^-Au
+ up = f,
(x,t)£BT
=
Bx(0,T),
0, dpBT
where 0 < a < 1, B is the unit ball of W1, p > a. Prove that
where C > 0 depending only on n, p, |/| Q ,a/2 ; B T and |U|O ; B T 6. Establish Schauder's estimates for solutions of the following initialboundary value problem of fourth order parabolic equation f du + A2u = f, dt du u = — = 0, ov u(x,Q) = 0 ,
(x,t)£QT (x,t)£dQ,x x £ £1.
=
Clx(0,T), (0,T),
Chapter 8
Existence of Classical Solutions for Linear Equations In this chapter, we establish the existence theory of classical solutions for linear elliptic and parabolic equations of second order.
8.1
Maximum Principle and Comparison Principle
The existence of classical solutions is based on Schauder's estimates. In addition, the L 00 norm estimate on solutions is also needed. In this section, we introduce the maximum principle, which will be used to establish the L°° norm estimate and comparison principle on classical solutions. 8.1.1
The case of elliptic
equations
Consider the following linear elliptic equation Lu=—aij(x)DijU
+ bi(x)DiU + c(x)u = f(x),
x £
fi,
(8.1.1)
where Q, C W1 is a bounded domain, a^ = dji and for some constant A > 0, aij(x)^j
> A|£|2,
VfeR",ie(l.
Theorem 8.1.1 (Maximum Principle) Let c(x) > 0, bi(x) and c(x) be bounded inQ., u G C2(f2) n C(H) satisfy Lu = f < 0 (> 0) in Q.. Then sup u(x) < sup u+ (x) n an
(inf u(x) > inf u_ (x)), \ n an J
where u+ = max{it,0}, u_ = min{w,0}. 233
234
Elliptic and Parabolic
Equations
Proof. We first show that if / < 0, then the conclusion holds. If the conclusion were not true, then there would exist x° € Cl, such that u(x°) = rnaxu(x) > 0. n Thus (DijU(x°))nxn
< 0,
On the other hand, since (aij(x°))nxn Lu(x°) = -aij{x0)Diju{x0)
DiU(x°) = 0.
> 0, c(x°) > 0, we have
+ bi(x0)Diu{x0)
+ c(x°)u(x°)
> 0,
which contradicts f(x°) < 0 and hence the conclusion is valid for the case /<0. Now, we turn to the general case / < 0. If we may find an auxiliary function h G C2(Q) fl C(Q), which satisfies h > 0,
Lh<0,
in fi,
then for any e > 0, there holds L(u + eh) = Lu + eLh < 0 in 9,. So, according to the above proved conclusion, we infer sup{'u(a;) + eh(x)} < sup{u(x) + n an
eh(x)}+.
Thus supu(a;) <sup{u(x) +eh(x)} n n < sup{w(x) + eh(x)}+ an < sup u+ (x) + e sup h{x). an an Letting e —> 0, we get the desired conclusion. There are many functions with the above properties, for example, we may take h{x) = e a x i , where a > 0 is a constant to be determined. Noticing that Lh{x) =eaxi(-a2an(x) <e
QXl
2
+ ah(x)
+ c(x))
( - a A + a6i(a;) + c(x)),
x £ ft,
and bi(x) and c(x) are bounded in fi, we need only to take a to be sufficiently large, such that Lh < 0 in Cl.
Existence of Classical Solutions for Linear
Equations
235
As for the case of / > 0, we may consider — u instead of u, and get the desired conclusion. • Remark 8.1.1 From the proof of the theorem, we see that the condition that bi(x) (i = 1, • • • ,n) are bounded in fi can be replaced by the boundedness ofbi(x) for some i. Remark 8.1.2 Theorem 8.1.1 can also be proved by the following approach. Let v = u — supu+ and first show that v < 0 if c > 0. As for the an general case c > 0, we may let v = hw, and consider the equation for w, where h is an auxiliary function to be determined. Using the maximum principle, we can now establish the comparison principle. Theorem 8.1.2 (Comparison Principle) Let c(x) > 0, bi(x) and c{x) be bounded in Vt, v,w € C2(f2) l~lC(f2) satisfy Lv < Lw in Cl and v < w an an Then v(x) < w(x), Proof.
\tx £ Q,.
It suffices to use Theorem 8.1.1 by taking u = v — w.
•
By a suitable choice of the functions v and w in the comparison principle, we may obtain the a priori bound of solutions of the Dirichlet problem for equation (8.1.1). Theorem 8.1.3 Let c(x) > 0, bi(x) and c(x) be bounded in Q, u € C2(Cl) H C(fi) satisfy Lu = f in Q. Then sup|w| < sup|u| + C sup | / | , Q an n where C depends only on \, diamfi and the bound ofbi(x) in Q. Proof. Without loss of generality, we assume that / is bounded in 0; otherwise the conclusion is obvious. Set d = diamfi,
/3 = sup |6i|. a
Take a fixed point x° = (x°,x®,- • • ,a;°) € fi, such that x\ < xi,
Va; = {xi,x%, • • • ,xn) e fl.
Then 0 < Xi — i j < d,
Vx = (xi,X2,--•
,xn)
€0,.
236
Elliptic and Parabolic
Equations
Let ) sup |/|, w(x) = s u p | u | +g(xi), an
x € f2,
where a > 0 is a constant to be determined. Then w e C2(f2) n C(D.), and for any x £ il, w satisfies w(x) > g(xi) > 0, Lw(x) = - an(x)g"(xi)
+ bi{x)g'{x{) + c{x)w{x)
> - an(x)g"(xi)
+
=ea^-x°\a2an(x)
bi(x)g'(xi) - ah(x)) sup | / | n
>a(aA-/3)sup|/|. Choosing a = (j3 + 1)/A + 1 yields Lw(x) > sup |/|, n
Mx £ Q.
Similarly, we have L{-w(x)}
< - sup |/|,
Vx £ CI.
Thus L{-w{x)}
< Lu(x) < Lw(x),
Mx £ ft.
In addition, it is obvious that —w{x) < u(x) < w(x),
Vx £ dfl,
and so, from the comparison principle, the desired conclusion is valid for • c = e((/3+i)/A+i)d_ 8.1.2
The case of parabolic
equations
Consider the following linear parabolic equation Lu=—
Ou
aij(x,t)DijU
+ bi(x,t)DiU + c(x,t)u = f(x,t),
{x,i)£QT,
(8.1.2)
237
Existence of Classical Solutions for Linear Equations
where QT = fi x (0,T), f2 C R n is a bounded domain, ay = a,ji and Oij(x, t)tej
> 0,
V£ E R n , (x, t) € QT.
Theorem 8.1.4 (Maximum Principle) Let c(x, t) > 0 be bounded in QT, u e C2{QT) n C(QT) satisfy Lu = f < 0 (> 0) inQT- Then supu(a;,t) < sup n + ( x , t )
(infu(x,t) > inf
QT
^QT
dpQT
u-(x,t)\.
B
VQT
'
Proof. We first show the conclusion when / < 0. If the conclusion were false, then there would exist a point (x°,to) S QT\9PQT such that u(x°, to) = max u(x, t) > 0. QT
Thus
A M ( J ; 0 , to) = 0,
In addition, since {aij(x°,to))nxn Lu(x°,t0)
=
--^
u(x°, t 0 ) > 0.
> 0, c(x°,to) > 0, we have aij(x° ,t0)Diju(x0
,t 0 )
+ 64(x°, t 0 ) A « ( i ° , t 0 ) + c(x°, t0)u(x°, t 0 ) > 0, which contradicts f(x°, to) < 0, and hence the conclusion is valid if / < 0. Next we consider the general case of / < 0. Let h(x) = e~at, where a = supc(x.t) + 1. Then h £ C2(QT) n C(QT) and QT
Lh = — + ch = (-a + c)e~at < 0, at
h>0,
in QT-
Hence, for any e > 0, we have L(u + eh) = Lu + eLh < 0
in
QT-
According to the proved conclusion, it follows that sup{u(a;) t) + eh(x, t)} < sup {u(x, t) + eh(x, t)}+. QT
dpQT
Therefore sup u(x, t) < sup{w(a;, t) + eh(x, t)} QT
QT
238
Elliptic and Parabolic
Equations
< sup {u(x,t) +
eh(x,t)}+
dpQT
< sup u+(x,t)
+ e sup
dpQx
h(x,t).
dpQr
Letting e —» 0, we get the desired conclusion. For the case of / > 0, we need only to consider — u instead of u.
•
Remark 8.1.3 In Theorem 8.1.2, we merely assume the parabolicity condition rather than the uniform parabolicity condition for equation (8.1.2), and no boundedness condition for bi is assumed. If we do not assume the condition c(x, t) > 0, then the above maximum principle is invalid. However, we still have the following useful result. Theorem 8.1.5 Let c(x,t) be bounded in QT, U G satisfy Lu = f < 0 in QT and sup u(x,t) < 0. Then
C2(QT)
n C(QT)
9PQT
sup u(x,t) < 0. QT
Proof.
Let CQ = inf c(x,t) and set QT
=ecotu(x,t).
v(x,t) Then v satisfies
+ bi(x,t)DiV + {c(x,t) - CQ)V = eCotf(x,t)
— - aij(x,t)Dijv
< 0,
(x,t)eQT. Noticing that c(x,t) — CQ > 0 in QT, from Theorem 8.1.4, we obtain supv(x,t)
< sup v+(x,t)
QT
9PQT
= sup eCotu+(x,t)
=0,
9PQT
which leads to the conclusion of the theorem.
•
Applying Theorem 8.1.5, we may establish the following comparison principle. Theorem 8.1.6 in
QT,
v,w £
(Comparison Principle) Assume that c(x,t) is bounded
C2(QT)
n C(QT)
satisfy Lv < Lw in
QT
and v
< 9PQT
w
. Then dpQT
v(x,t) <w(x,t),
V(x,t)€QT-
Existence of Classical Solutions for Linear
239
Equations
Proof. We may take u = v — w in Theorem 8.1.5 to obtain the desired conclusion. • Similar to the case of elliptic equations, by suitably choosing the functions v and w in the comparison principle, we may obtain the a priori bound for solutions of the first initial-boundary value problem for equation (8.1.2). Theorem 8.1.7 Let c(x,t) > 0 and be bounded in C(QT) satisfy Lu = f in QT- Then
QT,
U
G
C2(QT)
l~l
s u p | u | < sup |u| + T s u p | / | . QT
dpQT
QT
Proof. Without loss of generality, we assume that / is bounded in QT; otherwise the conclusion is obvious. Let w(x,t) = sup |u|-Msup |/|, 9PQT
Then w G
C2(QT)
l~l
and for any (x, t) G
C(QT)
Lw(x,t)
(x,t) G QT-
QT
we have
QT,
= sup | / | + c(x, t)w(x, t) > sup |/|. QT
QT
Similarly, for any (x,t) G QT, we also have L{-w(x,t)}
< -sup|/|. n
Thus L{-w(x,t)}
< Lu(x,t)
< Lw{x,t),
V(x,t) G QT-
In addition, it is obvious that —w(x, t) < u(x,t) < w{x,t),
V(x,t) G
8PQT,
from which and the comparison principle we get the desired conclusion. • Theorem 8.1.8
Let c(x,t)
be bounded in QT, CQ = min{0,inf
u G C2(QT) n C(QT) satisfy Lu = f inQT-
QT
Then
sup|u| < e~coT [ sup | u | + T s u p | / | 1 . QT
\dpQT
QT
J
c(x,t)},
240
Elliptic and Parabolic
Proof.
Equations
Let v{x,t) = eCotu(x,t),
(x,t) € QT.
Then v satisfies dv — - aij(x,t)Dijv
+ bi(x,t)DiV + (c(x,t) - CQ)V =
eCotf{x,t),
{x,t)£QT. Noticing that c(x, t) - CQ > 0 in QT, from Theorem 8.1.7, we have sup|u| < sup \v\ + T s u p | e C o t / | , QT
&VQT
QT
that is sup|e C o t u|< sup |e C o *u|+Tsup|e C o t /|QT
9PQT
QT
It follows from Co < 0 that sup|u|<e
C T
° I sup | u | + T s u p | / |
QT
8.2
\9PQT
QT
D
Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations
In this section, we first investigate the existence of C2'a(fl) solutions and C2,a(fl) n C(fi) solutions for Poisson's equation and then investigate the existence of the same kinds of solutions for general linear elliptic equations. 8.2.1
Existence Poisson's
and uniqueness equation
of the classical solution
for
Consider the Dirichlet problem for Poisson's equation -Au = / , an
=<,,
i£fl,
(8.2.1) (8.2.2)
where (1 C 1 " is a bounded domain. We first prove the existence and uniqueness of its C2,a(Q) solution.
Existence of Classical Solutions for Linear
Equations
241
Theorem 8.2.1 Let dQ £ C°°, 0 < a < 1, / e C a (£2),
2
|/|Q;n
and fs converges to / uniformly on Q. as e —> 0. Consider the approximate problem -Au = fe(x),
xeCl,
u\ =0. Ian By the I? theory (Theorem 2.2.5 of Chapter 2), we see that the above problem admits a unique solution ue € C°°(fi). Prom the global Schauder's estimate (Theorem 6.3.1 of Chapter 6), we have |«e|2,a;fi < C ( | / e | a ; n + |u e |o ; n)-
Using the maximum principle (Theorem 8.1.3) yields k l o ; n < C|/ e | 0 i n < C | / e | a . n < C | / | a . n . Thus \u£\2,a-n < C | / | a ; n . The constant C in the above formula is independent of e. By ArzelaAscoli's theorem, there exists a subsequence of {u£}, denoted by itself, and a function u € C2'a(Q,), such that ue—*u,
Due^>Du,
D2uE^>D2u
uniformly on fi as e —> 0. Letting e —> 0 in the approximate problem, we see that u satisfies equation (8.2.1) and the boundary value condition u = 0 . So, we have proved the existence of solutions. The uniqueness an follows from the maximum principle. • In Theorem 8.2.1, it is assumed that the domain fi has C°° smooth boundary, which means that the theorem could not be applied even to Poisson's equation in square domain. In what follows, we will relax the
242
Elliptic and Parabolic Equations
restriction on the domain, but the solution space is enlarged to be C2'a(D.)n C(Q) in the same time. Definition 8.2.1 We call a domain 0 to have exterior ball property, if for any x° € dQ, there exists R > 0 and y e R n \fi such that ~BR(y)C\Ti = {a;0}. If such R can be chosen to be independent of x°, then the domain Q is said to have uniform exterior ball property. Theorem 8.2.2 Assume that Cl has the exterior ball property, and there exists a sequence of subdomains {Clk} with C°° boundary, such that £lk c flfc+i and dClk converges to d£l uniformly. Let 0 < a < 1, f £ C a (fi), (p € C 2 , a (fi). Then problem (8.2.1), (8.2.2) admits a unique solution u €
C2'Q(fi)nC(n). Proof. Without loss of generality, we may assume that tp = 0; otherwise, we set w = u—(p and consider the equation for w. Consider the approximate problem Au = /(a:),
x € fife,
= 0, which admits a unique solution Uk G C2'a(Qk) by Theorem 8.2.1. For fixed positive integer m, according to Schauder's interior estimate (Theorem 6.2.8) and the maximum norm estimate, we have |Wfc|2,a;fim
+ |Mk|o;fifc)
Vfc > m,
where C\ and C-x are independent of k. By a diagonal process, we may obtain a subsequence { u ^ } ? ^ of {u^^Li and a function u € C 2,a (f2), such that for any fixed m > 1, uki-*u,
Duki-+Du,
D2uki-^D2u
uniformly on Qm as i —> oo. Therefore, u satisfies equation (8.2.1) in fi. Now, we use the barrier function technique to show that u = 0 , that dn is for any fixed a;0 £ dD., u(x°) = lim u(x) = 0. For this purpose, it • El! X—>X°
suffices to construct a continuous function w(x) > 0, such that w(x°) = 0 and |u(aj)| < Cto(a;), a; £ fin Bs(x°).
Existence of Classical Solutions for Linear Equations
243
Such a function is called an exterior barrier function. Set = M(e-f}R
W (x)
c?2
_fll
„.|2\
-*-P\*-v\'),
i d ,
where R and y are the radius and the center of the exterior ball at the point x° respectively, /3 > 0 and M > 0 are constants to be determined. It is easily seen that the function w(x) has the following properties: i) w(x°) = 0, w{x) > 0 for all x G Ti\{x0}; ii) w G C2(Q) and for appropriate large /3 > 0 and sufficiently large M > 0, -Aw > 1 in fi. In fact, -Aw(x)
=M(Ap2\x
- yfe-®x-y\2
>Me-Wx-yf{4p2R2
-
2n(3e-^x-y]?)
- 2np),
x G n.
Next, we set vk{x) = uk(x) - \f\o-,nw(x),
x G Qfc
and proceed to show Vk(x) < 0 in f2fc. In fact, -Awfc(x) = -Auk(x)
+ \f\0-nAw(x)
< f(x) - |/| 0 ; n < 0,
i G fifc.
In addition, <0, - \f\o-,nw an, dak and so, from the comparison principle, we have Vfc
Uk
dak
Vk{x) < o, \fx e Ofc, that is uk{x) < \f\o-nw{x),
Vx G fife.
For any fixed x G fi, choosing m sufficiently large such that a; G flm, we have «k(aO < \f\o-,nw(x),
Vfc > m.
Taking k = hi and setting i —* oo lead to w(z) < |/|o;nty(ar),
Vx G ft.
Similarly, u(#) > -\f\o-,nw(x),
\/x G ft.
244
Elliptic and Parabolic
Equations
Summing up, we have \u(x)\ < \f\o-,UW(x),
Vx G ft.
Therefore, u(x°) — 0 and the existence of solutions is proved. The uniqueness follows from the maximum principle. • Remark 8.2.1 Theorem 8.2.2.
If 0. is a rectangle, then 0, satisfies the assumptions of
In Theorem 8.2.2, the boundary function ip G C2,a(Cl), but the solution obtained is only continuous up to the boundary. Is it possible to weaken the restriction on the boundary function
If tp G C(fi), then the conclusion of Theorem 8.2.2 is
Choose tpk G C°°(fi), such that \
VxGft, fc = l , 2 , - - - .
Consider the approximate problem -Au(x)
= f(x),
x G Clk,
By Theorem 8.2.2, the above problem admits a unique solution Uk G C 2 ' a (fifc). Using the interior estimate (Theorem 6.2.8 of Chapter 6) and a diagonal process, it is easy to prove that there exists a subsequence of {uk}, denoted still by {wfc}, and a function u € C 2,Q (fi), such that for any fixed m > 1, Uk—*u, Duk—^Du,
D Uk—>D u
uniformly on O m as k —» oo, from which it follows that u satisfies equation (8.2.1). Now, we verify u =
Vx G B,5(x0) n ft.
Existence
of Classical Solutions for Linear
Equations
245
Choose Ce > |/|o;fi + 1 such that \ip(x)-tp(x0)\<e
+ Csw(x),
Vxeil,
where w is the barrier function defined in the proof of Theorem 8.2.2. Thus \
Vx£il,
A; = 1,2,
Set vk(x) = uk(x) - Cew(x) - £ -
T
It
- tp(x°),
x € ilk-
Then -Avk = -Auk
Vk
dilk
+ CEAw = f + CeAw < f - C£ < 0,
=Uk
an i
-a™
1 ant
A;
- £ - - -
•Cew
=
x e ilk,
an* The comparison principle gives Vk{x) < 0,
Vx G fifc,
that is Ufc(z) < C £ w ; ( x ) + e + - +
Vx £ ilk-
For fixed i S ft, we choose m sufficiently large, such that x £ ilm. uk(x) < Cew(x)+e
+ - +
Vk > m.
Letting k —> oo yields u(x) < Ceiv(x)+e
+
So lim u(x) < e + ip(x°). X—>X°
By the arbitrariness of e > 0, we see that lim u{x) < (p(x ) .
Vx£il.
Then
246
Elliptic and Parabolic
Equations
Similarly, we obtain lim u(x) > ip(x°). x—>x°
Summing up, we have lim u(x) =
Thus the existence is proved. The uniqueness of solutions follows from the maximum principle. • 8.2.2
The method
of
continuity
Contraction Mapping Principle Let T be a contraction mapping on the Banach space B, that is, there exists 0 < 6 < 1, such that \\Tu-Tv\\<9\\u-v\\,
Mu,v£B.
Then T admits a unique fixed point, that is, the operator equation Tu = u admits a unique solution u € B. Proof.
For fixed UQ £ B, set Ui = Tui-i,
i = 1,2, ••• .
For any positive integer 1 < i < j , using (8.2.3), we obtain 3
| | U j - U t | | < ^ J \\uk -Ufc-l|| k=i+l 3
= ^2 WTuk-l
~Tuk-2\\
k=i+l
< J2 0*-1||ui-«o|| k=i+l ai
<^—^Iki — "oil -*0 (i->oo).
(8.2.3)
Existence of Classical Solutions for Linear
Equations
247
So, {ui} is a Cauchy sequence, and from the completeness of B, it converges to some u € B. Prom (8.2.3) we see that T is continuous, and so Tu = lim Tm = lim Uj+i = u. i—>oo
i—>oo
The uniqueness follows directly from (8.2.3).
•
Remark 8.2.2 From the proof of the theorem, we see that the conclusion is still valid if we replace B by any closed subset of B. The Method of Continuity Let B be a Banach space, V be a normed linear space, TQ and Ti be bounded linear operators from B to V. Set TT = (1-T)TO
+ TTU
r e [0,1].
/ / there exists some constant C > 0, such that \\u\\B < C\\TTu\\v,
UGB,T€[0,
1],
(8.2.4)
then Ti maps B onto V if and only if To maps B onto V. Proof. Let s € [0,1] and Ts maps B onto V. By (8.2.4), we see that Ts is injective and so the inverse map T " 1 : V —> B exists. For T £ [ 0 , l ] , « £ y , the operator equation TTu = v is equivalent to the equation Tsu = v + (Ts - TT)u = v +
(T
- s)(T0 - Ti)u.
Furthermore, since T~x exists, it is also equivalent to u = T~lv + (T - s)T-\T0
- 2\)u = Tu.
If |T_s|<(5s
C ( | | T o | | + ||T 1 || + 1)'
then from (8.2.4), we see that T : B —> B is a contraction map. According to the contraction mapping principle, for any s £ [0,1] satisfying \T — s\ < 8, the map TT is bijective. We decompose [0,1] into several intervals with their length less than 5. It is easy to see that if for some fixed To € [0,1] (in particular for ro = 0 or ro = 1), TT0 is bijective, then for all r € [0,1], TT is bijective too. • Remark 8.2.3 The method of continuity shows that the invertibility of a bounded linear operator can be deduced from the invertibility of another similar kind of operators.
248
8.2.3
Elliptic and Parabolic
Equations
Existence and uniqueness of classical general linear elliptic equations
solutions
for
By the method of continuity, we may extend the above results about the Dirichlet problem (8.2.1), (8.2.2) for Poisson's equation to the Dirichlet problem for general linear elliptic equation —aij(x)DijU + bi(x)DiU + c(x)u =f{x),
x e f2,
(8.2.5)
en =*>
^ a
where fl C 1 " is a bounded domain, aij,bi,c £ C (Q,), c > 0, ai7- = a^, and there exists some constants 0 < A < A, such that A|£|2 < o^aO&fc < A|£| 2 ,
V£ £ l " , i e
fl.
(8.2.7)
Theorem 8.2.4 Let dtt e C°°, 0 < a < 1, 0^,^,0, f € Ca(U), c > 0, ip € C 2 ' a ( 0 ) , ay = a-ji satisfy (8.2.7). Then problem (8.2.5), (8.2.6) admits a unique solution u e C2'a(Q). Proof. Without loss of generality, we may assume that tp = 0. Otherwise, we consider the equation for w = u —
= — Au,
L\u = — aij(x)DijU + bi(x)DiU + c{x)u. Consider the family of elliptic equations with a parameter r , LTU = {1-T)L0U
+ TL1U = f,
0
(8.2.8)
where the coefficients of second order term satisfy (8.2.7) with A, A taken as Ar = min{l, A},
A r = max{l, A}.
LT can be regarded as a linear operator from the Banach space B = {u e C2'a(Q) : u\dn = 0} to the normed linear space V = Ca(Ti). So the solvability of problem (8.2.8), (8.2.6) is equivalent to the invertibility of the operator LT. Let u € B be a solution of problem (8.2.8), (8.2.6). According to Schauder's estimates (Theorem 6.3.1 of Chapter 6) and the maximum norm estimates (Theorem 8.1.3), and noticing the assumption (p = 0, we have |u|2,a;n < C f l / U j n + Mo ; fi) < C | / | a i n ,
T G [0, 1],
Existence of Classical Solutions for Linear
Equations
249
that is \\u\\B
UGB, r e [0,1],
where C is a constant independent of r . When T = 0, problem (8.2.8), (8.2.6) is just problem (8.2.1), (8.2.2), which admits a unique solution u G B according to Theorem 8.2.1. This means that LQ maps B onto V. Using the method of continuity, we see that L\ maps B onto V too, and so problem (8.2.5), (8.2.6) admits a solution u G C 2 ' Q (0). The uniqueness can be proved by the maximum principle. • Using Theorem 8.2.4 and the barrier function technique, similar to the proof of Theorem 8.2.2, we obtain the following Theorem 8.2.5 Assume that Q has the exterior ball property, and there exists a sequence of subdomains {Ofc} with C°° boundary, such that fi^ C fifc+i and dQk approximates d£l uniformly. Let 0 < a < 1, aij,bi,c, f G Ca{Tl), c > 0,
If
Furthermore, we may prove the following theorem. Theorem 8.2.7 Let 0 < a < 1, 60, G C2'a, aij,buc,f G C a (H), c > 0,
Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations
In this section, we introduce the theory parallel to the second section for linear parabolic equations.
250
Elliptic and Parabolic
8.3.1
Existence the heat
and uniqueness equation
Equations
of the classical solution
for
Consider the first initial-boundary value problem du — -Au=f(x,t), u(x,t) =
(x,t)£QT,
(8.3.1)
(x,t)edpQT,
(8.3.2)
where QT = ft x (0,T), Q. C W1 is a bounded domain, T > 0. Theorem 8.3.1 Let dfl £ C°°, 0 < a < 1, f £ Ca'a'2(QT),
function w(x,t). Let (x°,to) £ 0PQT. The barrier function w(x,t) have the following properties: i) w(x°,t0) = 0, w(x,t) > 0 for all x £ QT\{x°,t0}; ii) w £ C2^(QT),
^-Aw>lm
should
QT.
Now, for the point (x°,to) at the lateral boundary, we choose w(x,t) — w(x), the barrier function constructed in the proof of Theorem 8.2.2, and for the point (x°,0) at the bottom, we choose w(x,t) = t. Clearly the function thus denned possesses the above properties. • Theorem 8.3.3 still valid.
/ / ip £ C(QT),
then the conclusion of Theorem 8.3.2 is
Existence of Classical Solutions for Linear
Equations
251
The proof is similar to that of Theorem 8.2.3 and the details are left to the reader.
8.3.2
Existence and uniqueness of classical general linear parabolic equations
solutions
for
Using the method of continuity, we may extend Theorem 8.3.1 for the heat equation to the first initial-boundary value problem du — - aij (x, t)DijU + bi(x, t)DiU + c(x, t)u = f(x,t), {x,t)GQT, u{x, t) = tp(x, t),
(x, t) G dpQT,
where fi C R™ is a bounded domain, aij,bi,c G Ca'a/2{QT), for some constants 0 < A < A, such that A|£|2 < OijfatMs
< A|£| 2 ,
(8.3.3) (8.3.4) a^ = aji and
V£ € R", (x,t) G QT.
(8.3.5)
Theorem 8.3.4 Let dQ G C°°, 0 < a < 1, aihbi,c,f G Ca'a'2{QT), 2+Q 1+a/2
If ip G C(QT),
then the conclusion of Theorem 8.3.5 is
We may further establish the following theorem. Theorem 8.3.7 Let 0 < a < 1, dQ G C2'a, aij,bi,c,f G Ca'a/2(QT), 2+a 1+a 2 ip G C > / (QT), a^ = aji satisfy (8.3.5). Then problem (8.3.3), (8.3.4) admits a unique solution u G C2+a'1+a/2(QT). Proof. Since dCl G C 2 , Q , Q has the exterior ball property, and all the conditions for Q in Theorem 8.3.5 are satisfied. So, according to Theorem 8.3.5 we see that problem (8.3.3), (8.3.4) admits a unique solution u G C 2 + a ' 1 + a / 2 ( Q r ) n C(QT). Then an application of Schauder's
252
Elliptic and Parabolic Equations
estimates (Theorem 7.2.24 and Remark 7.2.5 of Chapter 7) shows that u G C2+a
Exercises 1. Prove Theorem 8.1.1 by the method mentioned in the proof of Remark 8.1.2. 2. Let B be the unit ball in R". Assume that u G C2(B)nC(B) satisfies - A u ( i ) < 0,
x e B.
u{x) < u(x°),
x G B.
Let x° G dB with
Prove that
t;<*°»°. where v is the unit normal vector outward to dB. 3. Prove Theorems 8.2.5 and 8.2.6. 4. Let fi C R n be a bounded domain with appropriately smooth boundary, A G R, 0 < a < 1. Prove that there is exact one of the following alternatives: i) The homogeneous boundary value problem —Au + Xu = Oin Q,, u
an
-0
admits a nontrivial classical solution u G C2'a(Q,); ii) For any / G Ca(Q), the nonhomogeneous boundary value problem -Au + Xu = / i n Cl, u
an
admits a unique classical solution u G C 2 ' Q (f2).
= 0
Existence
of Classical Solutions for Linear
Equations
253
5. Consider the second initial-boundary value problem ' du ~Au + Xu = f, dt du = 0,
1h>
u(x,0) = Uo(x),
(x,t) &QT = Qx (0,T), (x,t)edflx(0,T), x € fi,
where A g R, (1 C 1 " is a bounded domain with appropriately smooth boundary, v is the unit normal vector outward to dCl. Prove that the problem admits at most a smooth solution u S C2'l{QT). 6. Prove Theorems 8.3.1 and 8.3.3. 7. Prove Theorems 8.3.4-8.3.6.
Chapter 9
Lp Estimates for Linear Equations and Existence of Strong Solutions
In the previous chapters we have investigated two classes of solutions, that is weak solutions and classical solutions of linear elliptic and parabolic equations. In this chapter, we consider another kind of solutions with intermediate regularity, called strong solutions. For this purpose, we need to establish the LP estimates. Just as the existence of classical solutions is based on Schauder's estimates, the existence of strong solutions is based on the LP estimates. We will first apply Stampacchia's interpolation theorem and the results on Schauder's estimates, to establish the LP estimates for Poisson's equation and the heat equation. On the basis of these estimates, we establish the LP estimates for general linear elliptic and parabolic equations, and establish the existence theory of strong solutions. It is worthy noting that the LP estimates can be established for equations in nondivergence form, but a crucial condition, i.e. the continuity assumption on the coefficients of second order terms is required. 9.1
LP Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions
In this section, we first introduce the LP estimates on solutions of Poisson's equation in cubes, and then apply these estimates to establish the LP estimates for general linear elliptic equations, and further establish the existence theory of strong solutions. 9.1.1
LP estimates
for Poisson's
equation in cubes
Consider the homogeneous Dirichlet problem for Poisson's equation 255
256
Elliptic and Parabolic
-Au = / , dQo
Equations
X G Qo,
(9.1.1) (9.1.2)
=0,
where Qo is a cube in R" with its edges parallel to the axes. To obtain the IP estimate on a solution u of problem (9.1.1), (9.1.2) in the cube Qo, we need to establish the estimate on D2u in the Campanato space £2'™(<2o)- We first establish the interior estimate. P r o p o s i t i o n 9.1.1 Let f G L°°(QQ), u G H2(Q0) n H^{Q0) be a weak solution of equation (9.1.1), x° G Qo, B2R0(x°) CC QO. Then [D2u]2tn.BRo{x0)
< C (||D 2 u|| L 2 ( B 2Ro(a; o)) + ||/||/,~(B 2 „ 0 (x°))) ,
(9.1.3)
where C is a constant depending only on n and RoProof. Let fe be the standard smooth approximation of / and u£ be the solution of the problem - A u £ = f£,
x G Qo,
= 0. dQo
By the L2 theory, uE is sufficiently smooth in B2R0(x°) and uE —> u,
in H2(B2R0(x0))
as e —> 0.
Therefore, to show (9.1.3), we need only to prove [ £ > V ] 2 , n ; B K o ( x ° ) < C (\\D V | | L 2 ( B 2 H O (X«)) + | | / e IU~(B 2 R o (xO)) J ,
where the constant C is independent of e. Owing to this reason, in what follows, we may assume that u is sufficiently smooth in B2R0{X°). For any x G BR0(X°), we have BRo(x) C B2R0(X°). From the proof of Schauder's interior estimate (Theorem 6.2.6 of Chapter 6), we see that for any 0 < p < R < RQ,
I.
\Dw(y) -
{Dw)xJ2dy
BP(x)
*t/
\Dw(y) - (Dw)XtR\2dy + C [
JBR(x)
+2
y J
f J JBR(X)
(f(y) - fx,R)2dy
JBR(X)
2
n
\Dw(y) - (Dw)XtR\ dy + CR \
IZ,~(SR(X))'
Lp Estimates for Linear
257
Equations
where w = A w ( l < i < n). So, the iteration lemma (Lemma 6.2.1 of Chapter 6) yields that for any 0 < p < R < Ro, [
{Dw)xJ2dy
\Dw(y) -
JBp(x)
^
C
(l) " ( /
(Dw)x>R\2dy
\Dw(y) -
Rn
\\f\\L°°(BR(x))
[
2
{x0)\
dy
JBp(x)r\BRo(x°)
\D2u{y) -
< f
(D2u)xJ2dy
JB„{x)
{^\\D2U\\2LHBRO{X))
(-^WDMUB^OV
+ ||/|ii- (fljlo(x)) ) + ll/lli~( W *o))),
where (D2u)B
{x)f)B
PK
=
{x0)
ft0v
i
D2u(y)dy.
/ \Bp{x)r\BRo{x")\
JBp{x)c\BRo(x°)
Therefore
lD2u\{iL%Ro(x°)
^ C (\\D2UWLHB2RO(X°))
+ ll/IU~(B 2Ko (x°))) ,
where C is a constant depending only on n and -Ro- The notation [-^n is defined in Remark 6.1.2 of Chapter 6. In addition, by virtue of [-]2,n < [•]2!n2) + C\\ • \\L>, we obtain (9.1.3). • Having the interior estimate in hand, we may further obtain the global estimate by the extension of solution. Proposition 9.1.2 Let f G L°°(Q0) and u G H2(Q0) n H^(Q0) be the weak solution of the Dirichlet problem (9.1.1), (9.1.2). Then \\D2u\\c^(Qo)
< C||/|U~Wo),
(9.1.4)
where C is a constant depending only on n and the length of the edge of Qo-
258
Elliptic and Parabolic
Proof.
Equations
Let Qo = Q(x°, R) = {xe R n ; \Xi - x°\ < R, i = 1,2, • • • , n}.
To prove (9.1.4), we extend the definition of u in the following manner, and denote the extended function by u. First, we make the antisymmetric extension of u with respect to the super planes Xi = x®+R and X\ = x\ — R, namely, define ' u(x), u(x) = <
if:r€<30,
-u(2x\ + 2R-xi,x2,-• ,xn), if xi £ (x1+R,x1 + 3R], \xi\
-u(2x1-2R-xi,x2,--,xn), if xi e [x1-3R,x1-R),
\xi\
Next, we make the antisymmetric extension with respect to the super planes x-i = x° + R and x2 = x® — R, • • •, xn = x„+R and xn = z° — R. Then we obtain a function u defined on Q(x°,3R). Repeating the above procedure for n times then yields a function u defined on Q(x°, 3nR). Similarly, we get the antisymmetric extension / of / on Q(x°,3nR). Obviously, / 6 L°°(Q(x°,3nR)). It is not difficult to check that u G 2 H (Q(x°, 3nR))r\Ho(Q(x°, 3nR)), and u is a weak solution of the equation xeQ(x°,3nR).
-Au = /, Owing to Qo = Q(x°,R)
c B^R(x°)
c B2VTiR(x°)
CC
Q(x°,3nR),
using Proposition 9.1.1 in <5(a;°,3"-R) leads to [D2u\2,n;Qo
<[D2u}2tn;B^R(x°)
+ ||/||L~(0(x0,3"R))J •
So, by changing a constant C depending only on n and R, we derive {D2u}2,n;Q0 < C (||i) 2 u|| L 2 ( Q o ) + | | / | | L - ( Q O ) ) • 2
(^^)
According to the L theory (Remark 2.2.2 of Chapter 2, which is still valid for cubes and can be proved by using the method similar to the proof of
Lp Estimates for Linear
Equations
259
Theorem 8.2.2 of Chapter 8), we have IMIi/'Wo) ^
(9.1.6)
Combining (9.1.5) with (9.1.6) yields (9.1.4).
•
To obtain the LP estimate for Poisson's equation in the cube QQ, we need Stampacchia's interpolation theorem. Definition 9.1.1 Let <3o be a cube in Rn with its edges parallel to the axes. For u € L1(Qo), if |U|»,Q 0
= sup \ / \uL \Qr\Qo\ JQC\QO
dx
UQDQO \
\ <5 is a cube parallel to Q0 \ J
< + oo, then we say that u € BMO(<2o)' We define the norm in BMO(Qo) by II ' llBMO(Qo)
=
II ' lU'CQo) + I • l*,Qo-
It is easily shown that BMO(Qo) is a Banach space. From the definition of Campanato spaces and the space BMO(Qo)> we see that £ 2 ' n (Qo) C £ 1>n (Qo) - BMO(Qo)Define the operator Ti : L°°(Q 0 ) -» BMO(Qo),
/ •-» DDtu,
where 1 < i < n, and u is a weak solution of problem (9.1.1), (9.1.2). The estimate (9.1.4) shows that Tt (1 < i < n) is a bounded linear operator from L°°(Q 0 ) to C2'n(Q0) C BMO(Q0). Stampacchia's Interpolation Theorem Let 1 < q < +oo. If T is a bounded linear operator both from Lq(Qo) to Lq(Qo) and from L°°(Qo) to BMO(Q0), namely ||Tu|| L , W o ) < C i | | u | | L , W o ) , WTU\\BMO(Q0)
<^2||«||L"O(Q0),
Vu e
L"{Q0),
VU G L ° ° ( Q O ) ,
p
then T is a bounded linear operator from L (Qo) to Lp(Qo), \\Tu\\LHQo)
< C||«||LP(QO)I
p
Vu e L (Q 0 ),
namely
260
Elliptic and Parabolic
Equations
where q < p < +00, and C depends only on n, q, p, C\ and C2 • For the proof of the theorem, we refer to [Chen and Wu (1997)] Appendix 4. Using Stampacchia's interpolation theorem, we may obtain Theorem 9.1.1 Letp>2,fe LP(Q0), U £ W2'P(Q0)nW^'p(Q0) Poisson's equation (9.1.1) in Qo almost everywhere. Then \\D2U\\LHQO)
satisfy
< C||/||iP W o ),
where C depends only on n, p and the length of the edge of Qo • Proof. By the L2 theory and the estimate (9.1.4), T; (1 < i < n) is a bounded linear operator both from L2(Qo) to L2(Qo) and from L°°(QQ) to BMO(<5o). Then, Stampacchia's interpolation theorem shows that Tj is a bounded linear operator from Lp(Qo) to Lp(Qo). • Remark 9.1.1 Under the conditions of Theorem 9.1.1, we may further use Ehrling-Nirenberg-Gagliardo's interpolation inequality to obtain \\u\\w2.r(Q0) <
C(II/IILP(Q 0 )
+ IMUp(Qo))-
Remark 9.1.2 The conclusion of Theorem 9.1.1 is still valid for the case 1 < p < 2. For the proof we refer to [Chen and Wu (1997)] Chapter 3. Remark 9.1.3 The conclusion of Theorem 9.1.1 can be extended to the general elliptic equations with the coefficients matrix of second order being constant and positive definite and throwing lower order terms. This is because Schauder's interior estimates which have been applied to prove (9.1.3) is still valid for such kind of equations, while the extension technique using in the proof of (9.1.4) is available to such equations too. 9.1.2
Lp estimates
for general linear elliptic
equations
Now, we turn to the general linear elliptic equations -aij(x)Diju
+ bi(x)Diu + c(x)u = f(x),
xeCl,
(9.1.7)
where fi c R" is a bounded domain, ay = a,-,, and for some constants A > 0, M > 0 OijixMj
> A|£|2,
V£ £ f , x e
fi,
(9.1.8)
LP Estimates for Linear
Equations
261
$ 3 IKIU°°(n) + Yl INU~(n) + l|c||L~(n) < M. i,j = l
(9.1.9)
i=l
Definition 9.1.2 Let w(i?) be a nondecreasing and continuous function defined on [0, +oo) and u(R) = lim u>(R) = 0. For a function a(x) on £2, R—*-0
we say that a(x) has continuity module u(R) on £2, if \a(x) -a(y)\
< w(\x-y\),
Vx,y&Q.
It is easy to verify that if a £ C(O), then a(x) has continuity module on Q. For the general linear elliptic equation (9.1.7), we have Theorem 9.1.2 Let dCl G C2, p > 1. Assume that the coefficients of equation (9.1.7) satisfy (9.1.8) and (9.1.9) and a^ G C(fi). If u G W 2 , p (fi)nW 0 ' p (fi) satisfies equation (9.1.7) almost everywhere in £1, then \\u\\w**p(n) < C(\\f \\Lp{n) + ||u||LP(n)), where C is a positive constant depending only on n, p, X, M, Q, and the continuity module of a^-. Proof. Similar to Schauder's estimates, we proceed to prove the theorem by three steps. S t e p 1 Establish the interior estimate. For any x° £ Cl, choose Ro > 0 such that QR = Q(x°,Ro) C Cl. Let 0 < R < Ro and 77 be a cut-off function on QR relative to BR/2, that is, 77 G C$°{QR) satisfying rj(x) = 1 in BR/2 and 0
|Vjj(aOI<|.
Let v = r\u. Then v G
W0'P(QR)
-aij(x)Dijv
\DiM*)\
^ §2>
xeQR.
and from equation (9.1.7) we obtain = g(x),
x G QR,
where g = -{hrj + 2aijDjrf)Diu{x)
- (crj + a ^ D y ^ u + 77/ in QR.
Rewrite the above equation as -aij(x°)Dijv
= F(x)=g(x)
+ h(x),
x G QR,
(9.1.10)
262
Elliptic and Parabolic
Equations
where h(x) = (ay(x) - a y (o: 0 ))Aju(x),
x e Qij.
Using the conclusion of Theorem 9.1.1 and Remark 9.1.2 to equation (9.1.10) (as pointed in Remark 9.1.3, for such kind of equations, the conclusion of Theorem 9.1.1 is still valid), we obtain \\D2V\\LP(QR)
< C0||F||LP(Qfl),
where Co depends only on n, p and R. In addition, IISIILP(QR) <M II^IUP(OH)
\jp
+ 1 J ll U lki.P(Q R ) + H/||LP(Q„),
<W(\/^)P2^IU»>(QH)>
where w(-) is the common continuity module of each a,ij (i, j = 1,2, • • • , n). Choose 0 < .Ro < &o such that Cow(v/ni?o) < 1/2. Then P^IU'WBO)
^ c (^(^2 + ^hll^'noflo) + ll/lli'(OHo)J •
Therefore, from the definition of v, we get that for any 0 < R < Ro, \\D2U\\LP(BR/2)
< C(||/|| L p ( n ) + ||u|| w i., ( n)).
(9.1.11)
Step 2 Establish the near boundary estimate. Let x° e dQ. Since <9fl S C 2 , there exists a neighborhood V of the point x° and a C 2 mapping $ : V —> Bi = Bi(0) such that
*(V) = Bi, v(£inv) = B+, v(danv) = dBfnB1. Denote by Qo the maximal cube contained in B^ with its edges parallel to the axes. After a coordinate transformation y = 9(x), equation (9.1.7) in ^~1(Qo) is transformed into the equation in Qo - a y ( j / ) A j « + bi(y)DiU + c(y)u = f{y), 2
(9.1.12)
d d where Di = ——, Da = ——-—, u = uf^ : (j/)) and the meaning of au, bi, dyi dyidyj c, f are understood similarly. It is easy to check that the coefficients and the right hand side function of equation (9.1.12) have the same properties as the corresponding ones of equation (9.1.7). To establish the near boundary estimate for equation (9.1.12), as we did for the interior estimate, we first cut off the function u, consider the equation for the function after cutting
Lp Estimates for Linear
263
Equations
off and throwing off the lower order terms, adopt the method of solidifying coefficients, and use the conclusion of Theorem 9.1.1 and Remark 9.1.2 (as pointed out in Remark 9.1.3, for such equation, the conclusion of Theorem 9.1.1 is still valid). After returning to the original equation with respect to the variable x, we finally obtain \\D2U\\LP{0R)
< C (||/||iP(n) + ll«lki.p(n)) ,
(9.1-13)
where 0 < R < R0, and Ro is a given constant, OR C il is such a domain which depends on R and for some constant a > 0 independent of R such that ft n BaR(xQ) C 0R. Step 3 Establish the global estimate. Combining the interior estimate (9.1.11) and the near boundary estimate (9.1.13) and using the finite covering argument, we get the following global estimate ||D 2 uj| £ p ( n) < C(||/|| L p ( n) + IMIwn.p(n)). Then, Ehrling-Nirenberg-Gagliardo's interpolation inequality implies ||w||vK2.p(n) < C(||/|| L P (n) + ||u||ip(n)). Remark 9.1.4 From the proof of Theorem 9.1.2, we see that the establishment of W2'p estimates on solutions depends essentially on the conditions a,ij € C(fl). For this reason, it should be careful in applications. In Theorem 9.1.1 and Theorem 9.1.2, we have mentioned a new kind of solutions which have weak derivatives up to the second order and satisfy the equation almost everywhere. Such kind of solutions will be called strong solutions. More general, consider the following nonhomogeneous boundary value condition u
fln^'
(9
-L14)
where
264
Elliptic and Parabolic
Equations
Prom Theorem 9.1.2 and Definition 9.1.3, it is easy to prove Theorem 9.1.3 Let dfl G C2, p > 1. Assume that the coefficients of equation (9.1.7) satisfy (9.1.8), (9.1.9) and ay G C(ty. If u G W2>p(Ct) is a strong solution of the Dirichlet problem (9.1.7), (9.1.14), then ||u||w2.p(n) < C (||/||Lp(n) + IMIw«.p(n) + IMUp(n)), where C is a constant depending only on n, p, X, M, fl and the continuity module of aij. 9.1.3
Existence and uniqueness ear elliptic equations
of strong solutions
for lin-
Now, we discuss the existence and uniqueness of strong solutions. As shown in Chapter 8, the existence and uniqueness of classical solutions are based on not only Schauder's estimates but also the L°° norm estimates on solutions themselves. Similarly, the existence and uniqueness of strong solutions are based not only the LP estimates but also the Lp norm estimates on solutions themselves. For general equations, the Lp norm estimates can be established by Aleksandrov's maximum principle (see [Chen and Wu (1997)]). While for special equations, for example, for the equation -Au + c(x)u = f(x),
xGfi,
(9.1.15)
the Lp norm estimate can be obtained by the methods similar to those in establishing the L2 norm estimate (see the following Theorem 9.1.4). However, such kind of methods can only be generalized to equations in divergence form and can only be used to treat the case p > 2. Theorem 9.1.4 Let dQ G C2>a, p > 2, c G L°°(fi) and c > 0. Then for any f G Lp(Cl), equation (9.1.15) admits a unique strong solution u G
w2>p(n)nwt>p(n). Proof. We first establish the LP norm estimates on strong solutions of equation (9.1.15). Let u G W2>P{Q)PWQ'V{SI) be a strong solution of equation (9.1.15). Multiplying both sides of equation (9.1.15) by |u| p _ 2 u and integrating the resulting relation over Q,, we have — / \u\p~2uAudx
Ja
+ / c|u|pda; = /
Jo,
Jn
f\u\p~~2udx.
Lp Estimates for Linear
265
Equations
Integrating by parts yields 4(P - 1) / |V(|u| p / 2 - 1 u)| 2 da;+ f c\u\pdx = f p2
Jn
Jn
Jn
f\u\p~2udx.
Then, using Poincare's inequality, Holder's inequality and Young's inequality with e leads to 4(P ~ 1) I \u\pdx + f c\u\pdx < f flu^dx fip2 Jn Jn Jn IP—i <\\f\\r,P(m\\u LP(n)||"llLP(n) <£ / \u\pdx + e-1/{p-l) Jn
[ \f\pdx, Jn
where ji > 0 is the constant in Poincare's inequality, e > 0 is an arbitrary constant. Owing to c > 0, the above estimate yields / \u\pdx
(9.1.16)
where C depends only on n, p, /u, ||c||ioo(n) and ^Using the a priori estimate (9.1.16), we immediately obtain the uniqueness of the strong solution. In fact, let ui,U2 £ W2'p(D,)r\W0'p(Cl) be two strong solutions of equation (9.1.15) and set v = u\ — u^. Then v£ W2'P(D.)C)WQ'P(Q.) is a strong solution of the homogeneous equation —Av + cv = 0,
x £ D,.
The estimate (9.1.16) gives \\v\\w2-"(n) 5= 0, which implies that v = 0 a.e. in Q, that is u\ = u 0, and Cfe converges to c weakly star in L°°(fl), fk converges to / in Lp(Cl). Consider the approximate problem -Auk
+ cku = fk{x),
Ufe
=0.
an
iGfl,
266
Elliptic and Parabolic
Equations
According to Theorem 8.2.7 of Chapter 8, the above problem admits a solution uk G C2'a(U) c W 2 ' p (fi)rW 0 1,p (ft). The estimate (9.1.16) implies ||wfc||w2.p(n) < C||/fe|Up(n), which shows that {uk} is uniformly bounded in W2'p(fl). Prom the weak compactness of the bounded set in W2'P(Q) and the compactly embedding theorem, there exists a subsequence of {ufc}, which converges weakly in W2'p(Cl) and converges strongly in W 1 , p (fi). Let u be the limit function. Then u G W2>p(£l)r\W01'p(Q) and it is easy to verify that u satisfies equation (9.1.15) almost everywhere. • Remark 9.1.5 The conclusion of Theorem 9.1.4 is still valid for the case 1 < p < 2. In addition, the smoothness condition on dCl can be relaxed to dSl G C2. For general linear elliptic equation (9.1.7), we have the following theorem, whose proof can be found in [Chen and Wu (1997)] Chapter 3. Theorem 9.1.5 Let dQ G C2, p > 1. Assume that the coefficients of equation (9.1.7) satisfy (9.1.8), (9.1.9), c > 0 and atj G C(Q). Then for any f G LP(Q), the Dirichlet problem (9.1.7), (9.1.14) admits a unique strong solution u G W2'P{Q). 9.2
Lp Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions
In this section, we introduce a parallel theory to the one in the first section for linear parabolic equations. 9.2.1
Lp estimates
for the heat equation
in cubes
Consider the first initial-boundary value problem for the heat equation Fin
^-Au=f, u
(x,t)eQT,
(9.2.1)
=0,
(9.2.2)
dpQr
where QT — Qo x (0, T), Q0 is a cube in W1 with edges parallel to its axes. Denote QR = QR(X°,
t0) = BR(x°)
x (t0 - R2, t0 +
R2),
Lp Estimates for Linear
QR
= QR(X°,0)
267
Equations
BR(X°)X(0,R2).
=
P r o p o s i t i o n 9.2.1 Let f G L°°(QT), u G Wl'l{QT)r\w\'l{QT) weak solution of equation (9.2.1), x° G Qo, B2R0(x°) CC QO- Then
be a
[D u]2,n+2;B H o (x°)x(0,T)
(\\D2U\\L2{B2RQ{XO)X{0,T))
+
| | / | | L ° ° ( B 2 H O ( X ° ) X ( 0 ,T)))
,
(9.2.3)
where C depends only on n, T and RQ . Proof. Let fe be a smooth approximation of / , ue be the solution of the problem Oil
—£• - Aue = / E ,
(x, t) G QT,
= 0. dpQr
From the L2 theory, ue is sufficiently smooth in B2R0{x°) X [0,T] and in W 2 2,1 (B 2Ro (a; 0 ) x (0,T)) as e -» 0.
u£ - • u,
So, in order to prove (9.2.3), it suffices to show \D2 UE}2>n+2iBRo
(x<>) x (0,T)
where the constant C is independent of e. Owing to this reason, in what follows, we may suppose that u is sufficiently smooth in B2R0(x°) X [0,T]. First, we establish the interior estimate. For any x G BR0(X°), 0< t < T, we choose 0 < Ro < Ro such that QRo(x,t) C B2R0(X°) X (0,T) C Q T From the proof of Schauder's interior estimate (Theorem 7.2.3 of Chapter 7), we see that, for any 0 < p < R < Ro,
I
\Dw(y,s) -
(Dw)XitJ2dyds
Qp(x,t)
\Dw(y, s) -
{Dw)x,t,R\2dyds
Qa(x,t)
+c
ii
(f(y, s) -
fx,t,R)2dyds
t)
JjQR(x
org
Qs(i,t)
\Dw(y,s) -
(Dw)x^R\2dyds
268
Elliptic and Parabolic
+ CRn
Equations
||/|lz,~(Q„(x,t))'
where w = Diu{l < i < n). By virtue of this and the iteration lemma (Lemma 6.2.1 of Chapter 6), we deduce that for any 0 < p < R < Ro, 2
\Dw(y,s) \±sujyy,oj -—
(Dw)xj,p\ {j_sui) xAp\
dyds
Qp(x,t)
-K
/
{Dw)x^R\2dyds
\Dw(y, s) -
K
JjQR(x,t)
Rn+2\\f\\UQR(x,t)))
+ ^pn+2
( ^ + 2 \\^\\h{QRix>t))
+ ii/iii-Wj,(..t))
Similar to the proofs of Theorem 7.2.4 and Theorem 7.2.5 of Chapter 7, we obtain \-u
U
h,n+2;Qko/2(x,t)
+ ||/||z,°°(QAo(x,t)))
+
||/||L°°(B 2 R O (X°)X(O,T))J
,
(9.2.4)
where C depends only on n and Ro. Similarly, we may obtain the near bottom estimate. For any x £ 5 f l o (x°), we choose 0 < Ro < Ro such that Q°R (x, 0) C B2Ro (x°) x (0, T) C QT- From the proof of the near bottom Schauder's estimate (Theorem 7.2.9 of Chapter 7), we see that for any 0 < p < R < Ro,
I
\Dw(y,s)\
dyds
Q?(».o)
-
C
(l)"
+ 4
\Dw(y,s)\2dyds
//o
(f(y,s)-fx,0,R)2dyds
+ C ff JJQ°R(X,O) n+i
^R'
\Dw(y,s)\2dyds
ff
+
CRn+2\\f\\l„{QO,0)), R
IQ°R(X,0) JJQ°(X,O)
where w = DiU (1 < i < n) and VX,O,R =
l n 0
,
nM
//
v(y, s)dyd,
LP Estimates for Linear
269
Equations
Therefore, the iteration lemma (Lemma 6.2.1 of Chapter 6) leads to that for any 0 < p < R < Ro, \Dw(y,s)\2dyds Q°(x,0)
-c(l)n+2(i/'o
\D™(y>s)\2dyds +
x0
Rn+2\\f\\l~(Q%(X,o^
U
h,n+2;Qlo/2(x,0)
+
||/||L~(B 2 R O (XO)X(O,T))J
.
(9.2.5)
where C depends only on n and Ro. Combining the interior estimate (9.2.4) with the near bottom estimate (9.2.5), using the finite covering argument and the relation [-]2,n < [ • ){2,L2) + C|| • || L 2, we obtain (9.2.3) immediately. D 1 1 Proposition 9.2.2 Let f G L°°(QT) and u e W%' {QT)C\w\' {QT) be a weak solution of the first initial-boundary value problem (9.2.1), (9.2.2). Then \\D2u\\c2,n+HQT)
< C||/||i«,WT)>
(9.2.6)
where C depends only on n, T and the length of the edge of QQ. Proof. Let Qo = Q(x°,R) = {x£Wl;\xi-x°i\
< R,i = 1,2, • • • , n } .
Similar to the proof of the corresponding conclusion for elliptic equations (Proposition 9.1.2), for fixed t G [0,T], we extend the definition of u(-,t) and f(-,t) to Q(x°,3nR) antisymmetrically, and obtain functions u and / inQ T (a; 0 ,3 n i?), where QT(x°,3nR) Obviously / G L°°(QT(x°, l
0
n
l
= Q(x°,3nR)
3nR)). n
Wl' (QT(x ,3 R))r\W2 {QT{x°,3 R))
x (0,T).
It is not difficult to check that u G and u is a weak solution of the
270
Elliptic and Parabolic
Equations
equation 3v ~ ^ - A u = /,
(x,t)GQT(x°,3nR).
Owing to Qo = Q(x°,R)
C B^R(x°)
c B2VER(x°)
use Proposition 9.2.1 in QT(x°,3nR)
Q(x°,3nR),
to get
u}2,n+2;Bv7[R(x0)x(0,T)
[£> U] 2 ,n+2;Q T <[D
CC
^||Z)2'U||Z,2(QT(a.0]3rlfi)) +
+ ||/||L-(S2v/ijfi(xO)x(0,T))) ||/||ioo(Qr(x0i3njR))J ,
which implies [D2u}2tn+2]Ql.
< C (\\D2U\\L2{QT)
+ ||/||LOO(QT))
(9.2.7)
with another constant C depending only on n and R. According to the L2 theory (Remark 3.4.1 of Chapter 3, although there the spatial domain is assumed to have C2 smoothness, the conclusion is still valid for cubes, which can be proved by the methods similar to those in Theorem 8.3.2 of Chapter 8), we have \Hwli(QT)
(9.2.8)
Combining (9.2.7) with (9.2.8) leads to (9.2.6).
•
Define an operator Ti:Loo(QT)-.BM0(gT),
/ >-> DDtu,
where 1 < i < n and u is a weak solution of problem (9.2.1), (9.2.2). The estimate (9.2.6) shows that Tj (1 < i < n) is an bounded linear operator from L°°(QT) to BMO(QT). Theorem 9.2.1 Letp>2 and u G W2'1 {QT)^Wl'x equation (9.2.1) almost everywhere in QT- Then \D2U\\LV{QT)
du
It
(QT) satisfy the heat
where C depends only on n, p, T and the length of the edge of QQ.
Lp Estimates for Linear Equations
271
Proof. Prom the L? theory and the estimate (9.2.6), Tt (1 < i < n) is a bounded linear operator both from L2(QT) to L2(QT) and from L°°(QT) to B M O ( Q T ) - Then, by Stampacchia's interpolation theorem, Tj is a bounded linear operator from LP(QT) to LP(QT) and \\D2u\\Lv{QT)
D
Remark 9.2.1 Under the conditions of Theorem 9.2.1, using EhrlingNirenberg-Gagliardo 's interpolation inequality, we further obtain \\U\\W2,I{QT)
< C(\\f\\Lp{QT)
+ ||U||LP«M)-
Remark 9.2.2 The conclusion of Theorem 9.2.1 is also valid for the case 1 < p < 2, whose proof can be found in [Gu (1995)J Chapter 7. Remark 9.2.3 The conclusion of Theorem 9.2.1 can be extended to parabolic equations with the coefficients matrix of second order derivatives being constant and positive definite and throwing all lower order terms. This is because Schauder's interior estimates, near bottom estimates and the extension used to prove (9.2.6) is also available to such kind of equations. 9.2.2
Lp estimates
for general linear parabolic
equations
Now we turn to the general linear parabolic equations du — -aij(x,t)DijU
+ bi(x,t)DiU + c(x,t)u = f(x,t),
(x,t) <E QT, (9.2.9)
where QT = ^ x (0,T), Q, c M" is a bounded domain, ay = a^ and for some constants A > 0, M > 0 MM)&&>A|£|2, n
VfeR
n
,MeQr,
J2 Hay IU~«JT) + 5Z INU~(QT) + ||C||L~(QT) ^ Mi,j = l
(9-2.10)
n
(9.2.11)
i=l
Similar to the case of elliptic equations, we may obtain the LP estimates for general linear parabolic equations.
272
Elliptic and Parabolic
Equations
Theorem 9.2.2 Let fi £ C2, p > 1. Assume that the coefficients of equation (9.2.9) satisfy (9.2.10) and (9.2.11) and aij £ C(QT). If u £ W2'1 (QT)^WJ,'1 then
(QT)
satisfies equation (9.2.9) almost everywhere in QT,
Hwtf-'Wr) -
C
(\\fh"(QT)
+ \\U\\LP(QT))
,
where C depends only on n, p, X, M, T, fi and the continuity module of Remark 9.2.4 The establishment of Wp'1 estimates on solutions is essentially depending on the condition a^ £ C(QT), and so, in applying such estimates one should take special care. Now, we consider the first initial-boundary value problem of equation (9.2.9) with the following boundary and initial value condition .
=¥>,
(9-2.12)
OPQT
where
W2'1(QT).
Definition 9.2.1 A function u £ Wp'1(Qx) is called a strong solution of equation (9.2.9), if u satisfies equation (9.2.9) almost everywhere in QT- If, * in addition, u —
Wu\\w$'\QT) ^ C
(|I/IU"(OT)
+ IMIWI'^QT) + IMU»(QT)) .
where C depends only on n, p, X, M, T, Q, and the continuity module of
9.2.3
Existence and uniqueness ear parabolic equations
of strong solutions
for lin-
To show the existence and uniqueness of strong solutions, besides the Lp estimates established in Theorem 9.2.2, we also need the U norm esti-
Lp Estimates for Linear Equations
273
mates on solutions themselves. Similar to the case of elliptic equations, for general equations, the Lp norm estimates can be established by using the Aleksandrov's maximum principle (see [Gu (1995)]). While for some special equations, for example, for the equation 3u — -Au
+ c{x,t)u = f(x,t),
(x,t)EQT,
(9.2.13)
we may utilize the methods similar to those utilized in establishing the L2 norm estimates (see the following Theorem 9.2.4). However, such kind of methods can only be generalized to the equations with divergence form and can only be used to treat the case p > 2. Theorem 9.2.4 any f G LP(QT),
Let dQ G C2'a, p > 2 and c G L°°{QT). Then for equation (9.2.13) admits a unique strong solution u G
Proof. Let u G (9.2.13). Setting
W2'1(QT)<~\WI'1{QT)
be a strong solution of equation
w(x,t)=e-mu{x,t), where M =
\\C\\L*°(QT).
9w — at =e~m
{x,t)eQT,
Then w G W2A (Qr)rWp' 1 (QT) and w satisfies . .,, . Aw + (M + c)w
(J^-Au
+ cu\ = e~Mtf,
a.e. in QT
with M + c > 0. Hence we may assume that c > 0 in equation (9.2.13). We first establish the Lp norm estimates on strong solutions for equation (9.2.13). Assume that u G W2'1 (QT)r)Wl'1 (QT) is a strong solution of equation (9.2.13). Multiplying both sides of equation (9.2.13) by |w| p_2 w and integrating over QT, we obtain //
= If
\u\p-2u-^dxdt
- jf
\u\p'2uAudxdt
+ ff
c\u\pdxdt
f\u\p~2udxdt.
JJQT
Integrating by parts with respect to the spatial variable yields
274
Elliptic and Parabolic Equations
I
f\u\p-2udxdt.
QT
Using Poincare's inequality, Holder's inequality and Young's inequality with e, we get - / \u\p Tdx + 1 ^ — H
//
PJn
JJQT IQT
MP2
o
\u\pdxdt + ff
c\u\pdxdt
JJQT JJQT
f\u\p-lcdxdt
l
^II/IIL»«T)II U IILP(QT) <£
\u\pdxdt + e-ltto-V
ff JJQT
ff
\f\vdxdt,
JJQT
where /x > 0 is the constant in Poincare's inequality, e > 0 is an arbitrary constant. Owing to u €.W),'1{QT)
and c > 0, the above estimate yields
/ / \u\pdxdt
JJQT JJQJ
with C depends only on /x and p. Combine this with Theorem 9.2.2 to get NI^.1(QT)
(9.2.14)
with C depending only on n, p, /x, ||C||/,<=O(QT), T and f2. From (9.2.14), we immediately obtain the uniqueness of the strong solution. In fact, assume u\, u-i G W2'1 (QT)CWI'1
(QT)
are two strong solutions
of equation (9.2.13). Set v = ui - u2. Then v G W^1 (QT^W1/[QT) strong solution of the homogeneous equation Ov — -Av
+ c(x,t)v = 0,
is a
(x,t)GQT-
According to the estimate (9.2.14), we have ||«||w a ' 1 (Q r ) ^ 0- Therefore v = 0 a.e. in QT, that is ux = it2 a.e. in QTFinally, we prove the existence of strong solutions. Let ck,fk 6 aa 2 C ' / (QT), Ck > 0, and ck converges to c weakly star in L°°(QT), fk converges to / in LP(QT)Consider the approximate problem —
- Aufc +ck(x,t)u = 0.
Uk
dQT
= fk(x,t),
(x,t) G QT,
LP Estimates for Linear
275
Equations
According to Theorem 8.3.7 of Chapter 8, the above problem admits a solution Uk G C2'a(QT). Thus the estimate (9.2.14) gives INHW^QT)
- CWfk\\LP(QT)'
which implies that {u^} is uniformly bounded in
W2'1(QT).
Prom the
21
1
weak compactness of the bounded set in W ' (QT) and WP' {QT) and the compactly embedding theorem, there exists a subsequence of {wfe}, which converges weakly in W2'1(QT) and Wp^iQr), and converges strongly in LP(QT). Let u be the limit function. It is easy to verify that u G Wp'1 (QT)^W);1 (QT) and u satisfies equation (9.2.13) almost everywhere in QT. • Remark 9.2.5 Different from the case of elliptic equations 9.2-4)> the restriction condition c > 0 is not required.
(Theorem
Remark 9.2.6 The conclusion of Theorem 9.2.4 is still valid for the case 1 < p < 2. In addition, it suffices to require dfl G C2. For general linear parabolic equation (9.2.9), we have the following theorem, whose proof can be found in [Gu (1995)] Chapter 7. Theorem 9.2.5 Let dCl G C2, p > 1. Assume that the coefficients of equation (9.2.9) satisfy (9.2.10) and (9.2.11), and a^ G C{QT). Then for any f £ LP(QT), the first initial-boundary value problem (9.2.9), (9.2.12) admits a unique strong solution u € W2'1 (QT) •
Exercises 1. Check u G H2(Q(x°,3nR)) in Proposition 9.1.2, and judge whether the following arguments are valid: i)IiueC1'a(Q(x°,R))&ndu = 0, thenu € C1'a(Q{x°,3nR)); Q ii) If u € C2(Q(x°, R)) and u
= 0 , then u G H2(Q(x°, dQ(x°,R)
_
V
V
3nR)). "
2. Prove that a(x) has a continuity module in Q if and only if a(x) G C(£l), where CI is an bounded open set in R™. 3. Let CI C E™ be a bounded domain with appropriately smooth boundary, A G E, p > 1. Then there is one and only one of the following alternatives:
276
Elliptic and Parabolic
Equations
i) The boundary value problem for the homogeneous equation —Au + Xu = 0 in il,
u an
0
admits a nontrivial strong solution u € W2'P(Q,) n WQ'P(CI); ii) For any / e Lp(il), the boundary value problem for the inhomogeneous equation —Au + Xu = f in CI, u
= 0 an admits a unique strong solution u € W2'p{Cl) D W0'P(Q).
Chapter 10
Fixed Point Method
The approaches based on fixed point theorems have very important applications in the investigation of partial differential equations, especially the nonlinear differential equations. In this chapter, as an example, we apply such a method to the solvability of quasilinear elliptic equations.
10.1
Framework of Solving Quasilinear Equations via Fixed Point Method
In this section, we describe the basic framework of fixed point method in solving quasilinear equations. 10.1.1
Leray-Schauder's
fixed
point
theorem
Leray-Schauder's Fixed Point Theorem Let U be a Banach space, T(u,a) be a mapping from Ux [0,1] to U satisfying the following conditions: i) T is a compact mapping; ii) T{u, 0) = 0, \/u £ U; Hi) There exists a constant M > 0, such that for any u € U, if u = T(u, a) for some a € [0,1], then \\u\\u < M. Then the mapping T(-, 1) has a fixed point, that is, there exists u G U, such that u = T(u, 1). 10.1.2
Solvability
of quasilinear
elliptic
equations
We first consider the Dirichlet problem for quasilinear elliptic equations - diva(:r, u, Vw) + b(x, u, Vu) = 0, 277
x € 9.,
(10.1.1)
278
Elliptic and Parabolic
dQ
Equations
(10.1.2)
= ¥>>
where a = {a\, a
OTjj
\a,i(x,z,Q)\ dai drij
(10.1.4)
(10.1.5)
+ |o i |<MN)(i + l'7l),
(10.1.6)
+ |6|<M(N)(1 + H2),
(10.1.7)
-b(x, z, r])sgnz < A(\n\ + h(x)),
(10.1.8)
dai
8z dai dxj
where i, j = 1,2, • • • ,n, X, A > 0, g € Lq(Cl)(q > n), h G L 9 *(0), q* nq/(n + q) and /x(s) is a nondecreasing function on [0, +oo). A typical example of such equations satisfying (10.1.3)-(10.1.8) is -div(a(u)Vu) + b(u) = f(x),
(10.1.9)
x e ft,
where o(u) = (u2 + l ) m / 2 (m > 0),
b{u) = lu^u
(7 > 1)
and / e Ca(ft). For simplicity, we consider only the Dirichlet problem with the homogeneous boundary value condition (10.1.10) = 0. an T h e o r e m 10.1.1 Let 0 < a < 1, dfl € C2'a, a e C1'01, b € Ca,
Fixed Point
279
Method
Proof. We do not intend to give a detailed proof for such a general theorem. For simplicity, we merely consider problem (10.1.9), (10.1.10). The proof of the conclusion i) is quite lengthy, which will be completed in the subsequent sections. Here we merely give the proof of the conclusion ii) by assuming that the conclusion i) is valid. Choose U = C 1 , a (fi). For any v £ U, 0
- (1 - a)Au + ab{v) = af(x),
x £ Q,,
(10.1.11)
(10.1.12) = 0. an By the Schauder theory for linear equations, this problem admits a unique solution u £ C2'a(D.). Define the mapping T : U x [0,1] -+C/, (v,a) \-*u. In what follows, we check the properties of the mapping T. Firstly, since C 2 ' a (fi) can be compactly embedded into C 1 , a (fi), T is compact. Secondly, when a = 0, (10.1.11), (10.1.12) reduces to the homogeneous Dirichlet problem for Laplace's equation Au = 0, « an
= 0
x £0,,
>
which has only a trivial solution. Therefore T(u, 0 ) = 0 ,
Vv£U.
Finally, assume that u is a fixed point of the mapping T for some a £ [0,1], namely, u is a solution of the problem crdiv(a(u)Vu) — (1 - a)Au + crb(u) = af(x), an
x £Cl,
= 0.
Then according to the conclusion i), there exist 0 < (3 < 1 and a constant M > 0 independent of u and a, such that |u|i,/3;n < M.
(10.1.13)
280
Elliptic and Parabolic
Equations
In applying the conclusion i), it should be noted that, all elements appeared in the structure conditions which the constant M depends on can be chosen to be independent of a € [0,1]. Now, we rewrite the equation of u into the nondivergence form -(aa(u)
+ 1 - a)Au = cr(a'(w)|Vu|2 - b(u) + f(x)).
(10.1.14)
Owing to the estimate (10.1.13), the coefficients of equation (10.1.14) belong to C a "(ft). So, according to the Schauder theory, we conclude that u S C2'af}(Q.), and there exists a constant C independent of u and a, such that |w|2,a/3;0 <
C,
which implies ||u||y = Ml,a;f2 < C. Summing up, we have proved that the mapping T(u,a) satisfies all conditions of Leray-Schauder's fixed point theorem, and so, there exists u € U such that T(u, 1) = u. Then, from the definition of the operator T and the classical theory for linear elliptic equations, we get further that
u£C2'a(ty. 10.1.3
D
Solvability
of quasilinear
parabolic
equations
Now we turn to the first initial-boundary value problem for quasilinear parabolic equations du — - diva(x, t, u, Vu) + b(x, t, u, Vu) = 0,
(i,t)eQr,
=
(10.1.15) (10.1.16)
dpQT
where a = (ai, 02, • • • , a n ), Q, C M.n is a bounded domain, T > 0, QT = ft x (0,T). Assume that a(x,t,z,r)), b(x,t,z,rj) satisfy the following structure conditions: A(|z|)|£|2
V^el",
(10.1.17)
+ b2,
(10.1.18)
j O i l ^ / i d z D a + M),
(10.1.19)
Fixed Point
da* dxj
Method
281
+ |6|<M(|Z|)(1+|T?|2))
(10.1.20)
where i,j = 1,2, ••• ,n, A(s),A(s) are positive continuous functions on [0, +oo), 61,62 are positive constants and /x(s) is a nondecreasing function on [0,+oo). The equation du —
div(a(u)Vu) + b(u) + c(x, t)u = f(x, t),
(x, t) G QT
satisfies all the structure conditions (10.1.17)-(10.1.20), where a(u) = (u2 + l)m'2
(m > 0),
b(u) = {u^u
(7 > 1)
c,feCa>a/2(QT).
and
Theorem 10.1.2 Let 0 < a < 1, <9ft G C2'a, a G Cl
diva(:r, t, ip, Vy) + b(x, t, (p, Vy) = 0,
when x G dQ., t = 0.
Assume that equation (10.1.15) satisfies the structure conditions (10.1.17)(10.1.20). Then i) There exist a constant 0 < (3 < 1 and a constant M > 0 depending only on n, 61, b2, X(s), A(s), fi(s), \
\VU\0I0/2.QT
< M;
ii) Problem (10.1.15), (10.1.16) admits a solutionuG
C2+a'1+a/2{QT).
Proof. The proof of the conclusion i) is much complicated, here we do not intend to present. Assuming the conclusion i), the proof of the conclusion ii) is similar to that for elliptic equations, in which, instead of the space U = C1'a(Q) we choose U = {u;u,Vu We omit the details.
£
Ca'a/2(QT)}. •
282
10.1.4
Elliptic and Parabolic
The procedures
of the a priori
Equations
estimates
As shown in the previous section, to prove the existence of solutions for quasilinear equations, by means of the fixed point method, it suffices to establish the a priori estimates stated in Theorem 10.1.1 i) for solutions in C2'a{U) and Theorem 10.1.2 ii) for solutions in C2+a>1+a/2(QT). We will do this in the subsequent sections. However, merely elliptic equations will be considered and for clarity of the expression, we merely discuss equation (10.1.9) with the homogeneous boundary value condition (10.1.10). Moreover, all arguments are presented for the case n > 2; the discussion of the case n = 1 needs some modification, although it is much simpler on the whole. Let u e C2'a(Ti) be a solution of problem (10.1.9), (10.1.10). To obtain the required a priori estimate, we proceed to establish the following estimates successively. i) Maximum estimate ||«||L°°(n) < M. It can be obtained by the maximum principle, the Moser iteration or the De Giorgi iteration; ii) Holder's estimate [w]a;n < M. The main approaches are based on Harnack's inequality and Morrey's theorem; iii) Boundary gradient estimate sup|Vu| < M. The main approaches an are based on barrier function technique; iv) Global gradient estimate sup |Vu| < M. It can be derived by Bernn stein approach; v) Holder's estimate for gradients [Vu] a; n < M. The main methods are based on Harnack's inequality and Morrey's theorem.
10.2
Maximum Estimate
Several methods can be applied to establish the maximum norm estimate. We present one of them, which is based on the De Giorgi iteration technique. Theorem 10.2.1 (10.1.10). Then
Let u € C2
sup|u| < C||/||L°°(fi), n where the constant C depends only on n, m, j and fi.
283
Fixed Point Method
Proof.
We need only to show suptt < C||/|| LO o(n), n
since another part of the estimate can be obtained by considering — u. We adopt the De Giorgi technique. Set ip = (u — k)+, A(k) = {x e Q;u(x) > k}. Multiplying both sides of equation (10.1.9) by
Ju
Owing to (u — k)+
Jn
Jo.
= 0, integrating by parts yields
/ a(u)\V
where +00,
2
n = 1,2
2n
. n-2
,
n>3,
1 1 - + - = 1. P «
The embedding theorem then implies that M\lp(A(k))
\^9?dx
<
C\\f\\L,iAik))\\(p\\Lp{Aik)),
J it
that is \\
C\\f\\L^n)\A(k)\^.
So, for any h> k, (h - k)\A(h)\^
< y\\LP{A{k))
<
or
\A(h)\<(CU^{Q)Y\A(k)\^.
C\\f\\Laola)\A(k)\^
284
Elliptic and Parabolic
Equations
Similar to the case of linear equations, by using the iteration lemma (Lemma 4.1.1 of Chapter 4), we achieve SUpU
10.3
Interior Holder's Estimate
In this section, we apply Harnack's inequality to estimate the interior Holder norm of solutions. Since the proof is much complicated, we will divide it into several steps. i) Estimate supu; BeR
ii) Estimate inf u (weak Harnack's inequality); BeR
iii) Prove Harnack's inequality; iv) Estimate [u]a. The following theorems present the details of the above steps. Theorem 10.3.1 Letu S C 2 , "(!?#) be a nonnegative solution of equation (10.1.9) in BR, u = u + F0, F0 = i? 2 ||/||L~ ( B f l ) . Then for any p > 0, 0 < 9 < 1, we have sup u < C ( BeR
/ B
\\ R\JBR
updx )
,
J
where the constant C depends only on n, m, j , (1 — 9)~l and
||U||X,°°(B R )-
Proof. Without loss of generality, we assume that R=l. First consider the case p > 2. Let £(x) be the cut-off function on B\. Multiplying both sides of equation (10.1.9) by C 2 w p_1 , integrating the resulting relation over fi and then integrating by parts, we have /
a(u)Vu-V{<;2up-1)dx+
JBi
[
b(u)t2up-1dx=
JBi
f(?up~ldx.
[ JBi
Noticing the structure of a(u),b(u) and using the boundedness of u and Cauchy's inequality with e, we obtain (p-1) / <-2
[ JB!
C2up-2\Vu\2dx
C,a{u)up-1Vu-VC,dx-
f JBi
b(u)(2up-1dx+
[ JBi
f
Fixed Point
<£ /
<;2up-2\Vu\2dx
+-
285
Method
|VC| 2 u p dx + i /
f
+ U \f\2<2n"-2dx, where e > 0 is an arbitrary constant. Thus /
(2up-2\Vu\2dx
\V<;\2updx + C [
[
JBi
JB!
(?updx
JBi 2
+ C [
2 p 2
\f\ C u - dx.
(10.3.1)
JBi
Noticing that u > F0 =
H/HLO^BJ)
implies
\f\2<;2up-2dx
/
C2updx,
< f
JBi
JBx
from (10.3.1), we see that <:2up~2\Vu\2dx < C(l + sup |VC|2) /
/ JBi
Bi
updx.
JBi
Therefore /
|V(Cu p / 2 )| 2 da;
JBX
Si
updx.
JBX
The embedding theorem then implies 1/9
( f \JBI
C2"upqdx)
< C(l + sup|V<| 2 ) /
)
Bi
updx,
JB!
where +oo,
Kq<{
n = 1,2,
n
n-2
,
n > 3.
Thus, applying the standard Moser iteration technique, we get the conclusion for the case p > 2. As for the case 0 < p < 2, similar to the proof of the corresponding conclusion about Harnack's inequality for solutions of Laplace's equation (Theorem 5.1.3 of Chapter 5), we may use the result for p — 2 to get the desired conclusion. •
Elliptic and Parabolic Equations
286
Theorem 10.3.2 Letu G C 2 , Q ! ( B R ) be a nonnegative solution of equation (10.1.9) in BR, u = u + F0, FQ = R2\\f\\L°°(BR)- Then there exists p0 > 0, such that for any 0 < 0 < 1, we have \l/PO
kL<"*)'
inf u > C ,' . _, BeR
\
\Bf
where C depends only on n, m, 7, (1—9)
and
||U||£,OO(B B ).
Proof. Without loss of generality, we assume that Fo > 0, otherwise, we replace Fo by F0 + e. Let R = 1, and ( b e a cut-off function on B(0+iy2Multiplying both sides of equation (10.1.9) by £2u~(p+1\ similar to the proof of Theorem 10.3.1, we may obtain sup u~p
u-pdx.
J *'B(e+i)/2
Thus -1/p
MX *" ™[L *~PdxL H [L
inf u > „ . I / u Be -CVp\JBlt+1„
v
dx
\JB(e+i)/2
1/p
/
\ 1/p
n
updx )
CM
\ J D(.e+i)/2 W £>(S + l)/2
J
• D (e+i)/2 •/-£>(e + l)/2
.
\ J D(e+i)/2 V-0(0 +1
/ /
So, to show the conclusion of the theorem, we need only to prove that for some po > 0, ePo^dx
/
< C,
where
w = lnu - —
- /
|-B(9+3)/4|
JB(o+i)/2
\nudx. JB(e+3)/4
The remainder of the proof is almost similar to the linear case with some modifications. • Combine Theorem 10.3.1 and Theorem 10.3.2 to get Harnack's inequality. Theorem 10.3.3 Letu G C2,C"(BR) be a nonnegative solution of equation (10.1.9) in BR, u = u + F0, F0 = R?\\f\\L°°(BR)- Then for any 0 < 0 < 1, sup u < C inf u, BeR
BsR
where the constant C depends only on n, m, 7, (1 — 6)~l and
||U||Z,°°(B B )-
287
Fixed Point Method
Similar to the linear case, by virtue of Harnack's inequality we immediately deduce Holder's estimate. Theorem 10.3.4 Let u € C 2 , a (fi) be a solution of equation (10.1.9). Then for any Cl' CC f2, there exists 0 < 0 < 1 such that [u]p;n- < C, where the constant C depends only on n, m, 7, ||w||z,°°(n), H' and Q. 10.4
Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation
The derivation of the boundary estimates for quasilinear equations is much delicate and complicated. In this section, we center our efforts on Poisson's equation to present the key ideas in getting such estimates. A discussion for general equations will be given in the next section. Theorem 10.4.1 Let 0. C M" be a bounded domain with uniform exterior ball property. Assume that u S C2'a(Q) satisfies —Au = / in Q,,
= 0. an
Then sup|Vu|
xeD,
(10.4.1)
where D is the set of some neighborhood of x° intersecting with fi. Assume that, for the moment, such a function w±(x) exists. Then w~(x) — w~(x°)
u(x) — u(x°)
w+(x) — w+(x°)
Letting x tend to z° along the normal direction of x° gives dw~ dV
du
dw+ 0u
(10.4.2)
288
Elliptic and Parabolic Equations
where u is the unit normal vector inward to <90. If w±(x) -Cl/loiO <
dw dV
dw+ dv
p0'
satisfies
< C|/| 0 i n
(10.4.3)
with the constant C independent of x°, then from (10.4.2) we have du
Since u dn obtain
< C|/| 0 ; n.
0 implies that all the tangential derivatives of u are zero, we
|Vu|
By virtue of the maximum principle, to get (10.4.1), it suffices to require - A w " < -Au 8D
in D,
<W+
w
< -Aw+,
dD
(10.4.4) (10.4.5)
dD
We will seek such a function from the family of functions of the from W+(x)
= A (
r - •:
1
;
r
|
where y is the center of the exterior ball at the point a;0 with uniform radius r, and A > 0 is to be specified below. For any A > 0, we have u\ < w+ an \an Since
- A u ^\x-y\ . ^ . U~ n+l
A{n l)
'
(r + d i a m n ) n + 1 '
if we choose A such that A{n-l) ^IJ.I (r + d i a m f t ) ^ 1 " l 7 | 0 ; " '
A>
(r + d i a m ^ ) ^ 1 — |/| 0 i n,
Fixed Point
289
Method
then we obtain -Aw+
> |/| 0 i n > / = - A « ,
in SI.
So, we choose (r + diamfi)" + 1 I/loinn-1
(10-4.6)
With such a choice of A, we have, moreover, dw+ 9?
A(n - 1) ir* it
(r + d i a m n ) " + 1 . ,. ~~ —|/|o;n, fit
which shows that w+(x) satisfies (10.4.3) too. Similarly, if we choose ~(x) — ** , *• I
1
,
\j.n-\
,
1
,
,
|x_^|n-l
where A is the constant in (10.4.6), then w~(x) satisfies (10.4.4), (10.4.5) and also (10.4.3) with CI in place of D. D Remark 10.4.1 The above method can be used to derive the boundary Holder's estimate. In fact, it follows from (10.4-1) that \u(x) - u(x°)\ < lur^x) - w±(x°)\ 10.5
< [ u r ^ l x - x°\a,
x e SI.
Boundary Holder's Estimate and Boundary Gradient Estimate
In the previous section, we have used the barrier function technique to establish the boundary estimate for Poisson's equation. Such an important technique can also be applied to the general quasilinear equations with divergence structure. Now, we consider problem (10.1.9), (10.1.2). First establish the boundary Holder's estimate. Theorem 10.5.1 Let SI C Rn be a bounded domain with uniform exterior ball property. Assume that u £ C2'a(Sl) is a solution of problem (10.1.9), (10.1.2). Then for any x° € dSl and x1 £ SI, we have K x 1 ) - u(x°)\ < Qx1 - x T / ( a + 1 ) ( M a ; a n + 1), where the constantC depends only onn, m, 7, |/|o;fi) Mo ; n, a> diamfi and the uniform radius p of the exterior ball of SI.
290
Elliptic and Parabolic
Proof.
Equations
We merely prove uix1) - u(x°) < Qx1 - z°p /(a+1) (M<*;9fi + 1);
(10-5.1)
another part of the theorem can be proved similarly. Without loss of generality, we make the following two assumptions: i) Assume l a ; 1 - ^ 0 ! =dist(x 1 ,dft). Otherwise, we may choose a point y £ dD, such that \xl —y\= dist(x 1 , dfl). In this case, .1
y \ <
l x
l
™0,
\x — y\ S \x — x \ , 1 \y -x°\<\y-x \ + Ix1 - a;01 < 21a:1
y.0\
If we have proved ^a; 1 ) - u(y) < C\xx - y\a^a+l)([u]a]9Q
+ 1),
then u(x*) — u(a;°) =u(x1) — u(y) + u(y) — u(a;0)
x°\a[u}am
+ l).
ii) Assume \xl — x°\ < p. This is because the constant C in (10.5.1) is allowed to depend on p, if Ix1 — a;°| > p, then the desired conclusion can be derived directly from |u|o;n < C. The basic idea of the proof of (10.5.1) is similar to the one for Poisson's equation in §10.4, namely, to construct a barrier function w(x) such that w(x°) = u(x°) and u(x) < w{x),
xeD,
(10.5.2)
where D is the set of some neighborhood of a;0 intersecting with Cl. According to the maximum principle, to get (10.5.2), it suffices to construct a second order elliptic operator L such that Lu < Lw, u
(10.5.3) (10.5.4)
< w dD
in D,
dD
Fixed Point
291
Method
We expect to seek a suitable barrier function from the family of functions of the form w(x) = tp(\x — y\ — r),
(10.5.5)
where r € (0, p] is a constant to be specified, y is the center of the exterior ball at the point x° with radius r, tp is a function to be determined. We choose D = {x £ K n ; \x - y\ < r + 6} n fi with 5 > 0 to be determined. The proof of (10.5.1) will be completed in the following steps. S t e p 1 Construction of the operator L. Rewrite equation (10.1.9) as - o ( u ) A u - a'(u)|Vu| 2 + b(x, u) = 0,
(10.5.6)
where b(x,u) = b(u) — f(x). Owing to |u|o;n < C, there exist constants Mo, Mi > 1 depending only on m, 7, |/|o ; n and |u|o-,n, such that 1 < a(x) < fio,
(10.5.7)
2
(10.5.8)
-a(x)Au
< /xi(|Vu| + 1),
where a(x) = a(u(x)),
x e fi.
Define Lv =
-5(X)AD
- m(\Vv\2
+ 1).
Then from (10.5.6) and (10.5.8) we have Lu < 0,
in D.
(10.5.9)
S t e p 2 Construction of the function tp. A direct calculation shows that dw_ _Xj-yj
,
dxi \x — y\ 2 d w _ (xi - Vi)2,,, ( 1 dx2 \x-y\2 \\x-y\ . ,,, n — 1 ,, AW =1p" + rip', \x-y\
_ (xj-yi)2\ \x - y\3
,
292
Elliptic and Parabolic
Equations
\Vw\2 =(ip')2Thus Lw = - a ( x ) V " - a(x)^——tf - ^[{ip')2 + 1]. If we require ip' > 0,
V" < 0,
(10.5.10)
then by (10.5.7) Lw
> _ ^ _ ("-^)/*y _ m [ ( v / ) 2 +
x]
l = -(^) 2(V>') f 7 4 + l !r-V-' ^ £ +m+( V M O 2
2
V
with r to be determined. If, in addition, we require V / > ^ ^ Mir
+ l,
(10.5.11)
then
Therefore, in order that w satisfies (10.5.3), since (10.5.9) holds, we only need Lw > 0, which is valid if we require tp to satisfy ip" + 2li1(ip')2 < 0 .
(10.5.12)
Summing up, in order to get (10.5.3), it suffices to construct a function ip(d) on [0,5] satisfying (10.5.10), (10.5.11) and (10.5.12). It is not difficult to check that, for any k > 0, the function V»(d) = ^ - l n ( l + Jfcd), 0
k
* 2fj,i(l + kd) ~ 2fiiS
kS
(1 + kS)'
Fixed Point Method
we have ib' >
293
-, if 4/Xi<5 k>\.
(10.5.14) o
If we choose 0
< 5^
777 TT"^: v (10.5.15) 4[(n - l)/x0 + Hir] then xp satisfies (10.5.11). Therefore, if we choose r e {0,p}, 5 satisfying (10.5.15) and k satisfying (10.5.14) in turn, then the function V given by (10.5.13) satisfies (10.5.10), (10.5.11) and (10.5.12), and hence the function w given by (10.5.5) satisfies (10.5.3). Step 3 Construction of the required barrier function. The function w(x) = ip(d(x)) = -— ln(l + kd(x)), 2/xi
d(x) = \x — y\ — r
constructed above may not satisfy (10.5.4). In order to get a function satisfying both (10.5.3) and (10.5.4), we consider w(x) = w(x) + u(x°) + (3r) a [u] a ; a n . Since Lw — Lw > 0, we have Lu < Lw,
in D.
(10.5.16)
Set v(x) = w(x) - u(x) = w(x) - (u(x) - u(x0)) +
(3r)a\u]a.dn.
If we require 5 < r, then \x-x°\
< \x-y\
+ \y-x°\
< 3r,
Mx&D.
Hence v{x) > -\u{x) - u{x0)] + (3r)a[u]a;dn
> 0,
Vz G OQ D D.
In order that v(x) > 0 in 3D n Vt, it suffices to require w i.e.
4>(5) = ^-ln(l
+
k5)>2\u\0.n
^
> 2|u|o ; n,
294
Elliptic and Parabolic
Equations
or
Therefore, if S satisfies (10.5.15) (and hence 5 < r) and
fc>i(e4"l|ul°'n-lV then v
(10.5.17)
> 0 namely dD
dD
(10.5.18)
< W dD
By virtue of (10.5.16) and (10.5.18), we have ' Lv = -a(x)(Aw
- Au) - /xi(|Vw| 2 - |Vu| 2 )
-a(x)Av - fn\7(w + u)-Vv>0, dD
in D,
>0.
The maximum principle for linear equations then yields v(x) > 0,
x £ D,
that is u(x)-u(x°)
< J - ln(l+k(\x-y\-r))
+ {3r)a[u)a,dn,
Vz € £>. (10.5.19)
Step 4 Establishing Holder's estimate. For fixed x1 € fi, the choice of D depends on r and 5 in the above discussion. We will further see in the following argument, that 5 may depend on r, while r depends on xl. Hence D depends on x1. However, at the end of the proof we will prove that under the condition la; 1 -a; 0 ] < C(p)
(10.5.20)
with some constant C(p) depending only on p, there holds x1 € D. So, it suffices to prove (10.5.1) in the case x1 £ D. This is because, if xi £ D, then la;1 — a;°| > C(p), and since the constant in the right hand side of (10.5.1) is allowed to depend on p and |u|o ; n, (10.5.1) holds clearly. Let xi £ D and take x = x1 in (10.5.19). Since we have assumed that 1 jx — a:°|= dist(a; 1 ,9n) at the beginning of the proof, \x1—y\—r = ja;1—x°|,
Fixed Point
295
Method
and hence uOc1) - u(x°) < J - m(l + fcla;1 - x°|) + (3r)a[u}a.,an < ^ - | a ; 1 - x 0 | + (3r)o[«]a;en. If we choose jb = L4/nMo i n j o
(10.5.21)
then uix1) - u{x°) <^\xxd
x°\ +
(3r)a{u]a,an.
If, in addition, 6 = kir,
(10.5.22)
then the above inequality becomes uix1) - u(x°) < -^{x1
- x°\ + ( 3 r ) > ] Q ; a n .
Furthermore, if r = k2\xl -x°\l/{a+1\
(10.5.23)
then we may further obtain uix1) - u(x°) K-^-lx1K1K2
x°\a^a+1\[u}a.tdn
+ 1),
which is just the desired conclusion (10.5.1). Now, we first choose k
296
Elliptic and Parabolic
Equations
Thus la;1 - x°| < (fcifc 2 ) (a+1)/a implies la;1 - a;°| < S. So, if we choose C(p) = (k1k2)(a+1)/a in (10.5.20), then the condition (10.5.20) implies x1 € D. The proof of (10.5.1) is complete. • Similarly, we may establish the boundary gradient estimate. Theorem 10.5.2 Let D, c R™ be a bounded domain with uniform exterior ball property. Assume that u £ C2>a(Q,) is a solution of problem (10.1.9), (10.1.10). Then du
where v is the unit normal vector outward to dQ,, C depends only on n, m, 7) |/|o;fi, distfi and the uniform radius of the exterior ball of CI. Proof.
By (10.5.19), we have
u(x) - u(x°) < - ! - ln(l + k(\x -y\Since u
r)) + (3r) a [u] a ; an,
Vx € D.
= 0 implies [w^an = 0, we see that an
u(a;) - u(x°) < -— ln(l + k(\x -y\2/Xi
r)) < C(\x -y\-
r),
Va; G D.
Similarly we may estimate the lower bound of u{x) — u(xo) to obtain an opposite inequality, and hence u{x) — u(x°)
\x-y\ \x — x01
'
VxeD.
Letting x tend to x° along the normal direction at the point x°, and noticing that for such x, \x — y\ — r = \x — x°\, we immediately get the conclusion of the theorem. •
10.6
Global Gradient Estimate
In this section, we apply the Bernstein approach to get the global gradient estimate. To present the main idea of this approach clearly, we first discuss Poisson's equation.
297
Fixed Point Method
Theorem 10.6.1 Let fi C R n be a bounded domain with uniform exterior ball property. Assume that u G C2'a(Q) satisfies —Au = f inQ,
u
an
= 0.
Then sup |V«| < C(|/| 0 ; n + |V/| 0 i n + |u| 0; n), n where C depends only on n and the uniform radius of the exterior ball of
n. Proof.
The main idea is to apply the sign rule to the function w(x) = | V u | 2 + u 2 ,
xefi
to estimate its maximum. This is the so called Bernstein approach. Since u G C 2 , a (fi), there exists x° G Q, such that w(x°) = maxra. If n x° G dSl, then due to the boundary gradient estimate, the desired conclusion is obviously valid. So, without loss of generality, we assume that x° G fi. Then Aw(x°) < 0. A direct calculation shows that -Aw
= -A(J2(Diu)2+uA n
n
(Diju)2
= - 2 ^2 DiuDiAu -2^2 n
~
2uAu
- 2|Vu| 2
n
=2 Y, DiuDif -2^2
(Dijuf
+ 2uf - 2|Vu| 2 .
Thus |Vu(z°)| 2 n
< ^2 Diu(x0)Dif(x0)
+
u(x°)f(x°)
t=i
<J|V M (x°)| 2 + i | V / ( z ° ) | 2 + \u\x°)
+
\f\x°)
and hence \Vu(x°)\2
< |V/(z°)| 2 + u2(x°) +
f2(x°).
298
Elliptic and Parabolic
Equations
Therefore, for any x G Q, we have |Vu(:r)| 2 <w(x)
<w(x°) =\Vu(x°)\2+u2(x°) <\Vf(x°)\2
2u2(x°)+f2(x0)
+
<|V/|g.n + 2Hg.n + |/|g.n, from which we get the conclusion of the theorem.
•
Now, we turn to problem (10.1.9), (10.1.10). Theorem 10.6.2 Let Cl c l ™ be a bounded domain with uniform exterior ball property. Assume that u € C2'a(Q.) is a solution of problem (10.1.9), (10.1.10). Then sup|Vu| < C, where C depends only on n, m, 7, |/|o ; n, |w|Q;n, distfi and the uniform radius of the exterior ball o/fi. Proof. Let
= 0.
Integrating the first term by parts, we further have - [ Dk(a(u)DiU)Diipdx Jn
+ f (b(u) - f{x))Dk
i.e.
/ a{u)DikuDnpdx Jn
+ [ f^D^dx Jn
= 0,
Vy> G C£°(fi),
(10.6.1)
where fl = a'(u)DkuDiU - 5ik(b(u) - / ) ,
in ft.
Since C$°(Q.) is dense in H&(Q), (10.6.1) is valid for any (p G H^(Cl). By the structure conditions on a(u) and b(u), the maximum estimate on u and the boundedness of / , we get \fl\
+ \Du\2),
in ft.
Fixed Point Method
299
Let v = |Du| 2 . By virtue of u e C2'a{Vt), there exists x° € fi, such that N = y/v(x°) = max|Dw|. If a;0 G dil, then due to the boundary gradient estimate, the desired conclusion is obviously valid. So, without loss of generality, we may assume that i ° e f i and N > 1. Let R = I/TV and £ be a cut-off function on BR(x°) satisfying <;eCZ°(BR(x0)),
C(x°) = l,
0
|Z>C(z)l < I = 2JV.
Let 0 < i p £ Co°(fi). Choose Q2ipDku as the test function in (10.6.1), and sum with respect to k, we have / (2(pa(u)DikuDikudx
Jn
+ J (^a{u)DiV
+ /
Jn
C,2(pfkDikudx
+ flkDku^j
(2CPAC
+ C2D^)dx
= 0.
Applying the structure conditions of a(u) and setting w = (2v, we obtain / £2
+ /JD fc u)CAC) V>dx DiVdx
< 0.
Thus, by Cauchy's inequality with e, we have / (a(u)DiW - 2Ca(u)vDiC + 2C?PkDku) Dupdx
V0 < ^ e C0°°(ft),
which shows that w is a weak subsolution of some linear equation in flu = BR(X°) n fl. According to the maximum principle for weak subsolutions (Theorem 4.1.2 of Chapter 4. Although in that theorem the conclusion is established for Poisson's equation, the similar proof is available to general elliptic equations with divergence structure), for p > n, we have supu; < suPw + C(\\f\\LP.{nR) + OR
dOR
\\ghHoR))\fln\1/n-1/p,
300
Elliptic and Parabolic
where p» = — — , g = {gi,g2,---
Equations
,9n),
Tl ~f~ P 9i
= 2Co(u)i;AC " 2C2fkDku,
in O,
(i = 1,2, • • • , n)
/ = C 2 | / j | 2 + |£>C|2(l + U?u|2),
in O.
Since
ll/llL-(nH)
<2Ua(u)vDiC\\LHnR)+2\\eflDku\\L^R) 2
+ CN\\1 + |Z*i| 2 |Up (njl )
\Du\2\\LHnR)
+
CN\\\Du\2\\LP{nR),
it follows that supw< sup w + c(lv + ft* dnr\BR(x°)
N1+n/p\\\Du\2\\LP.{nR)
+ Nn/p\\\Du\2\\LHaR)).
(10.6.2)
Now, we estimate ||Du|| i 2(fj R ). Let £ be a cut-off function on B2R(X°) relative to BR(x°), that is £ G C§°(B2ie(a;0)), £ = 1 in Bfi(x°), and 0 < £(z) < 1,
|£>£(z)| < ^ = C7V,
l £
B 2fl (cc 0 ).
Multiplying both sides of equation (10.1.9) by (,2(x)(u(x) - u(x0)), integrating over fi and then integrating by parts, we obtain /
a(u)Z2\Du\2dx - J
+2 I
a(u)£(u(x) - u(x°))Du • D£dx
a(u)£2(u(x) - u{x°))Du • vds
JdQ2R
+ f J Hop
(b(u) - f)Z2{u(x) - u{x°))dx = 0,
Fixed Point
301
Method
where v is the unit normal vector outward to 8Q.2R- Applying Cauchy's inequality with e and using |«(x) - u(x°)| < [u]a.n(2R)a,
Vx e Q2R,
we have l ^ l l ^ ( n „ ) <; //
e\Duf $2\Du\2dx
JQ2R,
+ NRn~1+a
+ Rn+a)
<
CN2-n-a.
Thus \\\Du\2\\L,.(na)
<
< CiV2-("+^%
N2^-W*\\DU\\%^R)
IH^| 2 |U P ( n K ) < N>W\\Du\\%>(nit)
< CiV2-("+^.
Substituting these into (10.6.2) gives N2 = w(x°) = sup w < sup w + C{\ + N2~a/P' nR anr\BR(x°)
+ JV 2 - Q / p ).
Using Young's inequality with e, we obtain N2 <
sup
w + C,
dClf)BR(x°)
which implies sup \Du\ =N n
with another constant C. 10.7
O
Holder's Estimate for a Linear Equation
To establish Holder's estimate for gradients of solutions of equation (10.1.9), we first investigate a special linear equation with divergence structure. 10.7.1
An iteration
lemma
We first introduce a useful iteration lemma. Lemma 10.7.1 Let <3>(p) be a nonnegative and nondecreasing function defined on [0, RQ] satisfying $(p) < A [ ( - 0 " + e] $(fl) + BR13,
\/0
302
Elliptic and Parabolic
Equations
where A, B > 0, 0 < j3 < a. Then there exist constants £0 > 0 and C > 0 depending only on A, a, (3, such that for any 0 < e < £o, there holds V0 < p < R <
*(/o) < C Proof.
RQ.
By assumption, for any r £ (0,1), $(TR)
< Ara{l
+ eT~a)$(R)
+ BR13,
\/0
R0.
Without loss of generality we may assume A > 1. First, choose a real number v £ (/?, a) and then choose r such that 2Ar Q = TV or
— {-^}c(M,. Finally, choose £o > 0 such that £oT _ a < 1 or
f
ln(2A) I
0 < £0 < exp -^ - a — i \ a-v
J
\.
For such selected v, r and £Q, when 0 < e < £o, we have <2ATa^(R)+BRf}
+ BR0,
V0
R0.
The remainder of the proof is completely similar to that of Lemma 6.2.1 of Chapter 6. • Remark 10.7.1 Chapter 6. 10.7.2
Morrey's
When e = 0, Lemma 10.7.1 reduces to Lemma 6.2.1 of
theorem
In Chapter 6, we have applied the Campanato spaces to describe the integral characteristic of Holder continuous functions (Theorem 6.1.1 of Chapter 6). Now, we introduce Morrey's theorem, which can also be used to describe the integral characteristic of Holder continuous functions. Morrey's Theorem Let p > 1, 0 < a < 1, and Q C Rn be a bounded domain with appropriately smooth boundary (such as dQ. € Cl>a). i) Ifu€ W^(Rn) and for any i £ l " and p> 0, there holds [ JBp(x)
\\7u(y)\Pdy
303
Fixed Point Method
thenueCa(Rn). ii) If u & W1,P(Q) and for any x G ft and 0 < p < diamft, there holds
\Vu(y)\pdy
f
where Qp{x) = Bp(x)C)Q, then u € C Q (ft). Proof. The proof of the above two conclusions are quite similar, and we only prove the second one. Let x G ft, 0 < p < diamft. From Poincare's inequality, we have f \u(y) - uxJpdy Jn„(x) which implies u G Cp'n+pa(fl), 6.1.1 of Chapter 6. 10.7.3
Holder's
< Cpp f \Vu{y)\pdy < Jn„(x)
Cpn+pa,
and so u G C Q (ft) according to Theorem •
estimate
Consider the linear equation -div(5(i)Vu) = div/,
i€l"+,
(10.7.1)
where a G C Q (R+), /*G L°°(R^,R n ) and a{x) > A > 0. Theorem 10.7.1 Let u G C 1 , Q (R + ) be a weak solution of equation (10.7.1), (3 G (0,1). Then for any bounded domain ft CC R™, we have M/3;fi < C [\f\o-Ml + lu|0;K^J , where the constant C depends only on n, A, 0, |a|Q;R» and ft. Proof.
Let 0
< idist(ft,<9R!^)
and fix x° G ft. First consider the equation -div(a(z°)Vu) = div/,
i£l"+.
(10.7.2)
Let u G C 1 , Q (R + ) be its weak solution. Without loss of generality, we may assume that u is smooth in R™. Decompose u into u = U\ + Ui with u\
304
Elliptic and Parabolic
Equations
and U2 satisfying -div(a(x°)Vui) = 0 , Ul
x £ BR,
dBR
and -div(a(x0)Vu2) = div/, U2
= 0,
dBt
where 0 < R < R0 and BR = BR(X°). equations, for u\, we have / JB„
x € BR,
\Wui\2dx
I
Prom Schauder's estimate for linear |Vui|2rfa;,
0 < p < R < RQ.
JBR
To estimate U2, we multiply the equation of U2 by U2, integrate the resulting relation over BR, integrate by parts and use Cauchy's inequality with e. Then we have a(x°) J
\Vu2\2dx = - J
J BR
f-Vu2dx
•'BR
<£- [ 1
\Vu2\2dx + ±[ Z£
JBR
\f\2dx, J BR
where s > 0 is an arbitrary constant. From this and d(x°) > X, it follows that \Wu2\2dx < C [
\f\2dx < C\f\20.RnRn.
JBR
Thus, for any 0 < p < R < Ro, we have /
\Vu\2dx<2 [
\'Vu1\2dx + 2 [
JB„
JBP
^ ( i Y y R J
yRJ
-
C
( R Y
\Wu2\2dx
JBp
f J BR
\Vu1\2dx + 2 f
2 \Wu 2\ dx
JBR
f \Vu\2dx + C [ \Wu2\2dx JBR JBR 2 J \Vu\ dx + C\f\2.RlRn 2+2/3
Fixed Point Method
305
Applying the iteration lemma (Lemma 6.2.1 of Chapter 6) we further derive that, for any 0 < p < R < RQ, JB
\Vu?dx
C\f\lMlRn-^
\Vu\Hx +
In particular, by setting R = RQ, we deduce that, for any 0 < p < RQ, [
\S7u\2dx
JBP
P
\R0
Now, we estimate /
\Vu\2dx + \f%Rn).
(10.7.3)
+
JBRO
J
\S7u\2dx. Let £ be a cut-off function on
B2R0
JBRO
i.e., £ <E C0X{B2R0),
relative to BRo,
0
£ = 1 in BRo
and
|V£(z)|<£,
x€B2Ro.
Choosing £2it as the test function in the definition of weak solutions of equation (10.7.2), we obtain a(x°)£ 2 Vu • Vudx + 2 I
J •> B2R0
a{x°)£uVu • V£<£r
J BlRQ
i2f- Vudx -2 f
=- J J B2RQ
fit/- V£dx.
J B2R0
Then, Cauchy's inequality with e yields
f
\Wu\2dx
•'BRQ
\f\2dx + J
\ J B2RQ
2 u dx
JIBB ;
2RO
(10.7.4)
Substituting this into (10.7.3) leads to j
\Vu\2dx
0
Therefore, Morrey's theorem gives [U]/3;BR0
< C [\f\o-Rl + Mo;R") .
Finally we adopt the method of solidifying coefficients to equation (10.7.1). For this purpose, we rewrite the equation as —div(a(a;°)Vu) = divg,
306
Elliptic and Parabolic Equations
where g{x) = (a(x) - a(x°))Vu(x)
+ f(x),
x e Q.
By the conclusion proved above, we see that for any 0 < p < R < RQ, \Vu\2dx < C (^Y
f J Bp
•'•'
\Vu\2dx + C f
J J BR
J
\g\2dx, BR
which, together with \g\2<2\(a(x)-a(x°))Vu\2
+ 2\f\2
<2[a] 2 . R ni? 2 a |Vu| 2 + 2 | / ] 2 ; R ? ,
Vx G BR,
leads to that for any 0 < p < R < Ro, \Vu\2dx \Vu\2dx + CR2a
J BR
R20a]
2
\Wu\2dx +
[
C\f\lRlRn +
J BR
2
R.n-2+2/3
J \Vu\ dx + C\f\ .Ml
|Vu|2da;. Then the above inequality can be rewritten as
Let $ (p) = / JB„
$(p) < C [ ( I ) " + R20a] *(R) + C\f\2.RnRn-2+20,
V0
Thus, Lemma 10.7.1 yields n-2+2/3
*(p) < C
(I)
n-2+2/3
*(#) + 1/loW
,
VO
provided that R0 > 0 is small enough. In particular, setting iZ = Ro we conclude that for any 0 < p < Ro, [
\Vu\2dx < Cpn~2^
JBP
( ~ ^ s \R0
I " JBRO
l V u l 2 ^ + l / l o «+) • (10-7.5)
Similar to the proof of (10.7.4), we may obtain
\Vu\2dx < C (|/jg.R„ + |«|g.R„) . «0
J
Fixed Point
307
Method
Substituting this into (10.7.5) yields \Vu\2dx < Cpn-2+2P
(1/lg.Rn + |«|g. R ») ,
/ .Bp(x°)
\
' +
0
x°eO.
+/
From the arbitrariness of x° £Q and by using Morrey's theorem, we deduce M/3;BH 0 /2(X°) ^
C
(l/loiRJ + Mo;R$) ,
Vx0 € fi.
Then, we may use the finite covering argument to complete the proof of the theorem. • Similarly, we may establish Holder's boundary estimate, and prove the following Theorem 10.7.2 (10.7.1) with u
Let u G C 1 , a ( R + ) be a weak solution of equation = 0 and/3e
(0,1). Then
MfrRJ < C f l/|0jR!f. + MojRIM i where C depends only on |a| a; Rn, A, n and {3.
10.8
H o l d e r ' s E s t i m a t e for G r a d i e n t s
In the previous sections, we have obtained Holder's estimate for solutions and the maximum estimate for gradients of solutions of the Dirichlet problem (10.1.9), (10.1.10). In this section, we further establish Holder's estimate for gradients of solutions. 10.8.1
Interior
Holder's
estimate
for gradients
of
solutions
T h e o r e m 10.8.1 Let u e C 2 ' a ( 0 ) be a solution of equation (10.1.9). Then for any il' CCft, we have [Dku]a.n> < C, fc = l,2, ••• ,n, where C depends only on n, m, 7, |/|o ; n, |w|a;n, |Vu|o ; n, fi' and Q,. Proof. Let ip € C£°(ft). Multiplying both sides of equation (10.1.9) by Dk
308
Elliptic and Parabolic
Equations
obtain - / Di{a{u)Diu)Dkipdx + / (b(u) - f{x))Dkipdx = 0. Jn Jn Integrate the first term on the left hand side by parts to get - / Dk(a(u)Diu)Diipdx+ Jn
/ (b(u) - f(x))Dktpdx Jn
= 0,
i.e.
/ a(u)DikuDi
Jn
+ f fkD^dx
Jn
= 0,
V
where fk = a'{u)DkuDiU - Sik(b(u) - / ) ,
in fi.
This shows that Dku G C 1 , a (fi) is a weak solution of the equation —div(a(x)Vv) = div/(a;),
x G Q,
where a{x) = a(u(x)) G C a (H), / = (/fe\ /fe2 • • • , / £ ) G L°°(fi). Therefore, by interior Holder's estimate obtained in §10.7, we immediately obtain the conclusion of the theorem. • 10.8.2
Boundary tions
Holder's
estimate
for gradients
of solu-
Theorem 10.8.2 Let dQ, G C2'a and u G C2>a(U) be a solution of problem (10.1.9), (10.1.10). Then for any x° G dQ., there exists R > 0, such that \Dku\a.QR
k = 1,2, ••• ,n,
where CIR = QnBfl(x°), C depends only on n, m, 7, |/|o ; n, |w|Q;n, |Vu|o ; n and Q.. Proof. Step Prom x° and a
We divide the proof into four steps. 1 Local flatting. 9Q G C2'a, for fixed a;0 G dfi, there exists a neighborhood U of C 2 , a invertible mapping Vt : U —> Bi(0), such that 9{Unfl)
= flftO) = {2/ G Bi(0); j/„ > 0},
y(undn) = dB+n{y;yn = o}.
Fixed Point
Method
309
Then problem (10.1.9), (10.1.10) reduces locally in U n Q to -Dj(aij(y,u)Diu) u
= f(y,u,Du),
y € B+(0),
=0,
(10.8.1) (10.8.2)
dB+r\{y;yn=0}
where Di = -^—, u(y) = u(^/~1(y)).
It is easily seen that the metrics in
x-space and in y-space are equivalent. Step 2 Holder's estimate for tangential derivatives. Let 1 < k < n. Prom (10.8.1) and (10.8.2), it can be proved that is a weak solution of the linear equation -Dj(aij(y,u(y))DiDku)
= divg(y),
y € Bf
with the boundary value condition Dku
= 0. dB+n{y;yn=0}
It is easy to verify |a^(y,u(y))| a ; B + ( 0 ) < C and |5| 0 . B + (0) < C. By the boundary Holder's estimate obtained in §10.7, we obtain [Dku]a.B+{0) < C, fc = l , 2 , - - - , n - l . Step 3 Holder's estimate for normal derivative. From the conclusion of Step 2, we have
^2
f
\Diju\2dy < CRn-2+2a,
V0 < R < 1,
,B
i+j<2n ' R
d2 where Da = ——-—. Using equation (10.8.1), we further have dyidyj
L
\Dnnu\2dy
< CRn~2+2a,
V0 < R < 1.
/ \DDnu\2dy Bt
< CRn-2+2a,
V0 < R < 1.
Therefore J
Morrey's theorem then yields [DnU]a,Bt < C. Step 4
Returning to the original coordinate.
310
Elliptic and Parabolic
Equations
Returning to the x coordinate, we obtain lDku}a]u
n n ^ C>
k=
l,2,---,n,
from which we get the conclusion of the theorem. 10.8.3
Global Holder's
estimate
for gradients
• of
solutions
Combining the interior and the boundary Holder's estimates for gradients (Theorem 10.8.1 and Theorem 10.8.2), and using the finite covering argument, we obtain the following Theorem 10.8.3 Let dfl G C2
k = 1,2, ••• ,n,
where C depends only on n, m, 7, |/|o ; n, |w| a; n, |Vu|o ; n and£l. 10.9
Solvability of More General Quasilinear Equations
In the previous discussion, we have investigated the solvability of quasilinear elliptic equations (10.1.1) with structure conditions (10.1.3)-(10.1.8) and quasilinear parabolic equations (10.1.15) with structure conditions (10.1.17)-(10.1.20). However, there are many important quasilinear equations which do not satisfy such kind of structure conditions, for example, the quasilinear elliptic equation -div((|Vu| 2 + l f / ^ V u ) + c{x)u = f(x),
xeQ,
(10.9.1)
and the quasilinear parabolic equation ^
- div((|Vu| 2 + l ) " / 2 " 1 Vu) + c(x, t)u = f(x, t),
(x, t) G QT, (10.9.2)
where p > 1. In this section, we will illustrate the solvability of a class of more general quasilinear equations without proof. 10.9.1
Solvability tions
of more general quasilinear
elliptic
equa-
Consider the following Dirichlet problem for quasilinear elliptic equations - diva(x,u, Vu) + b{x,u, Vu) = 0,
x G Q,
(10.9.3)
Fixed Point
Method
311
(10.9.4)
an
where a = (01,02,-•• , a n ) , fl C R n is a bounded domain. Assume that a(x, z,rf), b(x, Z,TJ) satisfy the following structure conditions:
A(izD(i+\r,r2)\e < f ^ & < A(|Z|)(I+\vr 2 m 2 , ^ e (10.9.5) \a,i(x,z,0)\ dai
dz da dxi
(10.9.6)
+ |a i |
(10.9.7)
+ \b\
(10.9.8)
-b(x, z,r])sgnz < b0(\r)\p
1
(10.9.9)
+ h(x)),
where i,j = 1,2, • • • ,n, p > 1, A(s), A(s) are positive continuous functions on [0,+00), fi(s) is a nondecreasing function on [0,+00), g G Lq(Q), q > n/(p — 1), h £ Lq* (fi), q* = nq/(n + q), 60 is a positive constant. Equation (10.9.1) satisfies all of the above structure conditions provided c,f G C a ( 0 ) a n d c > 0 . Theorem 10.9.1 Let 0 < a < 1, d£l £ C2'a, a £ C 1 , a , b £ Ca,
C2'a(ty. 10.9.2
Solvability equations
of more
general
quasilinear
parabolic
Consider the following first initial-boundary value problem for quasilinear parabolic equations du - diva(x, i, it, Vzz) + 6(1, £, u, Vu) = 0, ~dt u = ip,
(x,t) G QT,
(10.9.10) (10.9.11)
dpQT
where a = (ai, 02, • • • ,a„), QT = O, x (0,T), Q c R" is a bounded domain, T > 0. Assume that a(x,t,z,r]), b(x,t,z,rj) satisfy the following structure
312
Elliptic and Parabolic
Equations
conditions:
\(\z\)(i+\vr2m2
den
< ^-tej < A(|ZD(I+\ri\ p - 2 m 2 , e e Rn, (10.9.12) 2
26(x,i,2,0) < 6 i | z | + 62, da< dai <*(W,H)(i + Wp), + |Oi|)(l + M) + dz dxj 96 P dr)j (I + M) + | 6 | < M ( N ) ( I + N ) , 96 db P+1 9xi
-^(i
+ N)<*(N,M)(i + M
)
(10.9.13) (10.9.14) (10.9.15) (10.9.16)
where i, j = 1,2, • • • , n, p > 1, A(s), A(s) are positive continuous functions on [0,+oo), 61,62 are positive constants, p,(s) is a nondecreasing function on [0, +00), * ( r , p) is a continuous function on [0, +00) x [0, +00) such that for any p £ [0, +00), \t(•, p) is nondecreasing on [0, +00), and as p —+ +00, $>(T,P) locally uniformly converges to 0 with respect to r . Equation (10.9.2) satisfies all of the above structure conditions provided that c,f £Ca'a'2{JQT). Theorem 10.9.2 Let 0 < a < 1, dfl £ C2>a, a £ C 1 , a , 6 £ C1,
diva(x, t,
when x £ 9 0 , t = 0.
Assume that equation (10.9.10) satisfies the structure conditions (10.9.12)(10.9.16). Then problem (10.9.10), (10.9.11) admits a solution u £ C2+a,l+a/2(Qr)_
Exercises 1. 2. 3. 4. 5. 6. 7.
Prove Theorem 10.1.2 ii). Complete the proof of Theorem 10.3.2. Prove Theorem 10.3.4. Prove Theorem 10.7.2. Prove Theorem 10.8.3. Establish the solvability of quasilinear parabolic equations. Prove Theorems 10.9.1 and 10.9.2.
Chapter 11
Topological Degree Method
The concept of topological degree was first introduced by L. E. J. Brouwer for continuous mapping in finite dimensional space. It was J. Leray and J. Schauder who generalized such a concept to the completely continuous fields in Banach spaces, and developed a complete theory of topological degree, which has been applied extensively to the investigation of partial differential equations and integral equations. In this chapter, we will illustrate the application of the topological degree method by a heat equation with strongly nonlinear source.
11.1
Topological Degree
In this section, we introduce the definition of topological degree and present its basic properties without proof. For the detailed theory we refer to [Zhong, Fan and Chen (1998)]. 11.1.1
Brouwer
degree
Let f2 be an open set of R™ and / be a mapping from fi to R n . Roughly speaking, the Brouwer degree is an integer valued function related to / and Q.. We first define it for / £ C ^ f y R " ) and then extend to / e C(fi;R n ). Assume that / = ( Z 1 , / 2 , - - - , / " ) £ C ^ t y R " ) . Then for any x £ ffc, the Frechet derivative operator of / at x, f'(x) : R™ —> R™ is a linear operator, and
\
Definition 11.1.1
UX
J
/
nxn
x £ 0. is called a regular point of / , if the Frechet 313
314
Elliptic and Parabolic
Equations
derivative operator f'(x) is of full rank; otherwise, we call x a critical point of / . y G R" is called a critical value of / , if there exists a critical point x G Cl of / such that f{x) = y; otherwise, we call y a regular value of / . Theorem 11.1.1 Let Cl be an open set o/R", f G C ^ f y R " ) . Then the Lebesgue measure of the set of critical values of f in R™ is equal to zero. Definition 11.1.2 Let fibea bounded open set of R", / G C 2 (fi;R"), p G M.n\f(dCl). We define the Brouwer degree deg(/, Cl,p) of the mapping / in Cl at the point p in the following way: i) If p is a regular value of / , then set deg(f,Cl,p) =
] T sgnJ/(x), xef-1(P)
where Jj{x) is the determinant of f'(x); ii) If p is a critical value of / , then choose a regular value p\ of / with Ibi — p\\ < dist(p, f(dCl)) and set deg(/,fi,p) = deg(/,fi,pi). It can be proved that deg(/, Cl,p) is independent of the choice of p\. Definition 11.1.3 Let Cl be a bounded open set of R n , / G C(f2;R n ), p G R n \/(0I2). Choose A G C 2 (fi;R") such that sup H / f r ) - / i ( a O | | < d i s t ( p , / ( 0 n ) ) . Define the Brouwer degree of / in CI at the point p by deg(/,fi,p) = deg(/i,fi,p). It can be proved that deg(/, Cl,p) is independent of the choice of f\. Some basic properties of the Brouwer degree are involved in the following theorem. Theorem 11.1.2 Let Cl be a bounded open set o/R", / G C(H; R"), p G M.n\f(dCl). The Brouwer degree deg(f,Cl,p) has the following properties: i) (Normality) 1,
pefl,
0,
p i fi,
deg(id, Cl,p) = where id denotes the identity map;
Topological Degree Method
315
ii) (Domain Additivity) Iffli, fi2 o-re two open subsets of tt with Cli D H2 = 0 and p 0 /(H\(Qi U fi2)), then deg(/,fi,p) = deg(/,fii,p) + deg(/,n 2 ,p); raj (Invariance of Homotopy) Let H : fi x [0,1] —> M" 6e a continuous mapping and denote ht(x) = H(x,t). Assume the mapping p : [0,1] —> 1 " is continuous andp(t) £ ht(dfl) for each t 6 [0,1]. Then deg(ht,fl,p(t)) is independent oft. Based on these basic properties, we may derive a series of important properties of the degree. For example, we have the following theorem. Theorem 11.1.3 (Kronecker's Existence Theorem) Let Q. be a bounded open set of Rn, f £ C(n;M n ), p e R n \/(<9fi). If p £ /(H), then deg(f,Q,p) = 0, and so, if deg(f,Q,p) / 0, then the equation f(x) = p must have at least one solution in Q. 11.1.2
Leray-Schauder
degree
Since many problems in analysis are referred to infinite dimensional space, it is natural to extend the Brouwer degree theory to the infinite dimensional case. However, owing to the lack of compactness of the unit ball in the infinite dimensional space, one cannot establish the degree theory for general continuous mappings. It was Leray and Schauder who found an important class of mappings in the investigation of partial differential equations and integral equations, that is the completely continuous perturbations of the identity mappings (also called the compact continuous fields), and applied the method of finite dimensional approximation to establish the degree theory for this class of mappings. Definition 11.1.4 Let X, Y be two normed linear spaces, D C X, A mapping F : D —> Y is said to be compact, if for any bounded set S C D, F(S) is a compact set in Y. If, in addition, the mapping F is continuous, then we call F a completely continuous mapping or a compact continuous mapping. Theorem 11.1.4 Let X, Y be two normed linear spaces, M be a bounded closed subset of X. Assume that the mapping F : M —> Y is continuous. If the mapping F is completely continuous, then for any e > 0, there exists a bounded continuous mapping F^ : M —» Yk with finite dimensional range
316
Elliptic and Parabolic
Equations
such that sap \\F(x) - Fk(x)\\ < e, where Yk CY is a finite dimensional space. Furthermore, if Y is complete, then the above condition is also sufficient. Definition 11.1.5 Let X be a real normed linear space, D c X. If the mapping F : D —> X is completely continuous, then / = id - F is called a completely continuous field in D, or a compact continuous field in D. Let X be a real normed linear space, fl be a bounded open set in X, F : tt —> X be a completely continuous mapping, / = id — F be a completely continuous field, p e X\f(dCl). By Theorem 11.1.4, there exist a finite dimensional subspace Xk c X, pk £ Xk and a bounded continuous mapping Fk : Q —» Xk, such that ||p - pfc|| + sup \\F(x) - Fk(x)\\ < dist(p, f(dQ)). x£fl
Denote fife = X fe nft, fk = i d - F k . Then /fc € C(f2fc,Xfe), pk e and hence the Brouwer degree deg(/fc,fifc,pfc) is well-defined.
Xk\fk(dnk),
Definition 11.1.6 Define the Leray-Schauder degree of the completely continuous field / in O at the point p by deg(/,fi,p)
=deg(fk,nk,pk).
It can be shown that deg(/, fi, p) is independent of the choice of Xk, pk and Fk. Since the Leray-Schauder degree is obtained by the approximation of the Brouwer degree, it can be proved that most properties of the Brouwer degree are retained. Theorem 11.1.5 Let Q be a bounded open subset of X which is a real normed linear space, f = id — F be a completely continuous field on CI, p € X\f(d£l). Then the Leray-Schauder degree deg(/, 0,p) has the following properties: i) (Normality)
pen,
i, deg(id,fi,p) =
o,
P
i U-,
Topological Degree Method
02
317
ii) (Domain Additivity) Iftoi, fi2 are two open subsets of to with to\ n = 0 andpg /(JT\(fii U to2)), then deg(/,n,p) = deg(/,Qi,p) + deg(/,fl 2 ,p);
Hi) (Invariance of Compact Homotopy) Let H : to x [0,1] —» X be a completely continuous mapping and denote ht(x) = x — H(x,t). Assume the mapping p : [0,1] —> X is continuous and p(t) £ ht(dto) for every t G [0,1]. Then deg(ht,to,p(t)) is independent oft. Theorem 11.1.6 (Kronecker's Existence Theorem) Let X be a real normed linear space, to be a bounded open subset of X, and f = id — F be a completely continuous field defined onto, p £ X\f(dCl). If p $ /(fi), then deg(/, £l,p) = 0. Thus, i/deg(/,fi,p) =/= 0, then the equation f(x) = p admits at least one solution in fi.
11.2
Existence of a Heat Equation with Strong Nonlinear Source
As an example in applications of the topological degree method, let us consider the heat equation with the strong nonlinear source ^.-Au=|ti|P,
(x,t)£QT
(11.2.1)
with the initial-boundary value condition u(x,t) = >p, {x,t) € dpQT,
(11.2.2)
where p > 1, QT = to x (0, T), Q, is a bounded domain in R" with dto G C2'a, a€ (0,1), T > 0 . If the right hand side of (11.2.1) is a function f(x,t) independent of u, namely, the equation ^ . - A « = /,
(x,t)€QT
(11-2.3)
is considered, then by the theory of classical solutions for the nonhomogeneous heat equation, the first initial-boundary value problem (11.2.3), (11.2.2) admits a unique solution u G C2+a'1+a/2(QT) and \u\2+a,l+a/2;QT
^
C
0 (\f\a,a/2;QT
+ \f\2+a,l+a/2;QT)
,
(11.2.4)
318
Elliptic and Parabolic
Equations
provided / G Ca>a'2{QT),
^Ca'a'2(QT),
x [0,1] (/, a) >-m,
where u £ C2+a'1+a^2(QT)
is the solution of the problem
UXl
— -Au
= crf,
(x,t)eQT,
u(x,t)=(p,
(x,t)edpQT-
We proceed to show that F is a completely continuous mapping. L e m m a 11.2.1
The mapping F is compact.
Proof. Assume that {fk}f=1 C Ca'a/2(QT), exists a constant M > 0, such that \fk\a,a/2;QT < M, Denote Uk = F(fk,ak), following problem
{o-k}f=1 C [0,1] and there
Vfc > 1.
that is uk G C2+a'1+a/2(QT)
is the solution of the
—— - Aufc = CTfc/fe, (a;, t) G Q r , uk(x,t)
= ip,
(x,t) £ dpQT-
By virtue of the classical theory, we have \Uk\2+a,l+a/2;QT
^ ^ 0 (l c r fe/fc|a,a/2;Q T +
\
with Co given in (11.2.4), which implies that { u f c } ^ is uniformly bounded in C2+a'1+a/2(QT). Therefore, there exists a convergent subsequence of {uk}kxL1 in Ca'a/2(QT), this means that the mapping F is compact. • L e m m a 11.2.2
The mapping F is continuous.
Proof._Assume that {fk}?=1 a 2 C ' " / ( Q T ) , a G [0,1], and
C C a - a / 2 ( Q T ) , K } £ ° = 1 C [0,1], / G
H m |/fc - f\a,a/2;QT
fc—>oo fc—»oo
= 0,
l i m <Jk = a.
319
Topological Degree Method
Denote uk = F(fk,o-k), u = F(f,a). By the definition of F, uk — u £ (j2+a,i+a/2^Q^ j s fae s o i u t i 0 n of the following problem — -Aw
= (<7fc/fc - af),
w(x,t) = 0,
(x, t) e QT, (x,t) e dpQr-
By the classical theory, we have |Ufc -
u\2+a,l+a/2;QT
C/U,C*/2;QT
(|/fc - f\a,a/2;QT
+ Wk - 0-||/U,a/2-,Q T ) + \°~k - ^ | | / | a , a / 2 ; Q T ) ,
where Co is the constant given in (11.2.4). Therefore l i m \uk - u\2+a,l
k—>oo
+ a/2;QT
= °>
which implies that lim \uk - 1iU,a/2;QT = 0.
k—»oo
Thus, the mapping F is continuous.
•
Combining Lemma 11.2.1 and Lemma 11.2.2, we see that the mapping F is completely continuous. T h e o r e m 11.2.1
Assume that (p G C2+a<1+a/2{QT)
and
M2+a,i+aAQT < ^ ( 2 ( p + l)C0)1/(1-p), where Co is the constant given in (11.2.4). Then problem (11.2.1), (11.2.2) admits at least one solution in C2+a'1+a^2{Q'r). Proof. Denote $(v) = \v\p. Since F is completely continuous and p > 1, it is easy to see that F ( $ ( - ) , 0 : Ca^2(QT)
x [0,1] -
Ca^2(QT)
is also completely continuous. According to the classical theory, solving problem (11.2.1), (11.2.2) in (j2+a,\+a/2(tQ^ j g equivalent to solving the equation u-F($(w),l) = 0
(11.2.5)
320
Elliptic and Parabolic
Equations
in Ca'a/2(QT). The latter will be solved by using the Leray-Schauder topological degree theory. To this purpose, we first choose R > 0 such that 0^(id-F($(-),a))(dBH(0)),
Vae[0,l],
(11.2.6)
where BR(0) is the ball of radius R centered at the origin in Ca>a/2(QT). If (11.2.6) holds, then by Theorem 11.1.6, in order to show that (11.2.5) has at least one solution in Ca'a^2(QT), we need only to show that deg(id - F($(-), 1), BR(0), 0) ± 0.
(11.2.7)
Furthermore, if (11.2.6) holds, then Theorem 11.1.5 in) yields deg(id - F($(-), 1), BR(0), 0) = deg(id - F($(•), 0), BR(0), 0). From the definition of F, it is seen that F($(-),0) : Ca'a/2(QT)
-> Ca>a'2{QT)
is a constant mapping, that is F{<5>{v),G)=u, where u € C2+a'l+a^2(QT)
Vu e
Ca'a/2(QT),
is the solution of the following problem
du — -Au
= 0,
u(x,t) = ip,
(x,t)eQT,
(11.2.8)
(x,t) e dpQT.
(11.2.9)
Consider the following mapping G(v,a)=v-au,
v£ Ca>a/2(QT),
a € [0,1].
If 0^G(dBR(Q),a),
VffG[0,l],
(11.2.10)
then applying Theorem 11.1.5 iii) and i) to G(v,a) yields deg(id - F($(-),0),B R (0),0) = deg(id,B R (0),0) = 1. Thus (11.2.7)holds. Consequently, if we can find R > 0 satisfying (11.2.6) and (11.2.10), then the proof is complete. We now show that if we choose R = (2(p +
l)C0)1/(1-p)
Topological Degree Method
321
with Co given in (11.2.4), then both (11.2.6) and (11.2.10) are satisfied. Assume that v G 8BR{0), i.e. v £ Ca'a^(QT) and \v\a,a/2;QT = R- F ° r any a S [0,1], the classical theory gives |F($(u),
In addition, I |"| P U,Q/2;QT =1 M" IO;QT + [ M P ]a,a/2;QT ^Mo ; Q T + P I W I O ^ 1 T H « . « / 2 ; Q T
^ R ,
thus |F($(u),CT)|2+a,1+a/2;QT < f l .
Therefore |F($(w),a)| a , a / 2 i Q T < fl, Vv e 0B«(O), Vcr e [0,1]. This shows F ( $ ( v ) , a ) ^ w,
W e 0B«(O), Vcr € [0,1]
and so (11.2.6) follows. Moreover, as the solution of problem (11.2.8), (11.2.9), the classical theory gives that u S C2+a'1+a/2(QT) and ma,a/2;QT
< \u\2+a,l+a/2;QT
^ Co\
Therefore \G{v,
=\V -
Vu\a,a/2;QT
>\v\a,a/2;QT
>R/2,
to
~
v\u\a,a/2;QT
e 0B fl (O), Va e [0,1],
which implies (11.2.10). The proof of Theorem 11.2.1 is complete.
•
322
Elliptic and Parabolic
Equations
Exercises 1. Consider the first initial-boundary value problem i ^ - A u [u(x,t)
= eu,
(x,t)£QT
=
=
Qx(0,T),
{x,t)edpQT,
where Q is a bounded domain in R n with dfi e C 2 - Q ,
f ^ - A u = K> [u(x,t)
= ip,
(x,t)£QT, (x,t)£dpQT,
where p > 1, QT = Q x (0, T), fl is a bounded domain in R" with dtt £ C2'a, a £ (0,1). Prove that for any
Chapter 12
Monotone Method
The method of supersolutions and subsolutions is a powerful tool in establishing existence results for differential equations. What is more, this method can also be applied to systems. The basic idea of this method is to use a supersolution or a subsolution as the initial iteration in a suitable iterative process, so that the resulting sequence of iterations is monotone and converges to a solution of the problem. The underlying monotone iterative scheme can also be used for the computation of numerical solutions when these equations are replaced by suitable finite difference equations. In this chapter, the method of supersolutions and subsolutions and its associated monotone iteration are introduced for a scalar heat equation and a system of coupled heat equations as two typical examples. Similar argument can be applied to the general parabolic equations and systems and also to elliptic equations and systems.
12.1
Monotone Method for Parabolic Problems
We consider the following nonlinear parabolic problem
— - A u = /(«), u(x,t)=g(x,i),
(z,t)eQr = (x,t)edpQT,
ftx(0,T),
(12.1.1) (12.1.2)
where fl C 1 " is a bounded domain with <9fi G C2
324
Elliptic and Parabolic
12.1.1
Definition
of supersolutions
Equations
and
Definition 12.1.1 A function u € C^iQr) solution of problem (12.1.1), (12.1.2), if ^-Au>f(u),
subsolutions n C{QT) is called a super-
(x,t)&QT,
u(x, t) > g(x, t),
(x, t) e dpQT.
Similarly, a function % £ C2'l{QT) n C(QT) is called a subsolution of problem (12.1.1), (12.1.2) if du,
-g-Au
(x,t)eQT,
&{x, t) < g(x, t),
(x, t) S dpQT.
For a supersolution u and a subsolution & of problem (12.1.1), (12.1.2), we say that the pair 'u, % are ordered if u(x,t)
>&(x, t),
(x,t) £ QT.
Definition 12.1.2 For any ordered supersolution and subsolution u, &, we define the sector (j/,u) as the functional interval (%,u) = {u e C(QT);&(x,t) 12.1.2
Iteration
< u(x,t)
and monotone
(x,t) s
QT}.
property
It is clear that every solution of problem (12.1.1), (12.1.2) in C2'1(QT) n C{QT) is a supersolution as well as a subsolution. Therefore, supersolutions and subsolutions exist unless the problem has no solution in C2,1(QT) n C(QT). To ensure the existence of a solution it is necessary to impose more condition on the reaction function / . A basic assumption is the following one-sided Lipschitz condition f(ui) - f(u2) > -c(ui - u2),
^,
(12.1.3)
where c is a constant and u, ^ are given ordered supersolution and subsolution. Clearly this condition is satisfied with c = 0 when / is monotone nondecreasing in E. In view of (12.1.3), the function F(u) = cu + f(u) is monotone nondecreasing in u for u € (%,ti).
Monotone
325
Method
Adding cu on both sides of equation (12.1.1) and choosing a suitable initial iteration u^ £ C2I1(QT)^C(QT), we construct a sequence {u^}'j£=0 successively from the iteration process duW dt
Au(fc) + cu(fe) = F(u{k~l)), u{k\x,t)
=g(x,t),
(s, t) £ QT,
(12.1.4)
(x,t) £ dpQT.
(12.1.5)
Since for each k > 1 the right side of (12.1.4) is known, the 1? theory and the maximum principle guarantee that the sequence {u^}^L0 is well denned. Prom the regularity of solutions of heat equations, u (l)
g Ca,a/2(QT^
u(k)
g C2+a,l+a/2(QTj
for
Jfc =
2
,3, ••• .
A natural choice of u^ is u^ = 'u and u^ = Q. Denote the sequences denned by (12.1.4), (12.1.5) with u(°> = u and «(°> = & by {uW}£L 0 and {u^}fc^o' an<^ refer to them as the upper sequence and the lower sequence of (12.1.4), (12.1.5), respectively. The following lemma presents the monotone property of these two sequences. Lemma 12.1.1 Let 'u, & be ordered supersolution and subsolution of problem (12.1.1), (12.1.2) and f satisfy (12.1.3). Then the sequences { u ^ } £ L 0 and {u^}j£Lo Possess the monotone property u(x, t) = M (0) (x, t) < u (fc) ( X) t) < u{k+1)(x, t)
{x,t)£~QT
(12.1.6)
for every k = 1,2, • • •. Proof.
Let
w(x,t) = u{0){x,t)-u{1)(x,t) Then w £ C2,1{QT)
H
C(QT)
=u(x,t)-u(-1)(x,t),
(x,t) £~QT.
is a solution of the problem
^ - A w + cw>F{u)-F(u)=0,
(x,t)£QT,
w(x, t) > g(x, t) - g(x, t) = 0,
(x, t) £ dvQT-
The maximum principle leads to w > 0 on QT, i.e. u{1){x,t)
< u{0)(x,t)
i{1)(x,t)
> u ( 0 ) ( x , t ) =%(x,t),
= u(x,t),
(x,t) £ QT.
Similarly {x,t) £Q T-
326
Elliptic and Parabolic
Equations
Let wil){x,t) Then w^ ——
e C2'\QT)
= u{1){x,t)-u(-1)(x,t),
(x,t)
£~QT.
n C(Q T ) satisfies
_ A ^ 1 ) + gwW = F ( u ) - F(%) > 0,
(x, t) e Q r ,
iu ( 1 ) (x,i) = s(z,t) - $ ( s , t ) = 0,
(x,t) e 9 P Q T .
Again, by the maximum principle, u/ 1 ) > 0 on QT, i.e. u ( 1 ) 0 M ) < u ( 1 ) (x,t),
(x,t) e Q r .
Then, we have u ( 0 ) ( M ) = ( %(x,t)
< u(x,t) =
u^(x,t),
(x,t) e QT. Suppose w ( f c _ 1 ) (^-0
< u^k-^(x,t),
(x,t)
eQT
for some k > 1. Then the function m'
l
»(i,t)=ii'il(i,()-5'wl(i,t),
(i,t)e5T
satisfies
5*
Aw(fc) + cw(k) = F(rf-k-1))
- F(uk) > 0,
w<>k) (x, t) = g(x, t) - g{x, t) = 0, The maximum principle implies that w^ u(k+1){x,t)
(a;, t) € QT, (x, t) e
dpQT.
> 0 on QT, i.e. (x,t)eQT.
Similar reasoning gives u{k+1)(x,t)
uik+1)(x,t)
Thus, by induction, the monotone property (12.1.6) follows.
(x,t)GQT. •
Monotone
12.1.3
Existence
327
Method
results
The relation (12.1.6) implies that the upper sequence {u^}^=0 is monotone nonincreasing and is bounded from below and that the lower sequence {uSk^}
= u(x,t),
k—»oo
lim u(k)(x,t)
= u(x,t),
(x,t) e~QT
(12.1.7)
fc—>oo
exist and satisfy u(x,t)
< u(x,t) < u(x,t)
(x, t) G QT.
We will show that both u and u are solutions of problem (12.1.1), (12.1.2). Furthermore, if there exists a constant c< c such that f(ui) - f{u2) < -c(ui ~u2),
&
(12.1.8)
then the solution is also unique in (j£,w). Theorem 12.1.1 Let'u, & be ordered supersolution and subsolution of problem (12.1.1), (12.1.2) and f satisfy (12.1.3). Then i) The upper sequence {u^}^L0 converges monotonically from above to a solution u and the lower sequence {yLk'}kxL0 converges monotonically from below to a solution u, and u(x,t)>u(x,t),
{x,t)£QT;
(12.1.9)
ii) Any solution u* G (j&,w) of problem (12.1.1), (12.1.2) satisfies u(x,t) < u*(x,t)
(x,t) G QT;
Hi) If, in addition, the condition (12.1.8) holds, then u = u and is the unique solution in (%,!<)• Proof. Let {u(fe)}£L0 b e e i t h e r {"(fe)}fcLo o r {u{k))T=o a n d « be u or u respectively. Since F is Holder continuous and monotone nondecreasing, the monotone convergence of {u^}^L0 to u implies that {F^^^^LQ converges to F(u) as k —> oo. As indicated above, we have u(l)
g Ca,a/2(QT^
„(fc) g
C
2 + a , l + a / 2 ( g T ) for fc = 2 , 3 , • • • .
Moreover, it follows from the maximum principle and Schauder's estimate that |u<*> | 2 + a , l + a / 2 ; Q T < C ( | f f l 2 + a , l + a / 2 ; Q T + l U
\O-,QT)'
fc
— 2, 3 , •
328
Elliptic and Parabolic
Equations
where C > 0 is a constant depending only on a, fl, T and / but independent of k. From the monotone property (12.1.6), {u^}'^L1 is uniformly bounded in C2+a<1+a>2{QT). Therefore, u £ C2+a'1+a^2(QT) is a solution of problem (12.1.1), (12.1.2). And (12.1.9) follows from the monotone property (12.1.6). If u* £ (%,tt) is a solution of problem (12.1.1), (12.1.2), then the functions u*, u are ordered supersolution and subsolution. Since the sequence {^fc)}fclo w ^ t n u ' ° ' = u* c o n s i s t s of the same function u* for every k, the above conclusion implies that u* > u. Similarly, by considering u, u* as ordered supersolution and subsolution, the same reasoning leads to u > u*. This proves ii). To prove hi), it suffices to show that u(x, t) < u(x, t),
(x, t) £ QT.
(12.1.10)
Indeed, the function w(x,t) — u(x, t) - u(x,t),
(x,t)eQT
satisfies —- - Aw = f(u) - f(u) > -cw,
(x, t) £ QT,
w(x, t) = g(x, t) - g(x, t) = 0,
(x, t) £ dpQT
and hence, by the maximum principle, w > 0 on QT and (12.1.10) follows immediately. • In the conditions (12.1.3) and (12.1.8) the constants c and c are not necessarily nonnegative. This is different from the case of elliptic problem. When / is a C1-function in (%,'u), we may take these constants as c = - min{/'(u(i, t)); u £ (%, u), (x, t) £ QT] and c = - m a x { / ' ( u ( x , t));u£
(%,u), (x,t) 6 QT}.
If / is Lipschitz continuous in (j^t/), namely there exists a constant K > 0 such that l/("i) - f(u2)\ < K\m -U2\,
ui,u2 £ (%,u).
Then we may take c = K and c= —K. This observation leads to Corollary 12.1.1 Letu, Q be ordered supersolution and subsolution of problem (12.1.1), (12.1.2) and f be a C 1 -function in (%,u). Then problem
Monotone
329
Method
(12.1.1), (12.1.2) has a unique solution in ( J J , U ) . Moreover this solution is the limit of the sequence defined by (12.1.4), (12.1.5) with either u^ = u or u^ = ^ . If f is Lipschitz continuous in (u,ti), the same conclusion holds. If both / and g are nonnegative functions, then the trivial function Q = 0 is a subsolution of problem (12.1.1), (12.1.2). Hence the existence of solutions is valid provided that there is a nonnegative supersolution. A sufficient condition is that for some constant p > 0, /(p)<0,
p>g(x,t),
(x,t)£dpQT.
(12.1.11)
This follows immediately from Definition 12.1.1 with u = p. By an application of Theorem 12.1.1 we have the following conclusion, which is quite useful in applications. Theorem 12.1.2 Let H be a nonnegative supersolution of problem (12.1.1), (12.1.2) and f be a C1-function in (0,u). If /(0)>0,
g(x,t)>0,
(x,t)edpQT,
then there exists a unique solution of problem (12.1.1), (12.1.2) in (0,11). If (12.1.11) holds for some constant p > 0, thenH = p is a nonnegative supersolution. To achieve the conclusion of Theorem 12.1.1, the existence of ordered supersolution and subsolution is necessary. In the following, we will show that under the conditions (12.1.3) and (12.1.8) any supersolution and subsolution of problem (12.1.1), (12.1.2) are ordered and u^k\ ySk^ are ordered supersolution and subsolution for each k = 1,2, • • •. Theorem 12.1.3 Let'u and u be a supersolution and a subsolution of problem (12.1.1), (12.1.2) respectively. Assume that f satisfies (12.1.3) and (12.1.8) for any U\ and U2 between)^ andU with u-2
(x,t)eQT.
Thus, 'it, ^ are ordered supersolution and subsolution of problem (12.1.1), (12.1.2). Moreover, vSk\ u^ are also ordered supersolution and subsolution for each k = 1,2, • • •. Proof.
Let c* = max{|c|, |c|},
w(x,t) ='u(x,t) — $i(x,t),
(x,t) £ QT,
330
Elliptic and Parabolic
Equations
where c and c are the constants in (12.1.3) and (12.1.8). Define c(x, t) = c*sgnw(x,t),
(x,t) G QT,
where sgn(-) is the sign function. Then dw — -Aw>
f(u) - /(%) >
-cw,
(x,t)
w{x, t) > g(x, t) - g(x, t) = 0,
eQT,
(x, t) G dpQT-
Since c is bounded on QT, the maximum principle implies w > 0 on QT, i.e.
u(x,t)
(x,t)GQT.
For each k = 1,2, • • •, by (12.1.4) and (12.1.3),
dt
Au
dt
Au< fe >
cu = -
(x,t)€QT, (fc)
F(M (fc_1) )
c(u(fe) - u ^ - 1 ) ) + /(u
+ /(«'(*h
(i,t)€Qr.
On the other hand, (12.1.5) and (12.1.6) imply u{k)(x,t)
> g{x,t),
u(k)(x,t)
u{k){x,t)
> u{k)(x,t),
(x,t) G dpQT
and
Hence u^k\ u^ 12.1.4
(x,t) G QT.
are ordered supersolution and subsolution.
Application
to more general parabolic
•
equations
The monotone method used above may be applied to the heat equations with more general reaction terms and even to uniformly parabolic equations of general form.
Monotone
Remark 12.1.1 du — -Au
331
Method
For the nonlinear parabolic problem = f(x,t,u),
u(x, t) = g(x, t),
(x,t)£QT,
(12.1.12)
(x, t) G dpQT,
(12.1.13)
we may apply the monotone method to get the same results as those for problem (12.1.1), (12.1.2). Here f G Ca'a^a(QT x R), and the conditions (12.1.3) and (12.1.8) are replaced by f(x, t, ui) - f(x, t, u2) > - c(ui - u2), (x,t) &QT, ^.
(12.1.14)
and f(x, t, ui) - f(x, t, u2) < - c(ui - u2), (x,t) £QT, &
(12.1.15)
respectively. Remark 12.1.2 The monotone method may also be applied to the uniformly parabolic equation of general form p,
n
n
— - ^2 aij(x,t)Diju+'^2bi(x,t)DiU+c(x,t)u S,J = 1
= f(x,t,u),
(x,t) G QT,
1=1
where aij,bi,c G Ca'a^2(QT), stants A, A, such that
o^- = a,ji and there exist two positive con-
n
We may also use the method of supersolutions and subsolutions to establish existence results for elliptic problems. Remark 12.1.3 The monotone method may be applied to the following nonlinear elliptic problem —Au = f(x,u),
x G f2,
u{x) = g(x),
x G dfl,
where Q C Rn is a bounded domain and dfl G C2'a, f G C a (ft x R) and g G C 2 ' Q (fi) with some 0 < a < 1. The conditions corresponding to
332
Elliptic and Parabolic
Equations
(12.1.12) and (12.1.13) are f(x,
U i ) - f(x,
U2) > —c(ui
—U2),
X € Q, & < U2 < U\ < U
and f(x,ux)
— / ( x , u 2 ) < —c(ui —u2),
x € fi, & < u 2 < u\
respectively. However, the constants c and c should be nonnegative due to the same reason as that for linear elliptic equations. Furthermore, this method may also be applied to the uniformly elliptic equation of general form -aij(x)DijU
+ bi(x)DiU + c(x)u = f(x,u),
x € f2,
where aij,bi,c € Ca(Cl), aij = ajj and there exist two positive constants X, A, such that A|£|2 < OijWZitj 12.1.5
Nonuniqueness
of
< A|£| 2 ,
V£ e l " , i e Q.
solutions
The existence theorem shows that if / satisfies the left-hand side Lipschitz condition (12.1.14), then problem (12.1.12), (12.1.13) has at least one solution in the sector (%,'u}. This solution is unique if / also satisfies the right-hand side Lipschitz condition (12.1.15). In particular, the existence of a unique solution of problem (12.1.12), (12.1.13) is guaranteed if / is a C1-function or a Lipschitz continuous function in u S (%,t/). However, this uniqueness result is ensured only with respect to the given supersolution and subsolution, and it does not rule out the possibility of other solutions outside the sector (%,tt). Furthermore, the uniqueness result may not hold when / is not Lipschitz continuous in u £ (u,u). In the following discussion, we give some examples to show that if / satisfies the condition (12.1.14) but fails to satisfy (12.1.15) then problem (12.1.12), (12.1.13) may possess more than one solution. Let us consider the one-dimensional problem Ut
^xx
= f(x, u), 0 < x < 7r, t > 0, u(x,t) = 0, (x,t)£ ({0,TT}X (0,+oo))U((0,7r)x {O}).
(12.1.16) (12.1.17)
Monotone
333
Method
Any nontrivial solution of problem (12.1.16), (12.1.17) must be spatially dependent. Consider the function f(x,u)
= u + 3sin 2 / 3 (x/2)u 1 / 3 ,
0 < x < n, u £ R.
Clearly this function is Holder continuous and is nondecreasing in u £ R. This implies that / satisfies (12.1.14) for all - c o < u-i < u\ < +oo with c = 0. We seek some ordered supersolution and subsolution of the form u{x, t) = pt3/2 sinx,
%(x, t) = -pt3/2
sinx,
0 < x < ir, t > 0
with p > 1. Since H and Q satisfy the boundary condition (12.1.17), it suffices to verify the differential inequality. In view of the relation t£t -ruxx
= P ( T ^ 1 / 2 + t3/2j sin a; = -pt1/2 sinx+ u,
0 < x < IT, t > 0,
'u is a supersolution if -ptl/2smx
+ u > u + 3sin 2 / 3 (x/2)(^ 3 / 2 sina;) 1 / 3 ,
0 < x < ir, t > 0.
This inequality is equivalent to p > p 1 ^ 3 , which is clearly satisfied by any p > 1. The same argument shows that ^ is a subsolution. Therefore, there exists at least one solution u of problem (12.1.16), (12.1.17) such that -pt3/2
sin x
< pt3/2 sin x,
0 < x < n, t > 0.
However, all the three functions ui(x,t)
= — t 3 / / 2 sinx, u2(x,t)
= 0, u 3 (x,t) = £ 3 / 2 sinx, 0 < x < 7 r , t > 0
are true solutions of problem (12.1.16), (12.1.17) in the sector (0,u). In fact, for each to > 0 the function ( 0, u(x,t) = < { (t — to) ' sinx,
when 0 < x < n,
0
when 0 < x < n, t > to
is also a solution, so that the problem has infinitely many solutions. This nonuniqueness result is due to the fact that / does not satisfy a righthand side Lipschitz condition (12.1.15) in (0,u). It should be noted that / is a C1-function in each of the intervals (—co,0) and (0,+co), so that the negative solution u\ and the positive solution u$ are unique in their respective sectors.
334
Elliptic and Parabolic
Equations
The nonuniqueness results for the one-dimensional model can be extended to problem (12.1.12), (12.1.13) in an arbitrary bounded domain f l c l " . Consider, for simplicity, the case where / ( x , 0 ) = 0,
xeO
(12.1.18)
(x, t) £ dpQT.
(12.1.19)
and g(x, t) = 0,
Then u = 0 is always a solution of problem (12.1.12), (12.1.13). To show the existence of another solution we define f{u) = sup{/(z, u)\ x € Q} and consider the Cauchy problem p'(t)=J(p(t)),
P(0)=Po
(12.1.20)
with po > 0. By the continuity of / there exists T* < +oo such that this problem has at least one solution p(t) in [0, T*). In the following theorem we give a sufficient condition on / for problem (12.1.12), (12.1.13) to have at least one positive solution in QT for any T < T*. Theorem 12.1.4 Let f be Holder continuous and satisfy (12.1.14) for 0 < «2 < u\, and let (12.1.18) and (12.1.19) hold. If there exist a constant (To and positive constants a, 7 with 7 < 1 such that f(x,u)>-o-0u
+ aur,
xeQ,u>0,
(12.1.21)
then for any T < T*, problem (12.1.12), (12.1.13) has the trivial solution u\ = 0 and a positive solution U2{x,t) in QT- In fact, there are infinitely many solutions to problem (12.1.12), (12.1.13). Proof.
Let )i(x,t) = e-fitq(t)
(x,t)GQT
with /3 = (T0 + Ao, where Ao > 0 is the smallest eigenvalue of the problem Acf)(x) + \
xed,
<j>(x) = 0 ,
x£
dfl,
Monotone
Method
335
(p is its corresponding normalized eigenfunction, and q, determined below, is a positive function with q(0) = 0. Since ^ = 0, ^ is a subsolution if 9VQT
t
3t
e^ (q'(t)-pq(t))(p(x)-e-> q(t)Act>(x)
(x,t) € QT,
which is equivalent to (q'(t) - aQq{t))4>{x) < ePffae-VqltMx)),
(x,t) e QT-
In view of the hypothesis (12.1.21), it suffices to find q > 0 such that (q'(t)-a0q(t))4>(x)
< ^t[-aoe-0tq{t)cj>{x)+a{^tq{t)4>{x)fl
(x,t) € QT
or, equivalently,
q'iW'-^x)
< ae^-^q^t),
(x, t) e QT-
Since 0 < (j) < 1 and 7 < 1, the above inequality is satisfied by any function q > 0 which is a solution of the Cauchy problem q'(t)=aq-i(t),
q(0) = 0.
(12.1.22)
A positive solution of this problem is given by ?(i) = ( < r ( l - 7 ) * ) 1 / ( 1 ~ 7 ) ,
*>0.
With this choice of q, ^ is a positive subsolution. We next seek a positive supersolution by letting u(x,t)=p(t),
{x,t)GQT,
where p is the solution of problem (12.1.20). Clearly, ti
> 0 and dpQr
ut - AH = p'(t) = f(p(t)) > fix,*),
(x, t) G QT.
This implies that p is a supersolution. By (12.1.21), the function z(t) = e0tp{t) satisfies the relation *'(*) =e<}t(p'(t) + 0p(t)) > e^(f(x,p(t)) ^ o - e ^ V ^ ) > vz^it),
t > 0.
+ (3p(t)) (12.1.23)
A comparison between (12.1.22) and (12.1.23) shows that z(t) > q(t) in [0, +00) and thus problem (12.1.20) has a positive solution p such that eptp(t) > q(t),
t > 0.
336
Elliptic and Parabolic
Equations
Therefore, the pair it (a;, t) = p(t), u(x, t) = e~/3tq(t)4>(x) are ordered supersolution and subsolution. Hence problem (12.1.12), (12.1.13) has at least one positive solution u2 in the section (e~,3tq(t)(j)(x),p(t)). This proves the existence of two solutions of problem (12.1.12), (12.1.13), u\ = 0 and u2. It is easily seen that for each to > 0 the function f 0, / u(x,t) = < [ u2(x, t-t0),
when x G ft, 0 < t < t0, when x e ft, t0 < t < T*
is also a solution. This shows that problem (12.1.12), (12.1.13) has infinitely many solutions. • It is seen form construction of the supersolution and subsolution in the proof of Theorem 12.1.4 that if the condition (12.1.21) is satisfied only for u £ [0, p] with some p > 0, then there is a Tp < T* such that the solution q of problem (12.1.22) exists and is bounded by p on [0,TP], This implies that 'u(x,t) = p(t) and ,%(£,£) = e~^tq(t)(j)(x) are ordered supersolution and subsolution in QTP • As a consequence, we have Corollary 12.1.2 Let the hypotheses of Theorem 12.1.4 be satisfied except that the condition (12.1.21) holds only for u € [0, p], where p is a positive constant. Then there exists Tp < T* such that all the conclusions in Theorem 12.1.4 hold in QT„12.2
Monotone Method for Coupled Parabolic Systems
The monotone method and its associated supersolution and subsolution for scalar equations, discussed in the previous section, can be extended to coupled systems of parabolic and elliptic equations. However, for coupled systems of equations, the definition of supersolutions and subsolutions and the construction of monotone sequences depend on the quasimonotone property of the reaction functions in the system. To illustrate the basic idea of the method, we consider a coupled system of two parabolic equations of the form dui
du2 ~dt
/i(«i,«2),
(x,i)eQT =
ftx(0,T),
(12.2.1)
Au2 = /2(ui,u 2 ),
(z,t)eQT =
ftx(0,T),
(12.2.2)
ux{x,t)
= gi(x,t),
(x,t)edpQT,
(12.2.3)
u2(x,t)
= g2(x,t),
{x,t)£dpQT,
(12.2.4)
Monotone
337
Method
where 0 C R" is a bounded domain with dil G C2>a, fi G C Q (R 2 ) and gi 6 C2+a'1+c,/2{QT) with some 0 < a < 1 for each i = 1,2. 12.2.1
Quasimonotone
reaction
functions
Let Jj (i = 1,2) be open sets of R. Definition 12.2.1 A function fi = / i ( u i , u 2 ) (i = 1,2) is said to be quasimonotone nondecreasing (quasimonotone nonincreasing) in J\ x Ji if for any fixed ut £ J,, /» is nondecreasing (nonincreasing) in Uj G Jj for Definition 12.2.2 A vector function f = ( / i , / 2 ) is said to be quasimonotone nondecreasing (quasimonotone nonincreasing) in J\ x J 2 , if both f\ and fi are quasimonotone nondecreasing (quasimonotone nonincreasing) in J\X Ji. If / i is quasimonotone nonincreasing and / 2 is quasimonotone nondecreasing in J\ x Ji (or vice versa), then f is said to be mixed quasimonotone. The function f is said to be quasimonotone in J\ x Ji if it has any one of the above quasimonotone properties. As usual, we call f a C 7 -function (0 < 7 < 1) in J\ x Ji if /1 G C, fi G C 7 . If fi(u\, •) is continuously differentiable in J 2 for any u\ G J\ and / 2 (-,u 2 ) is continuously differentiable in J\ for any u 2 G J 2 , then we call / = (/i) fi) a quasi C1-function in J\ x J 2 . If f is a C 1 -function or a quasi C1-function, then the three types of quasimonotone functions in Definition 12.2.2 are corresponding to |^->0,
|^>0,
(ui.ua) G J i x J 2 ,
|^<0, oui
|^<0, oui
(ui.ua) € J i x J 2
and -— < 0, ou2
—— > 0, au\
(ui,u 2 ) G J\ x Ji
(or vice versa)
respectively. These three types of reaction functions appear most often in many physical problems. 12.2.2
Definition
of supersolutions
and
subsolutions
Suppose the reaction function f = ( / i , / 2 ) defined in K2 possesses the quasimonotone properties described in Definition 12.2.2. Then we can extend
338
Elliptic and Parabolic
Equations
the monotone method for scalar equations to the coupled system (12.2.1)(12.2.4) using a supersolution and subsolution as the initial iterations. The supersolution and subsolution, denoted by u = (Si,ti2) and ji = (ji,,^0), respectively, are required to satisfy the boundary inequality u(x,t)>g{x,t)>)i(x,t),
(x,t)edpQT,
where g = (31,32)- The inequality u = (u\,u2)
> v = (vi,v2)
(12.2.5) means that
Wl > U2, Vi > V2.
Similar to scalar problems, the supersolution u and subsolution yi are defined by differential inequalities. However, the form of differential inequalities for u and u depends on the different quasimonotone property of f. For definiteness, we always consider the case that /1 is quasimonotone nonincreasing and f2 is quasimonotone nondecreasing when f is mixed quasimonotone. Definition 12.2.3 A pair of functions u = (ui, W2) and JJ = (%,,,%2) in C2,1(QT) n C(QT) are called ordered supersolution and subsolution of problem (12.2.1)-(12.2.4), if they satisfy u(x,t)
>)i(x,t),
(x,t)
eQT
and (12.2.5) and if ' Bui — - Aui ^
du /I(UI,BJ)
- Au2 - f2{uuu2)
>0>
-Q±
> 0> ^
- A^ -
fi(%vu2),
- A^2 -
f2(Mv&2),
(x,t) e Q r (12.2.6)
when {fi,f2) ( ^
is quasimonotone nondecreasing;
- A u i - /i(Si,Ai 2 ) > 0 > ^
- Attl -
h{jivu2), (a:,*) € Q T
^
- Au2 - f2(%vu2)
>0> ^
- A^ -
f2(uu^), (12.2.7)
when (fi,f2) ^ ^
is quasimonotone nonincreasing; and
- A«i - /i(«i,& 2 ) > 0 > % - - A ^ - A«2 - /2(ui,«2) > 0 > ^
- A^2 -
h(&vu2\
(a:,*) £ < 5 T
j ^ , ^ ) , (12.2.8)
339
Monotone Method
when (/i,/2) is mixed quasimonotone. Remark 12.2.1 It is seen from this definition that when (/i,/2) is quasimonotone nondecreasing, we can use the first and third inequalities in (12.2.6) to determine u and use the second and the fourth inequalities in (12.2.6) to determine ji independently; when (/i,/2) is quasimonotone nonincreasing, we can use the first and fourth inequalities in (12.2.7) to determine (ui,^ ) and use the second and third inequalities in (12.2.7) to determine (^,,^2) independently. Moreover, if{f\,f2) is mixed quasimonotone, then (u 1,'it2,&,,&?) must be determined simultaneously by all of the four inequalities in (12.2.8). Definition 12.2.4 For any ordered supersolution u = (wi,W2) and subsolution JJ = (j&1,2i2), we define the sector = {u = (ui,u 2 ) € C{QT);)i(x,t)
(JJ,U)
12.2.3
Monotone
(x,t) e
QT).
sequences
Suppose for a given type of quasimonotone reaction function there exist a pair of ordered supersolution u = (ui,H2) and subsolution JJ, = (J41,^i2). In the following discussion we consider each of the three types of reaction functions in the sector ( J J , U ) . In addition, we assume that there exist constants c, (i = 1,2) such that for every (ui,u2), (vi,v2) G (jj,u), (fi,f2) satisfies the one-sided Lipschitz condition j fi(u1,u2)-fi(vi,u2)>-c1(u1-v1), \ / 2 ( u i , U 2 ) - h(ui,v2)
whenui>ui,
> -c2(u2
-v2),
whenu 2
(122q\
>v2.
To ensure the uniqueness of the solution we also assume that there exist constants ct < ci (i = 1,2) such that for every (u\, u2), {v\,v2) G ( u , u ) with (ui,u2) > (vx,v2), f h(ui,u2)
- fi(v1,v2)
< - c i ( ( u i -vi)
+
{u2-v2)),
\f2{ui,U2)
- f2(vi,V2)
< - C 2 ( ( w i -V1)
+ (u2
-V2)).
It is clear that if there exist constants Ki > 0(i = 1,2) such that satisfies the Lipschitz condition \fi{ui,u2)
- fi(vi,v2)\
+ \u2
(ui,u2),(vi,v2)G{fi,u),
-v2\), (i = l,2)
(fi,f2)
340
Elliptic and Parabolic Equations
then both the conditions (12.2.9) and (12.2.10) hold with c4 = Kt and Ci •= -Ki. In particular, if (fi,f2) is a C^-function in ( J J , U ) , then the conditions (12.2.9) and (12.2.10) are satisfied. Let Fi(ui,u2)
= ciui +fi(ui,u2),
(ui,u2)
G (JJ,U)
(i = 1,2).
Then the condition (12.2.9) is equivalent to that Fi is monotone nondecreasing in Uj for i = 1,2. Starting from a suitable initial iteration u'°) = ( 4 , 4 ) G C2,1(QT) fl C(QT), we construct a sequence {uW}£L 0 = {(u[k) ,u2k))}%L0 from the iteration process ^
- A^ f c ) +cAk)
= Fi(«i*- 1 ) ,i4*~ 1 ) ),
(«,*) € O r ,
(12-2.11)
^ -
- A4 f c ) + c 2 4 f c ) = F 2 ( 4 f c - 1 ) , 4 f c - 1 ) ) ,
(*,*) G QT,
(12.2.12)
u[ f c ) (a: ) t)=5 1 (x,t),
(x,t) G dpQT,
(12.2.13)
4 * W ) = <72(x,0,
( s , t ) G d p Q r . (12.2.14)
It is clear that for each k = 1,2, • • •, the above system consists of two linear uncoupled initial-boundary problems, and therefore the existence of u(fe) = ( 4 , 4 ) is guaranteed by the L2 theory and the maximum principle. Furthermore, from the regularity of solutions of heat equations, u(l)
£
Ca,a/2(QT^
u (fc)
g
C
2 + a , l + a / 2 ( g r ) for jfc = 2, 3 , • • • .
Similar to the scalar case, to ensure that this sequence is monotone and converges to a solution of problem (12.2.1)-(12.2.4), it is necessary to choose a suitable initial iteration. The choice of this function depends on the type of quasimonotone property of (fi,f2). (I) Quasimonotone nondecreasing function. For this type of quasimonotone function it suffices to take either (^1,^2) or (% 1 ,^ 2 ) as the initial iteration (u[ ,u2 ). Denote these two sequences by {(u[ ,u2 )}fc^0 anc ^ { ( 4 , 4 )}S£o' respectively. The following lemma presents the monotone property of these two sequences. Lemma 12.2.1 For quasimonotone nondecreasing {fi,f2), the two sequences {(u\ ,u2 )}fcL0 and { ( 4 ' 4 )}A^=O possess the monotone property u{k){x,t)
<
rfk+1\x,t)
{x,t)eQT
(12.2.15)
Monotone Method
341
for every k = 0,1,- • •. Proof.
Let
wf (x,t) = ^ 0 ) ( x , i ) - u ^ f o t ) = u i (x,t) - u f ^ x , * ) ,
(a;,t)6Qr.
By (12.2.5), (12.2.6) and (12.2.11)-(12.2.14), a
(0)
2 g - - A w f + Qiwl0) > Fifafa)
- F i O E ^ . u f ) = 0,
^i 0 ) (x,i) > 9i(x,t) - gi{x,t) =0, The maximum principle leads to w\'
(x, t) e Q r , (x,t) £ dpQT-
> 0 on Q r , i.e.
^1)(x,t)
(i,t)6Qr
(» = 1,2).
(x,i)eQT
(i = 1,2).
Similarly lii1)(s,t)>«i0)(a:,t)=^>t), Let ^
( i , t ) = ^ 1 ) ( x , t ) - J i | 1 ) (*.*).
0M)eQr
(i = l,2).
Then, by (12.2.11)-(12.2.14) and the monotone property of Fu a
(i)
*%-
- Aw'* + * « , « = Fi(uf\u^)
- f i f c i 0 ^ ) > 0,
w(f)(x,t)=gi{x,t)-gi(x,t)=Q,
(x,t) e QT, (x,t) € dpQT-
Using the maximum principle gives w\ ' > 0 on QT, i.e. v^\x,t)
(x,t)eQT
(i = 1,2).
Thus, we have u{°)(x,t)
(x,t)£QT
(i = 1,2).
Suppose v^h~1)(x,t)<^k\x,t)
{x,t)eQT
(i = 1,2)
for some k > 1. Then, by (12.2.11)-(12.2.14) and the monotone property of Fi, the function wf\x,t)=u{k\x,t)-uf+l\x,t),
(x,t)eQT
(t = l,2)
342
Elliptic and Parabolic
Equations
satisfies the relation dt
— Awl
,(*) + £iw* = Fi(v
(fe-i) _(fc-
>o,
,"2
(x,, * ) 6
"V )-Fi(u[k\4k)) <3T,
wl (x ,*) = 9i(x, * ) - St fa,.*) = 0
(x,t) G 9p<5r-
This leads to the inequality wf+1)fa,i)
(x,t)GQT
(i = 1,2).
A similar argument gives
«f + 1 ) (M) > « f W ) , «ife+1)(M) <sf+1)fa.*)> fa.*) e QT (* = i>2). Thus (12.2.15) follows by induction.
D
Remark 12.2.2 From the proof of this lemma, we see that in the absence of a supersolution, the monotone nondecreasing property of the sequence {(u[ ,u2 )}fcLo remains true provided that the condition (12.2.9) holds for every bounded function (u 1 ,U2)- In this situation the sequence {bA >I*2 )}fc^=o either converges to some limit as k —> 00 or becomes unbounded at some point in QT. A similar conclusion holds for the sequence
{(u[k)M2k))}?=o(II) Quasimonotone nonincreasing function. When the reaction function (/i,/2) is quasimonotone nonincreasing, we choose (ui,%) or (ju ,^2) as the initial iteration (u[ ,u2 ) in the iteration process (12.2.11)— (12.2.14) and denote the corresponding sequences by {(u\ ',u2 )}j^L0 and {(^1 >*4 )}/K=o> respectively. The monotone property of these two sequences is presented in the following lemma. Lemma 12.2.2 For quasimonotone nonincreasing (/i,/2), the two sequences {(u\ ,u2 )}feLo and {(wj ,u 2 ) } ^ o Possess the mixed monotone property in the sense that their components u\ ' and u- satisfy the relation (12.2.15) for every k = 0, !,-••. Proof. { w °\x,t)
Let =u(i\x,t)
- u?\x,t)
= u1(x,t)-u{1\x,t),
(x,t) G QT
and w^\x,t)=u2l,(x,t)-u2u>(x,t)=u\1,(x,t)
-%Jx,t),
(x,t) G QT.
Monotone
343
Method
By (12.2.5), (12.2.7) and (12.2.11)-(12.2.14), ^ i - - Aii;<0) + S l < a
}
> F ^ u J
- Fi(izi 0 ) ,^ 0 ) ) = 0,
(x,t) e QT,
- F 2 ( » i , % 2 ) = 0,
(*, t) e Q r >
(0)
^ g - - A w f + c 2 ^ 0 ) > F2(uf\u20)) wf\x,t)
> gi(x,t) - gi{x,t) = 0,
(z,£)
w20)(x,t)
>g2(x,t)
(z,t) e dpQT-
-g2(x,t)
The maximum principle implies that u;^ iZ^fotJ^u^a:,*),
= 0,
edpQT,
> 0 on QT, i.e.
r^X) (ar, t) > u^0) (ar, t),
(x,t)
€QT.
u{2\x,t)
(x,t)
&QT.
A similar argument gives u
Let w\1\x,t)
= uli1)(x,t)-y^1\x,t),
{x,t)eQT
(i = 1,2).
Then, by (12.2.11)-(12.2.14), (12.2.9) and the quasimonotone property of Jii 3Wl
—^
A
(1) ,
AwJ ; + c l W i
(1) ;
n /-(0)
(0)\
^ / (0) - ( 0 ) \
= Fi(u\ ',u2 ') - F i ( u i ' , ^ ' )
= [ei(wi-Ai 1 ) + / i ( w i , A i 2 ) - / i f e 1 ^ 2 ) ] + [/iC^i^a) - / I ^ , « 2 ) ] > 0,
rf}
0) 0) A«,<» + e r f = FM A ) at
= [C2(W2 ~ ^ 2 ) + J ^ , ^ ) + [f2&vJl2)
(i,t) e Q r ,
- ^(Bi0).s40))
f2(UvJh2)]
- f2
(x,t) G Q r ,
t o ^ (a;, i) =gi(a;,t) -5i(o;,t) = 0 ,
(z,£) € d p Qr,
w2 (a;,*) = 02OM) -g2{x,t)
(x,i) e 9 P Q T .
= 0,
Using the maximum principle again gives wj ' > 0 on QT. Thus we obtain «J 0) (a;,t) < tij^(a;,t) < tZ^1'(a:,t) < u^ 0 ) (x,t), (x,t) &QT (t = l,2). The proof of the monotone property (12.2.15) can be completed by a induction argument similar to that of Lemma 12.2.1. •
344
Elliptic and Parabolic
Equations
(III) Mixed quasimonotone function. The construction of monotone sequences for mixed quasimonotone functions requires the use of both supersolution and subsolution simultaneously. When / i is quasimonotone nonincreasing and f2 is quasimonotone nondecreasing, the monotone iteration process is given by
^-^+c1ur=Fl{uri)Jti\ T(*)
,(k)
dU\
A
(k) ,
(k)
„ , (fc-1) _(fc-l)\
+ c2u{2k)
F2{urL\uri>),
a*(k)
du\ dt
Au(k)
-^+c24k)=F2(ut1\ut\
dt u[k'(x,t)
= Ui(x,t)
7(fc)/
k)
(x,t) = u2 (x,t)
=
x,t)
€
QT,
(12.2.16)
x,t)
e
QT,
(12.2.17)
a:,*) e
QT,
(12.2.18)
x,t)GQT,
(12.2.19)
gi(x,t),
x,t)edpQT,
(12.2.20)
=g2(x,t),
x,t)£dpQT,
(12.2.21)
and KT(°) .JT(°) 4 ) = (ui,u2),
..(0)N (u[(0)>,u 2 ') =
(uv%2).
(12.2.22)
It is seen from this iteration process that the equations in (12.2.16)(12.2.19) are uncoupled but are interrelated in the sense that the fc-th iteration (u[k\u2 ') or (u[ ,u2 ) depends on all of the four components in the previous iteration. This kind of iteration is fundamental in its extension to coupled system with any finite number of equations. The idea of this construction is to obtain the monotone property of the sequences shown in the following lemma. Lemma 12.2.3 For mixed quasimonotone {fi,f2), the two sequences k ( k) {(u[ \u 2 )}%L0 and {(u^,^)}^ given by (12.2.16)-(12.2.21) with (12.2.22) possess the monotone property (12.2.15). Proof.
Let
T(1), IU< 0) (x,t) =.^(°)i u?>(x,t) -uY>(x,t)
T(!), = ui{x,t)-u\L>{x,t),
(x,t) e QT.
By (12.2.8) and (12.2.16)-(12.2.22), ^ -
- A ^ 0 ) + £ < > > FiCSx,^) - F ^ , ^ )
= 0,
(x,t) e QT,
345
Monotone Method dw{0) £ - - Aw20) +c2w^
at
> F2(VUV2)
- F2(u?\u20))
= 0,
(x,t) G Q r ,
u;j0)(a;,*) > S i ( a : , i ) - s i ( a ; , i ) = 0 , 0)
w2 (x,t)
OM)
> g2(x,t) - g2{x,t) = 0,
The maximum principle implies that w\
u^OM^fW),
edpQT,
(x,t) G dpQT.
> 0 on <2T, i.e.
(M)eQr (* = l,2).
A similar argument gives l£\x,t)>di0\x,t),
{x,t)eQT
(i = 1,2).
Let wl1)(x,t)
= u(.1)(x,t)-v^\x,t),
(x,t)eQT
(i = 1,2).
Then, by (12.2.16)-(12.2.22), (12.2.9) and the mixed quasimonotone property of ( / i , / 2 ) , a (1)
^ - - Au,'1' + clW{1] = ^ i ° U 0 ) ) - fifcM*) = [ 2 1 ( ^ 1 - ^ ) + /i(ui>^2)-/i(l41,^2)] + [/i(%!^ 2 ) " /i(%!. "2)] > 0, ^21} ——
,„(i) A Ziw 2
(x,i) G Q r >
, „ u; ,.,(i) _—p i'(7!(u (0) ,u(0)v J—i p , ,(,u(0) ,u(o)s ) +c 2 2 2 1 2 2 1 2
= [c2(W2 -^4 2 ) + f2(ui,U2)
- /2(Ul,^ 2 )]
+ [/a(Si,A4 2 )-/ 2 Cl4 1 ,« 2 )] > 0 ,
(x,t)GQT,
w^\x,t)
= gi(x,t) -gi(x,t)
= 0,
(z,t) G dpQT,
w£\x,t)
= g2(x,t) - g2(x,t) = 0,
(x,t) G dpQr-
Using the maximum principle gives w\
> 0 on Q T . Thus we have
v!i0\x,t)
(x,t)eQT
(t = l,2).
Assume ^"^(x.t)^",-*^*.*)^^^.*)^^*"1^^*),
(x,*)GQT
(» = 1,2)
346
Elliptic and Parabolic
Equations
for some k > 1. Then, by (12.2.16)-(12.2.21), (12.2.9) and the mixed quasimonotone property of (/i,/2), the function w?\x,t)=u?\x,t)-u?+i\x,t),
(x,t)eQT
(t = 1,2)
satisfies the relation a CO
= [c1(uri> -*<*>) + / i ^ . a ? - 1 5 ) - /ip^U*" 1 *)] + [/i(u(1fc),i4fc"1)) - /i(5(ifc),l4fc))] > 0, a CO ^ - - A«4« +
rf
= F2{uri\utl))
=Uutx) - 4fc)) + Mutl)Mtl)) + [Mu^M^)
-
F2(rf\&)
- /adzi*"1',^)]
- f2(u[k\4k))}
fc)
(x,t) € QT,
> 0,
(x,t) e Q T ,
u4 (x,t) =ffi(x,t) - 5 i ( x , t ) = 0,
(x,i) e
w{k\x,t)
(a:,*)
= g2(x,t) - g2(x,t) = 0,
Using the maximum principle again leads to that w\ uf+l\x,t)
{x,t)£QT
dpQT,
> 0 on QT, i.e. (t = l,2).
A similar argument gives £+1\x,t)
>^k)(x,t),y^k+1\x,t)
The conclusion of the lemma follows by induction.
(x,t)eQT
(. = 1,2). D
The following lemma shows that the above construction of monotone sequences yields a sequence of ordered supersolutions and subsolutions for problem (12.2.1)-(12.2.4). L e m m a 12.2.4 Let (u\,U2), (^,,^i 2 ) be ordered supersolution and subsolution of problem (12.2.1)-(12.2.4) and (A,/2) be quasimonotone and satisfy (12.2.9). Then, for each type of quasimonotone (/i,/2), the corresponding iterations (u\ ,u~2 ) and O^i >^2 ) (^ = 1,2,• • •) given by Lemmas 12.2.1-12.2.3 are ordered supersolution and subsolution. Proof. First, consider the case where (/i,/2) is quasimonotone nondecreasing. Then, by (12.2.9) and (12.2.11)-(12.2.14), we have, for k =
Monotone
347
Method
1.2--
-k^*" 1 ' - J?) + fM^Mt1') - fMk)Mt1])] + [h(u[k\uri]) - fMk)Ak))] + fMk)Ak)) >h(uf\uik)),
(x,t)eQT,
»-(*0
= [fi.(4*-1)-^)) + /3(Bifc-1).4fc-1))-/a(Bi*-1),^))] >f2{u{k),u(k)), fe)
(i,t)6Qr,
u^ (a;,t) =5i(a;,t),
(a:,i) G dpQT,
u(k){x,t)
(x,t) £ dpQT,
=g2{x,t),
which shows that (u^ , u2 ) is a supersolution. The proof for the subsolution is similar. If (/i, f2) is quasimonotone nonincreasing, then from the construction of the sequence and using (12.2.9) and the quasimonotone nonincreasingness of(/i,/2),forfc = l , 2 . . - ,
-fetfi*-1' --ife)) + h(ut1)Ak~1)) - fMk),i£-l))] + [h(u[k)Ak-1]) - fMk\uik))] + fMk\y^k)) >fi(^k\i^k)),
(x,t)eQT,
k)
dui
+ [f2(u[k-l\u2k))
- h(u[k\uik))}
(x,t)£QT.
+
f2(u[k),U2k))
348
Elliptic and Parabolic
Equations
A similar argument gives Ft ( fe )
^
- Au[V
(x,t)€QT,
- Au^ >f2(u[k\4%
(x,t)eQT
a—(fc)
^
for k = 1,2, •••. Therefore, (u[ sujj ) and (uf\u2k^) are ordered supersolutions and subsolutions. Finally for mixed quasimonotone (/i,/2), (u\ ,u2 ) a n d (u[ \u2 ) are determined by (12.2.16)-(12.2.22). In view of (12.2.9) and the mixed quasimonotone property of (/i, / 2 ) , for k = 1,2, • • •,
=[a^1fc-i) - ^ + / 1 ( ^ - i ) , ^ - i ) ) - Adtfu*-1')] + [/iti'U*"1') -/ifltfU*')] +h(u[k\u^) >fi(uf\u^),
(x,t)eQT,
o-(fe)
^--A^^-c^+^^r 1 ),^- 1 )) = Uu{tl)
~ 4 f c ) ) + /a(Bi*- 1 ) .'4*- 1 ) ) -
>f2(u{k),U{2k)),
f2(u[k-1},uik))]
(x,t)£QT.
A similar argument gives a (*0
^ -
- A«jfc> ^ A ^ , ^ ) ,
(s,i) e Q T ,
^
- Auik)
(x,t) G Q T
forfc= 1,2, • • •. Hence (uj ,u2 ) and (u^ , u 2 ) are ordered supersolution and subsolution for mixed quasimonotone functions. This completes the proof of the lemma. • It is worthy noting that the iteration process stated above is not the only way to construct the monotone sequences. For example, for the case that (/i, A) is quasimonotone nondecreasing in (JJ, u ) , a different iteration
Monotone
349
Method
process is given by a CO ^ - - A u f ' + c ^ f = F^-V^-V), a
(x,t) G QT,
(12.2.23)
(x,t) G Q r ,
(12.2.24)
(x,t)£dpQT,
(12.2.25)
(x,0e9pQr-
(12.2.26)
(*)
^ -
- A4 f e ) + c 2 4 f c ) = F a ^ , ^ "
1 5
),
u(1fe)(x,i) = 5 i ( ^ 0 . 4
fc)
(x,0=52(x,t),
Compared with (12.2.11), (12.2.12), the difference of the present iteration process is that in determining u2 by (12.2.24), (12.2.26), we have to use u[ ' in addition to u2 • This kind of iteration is similar to the GaussSeidal iterative method for algebraic systems, which has the advantage of obtaining faster convergent sequences. It may be shown that the sequences thus defined possess monotone property when the initial iteration is either a supersolution or a subsolution. Lemma 12.2.5
Let (/i,/2) be quasimonotone nondecreasing in (jj,u).
Then the sequences {(u[ ,u2 (12.2.23)-(12.2.26) with (uf\u20))
)}T=o and {(^i '^2 )}V=o> obtained from
= (ui,u2)
and
(uf\u20))
=
(MV&2),
possess the monotone property (12.2.15) for every k = 0,1, • • •. Similarly, for the quasimonotone nondecreasing reaction function, we have Lemma 12.2.6 Let (/i,/2) be quasimonotone nonincreasing in ( J J , U ) . Then the sequences {(u\ ',u2 )}j£L0 and {(u[ ' ,u2 )}'j*L0, obtained from (12.2.23)-(12.2.26) with (uf\u20))
= (ui,&2)
and
{u^ ,u20)) =
(&vu2),
possess the monotone property (12.2.15) for every k = 0,1, • • •. In the case of mixed quasimonotone (/i, f2), a modified iteration process for {(u[k\u2k))}kLo
^ L _ - Au[V
and {(u(1fc),«3fc))}2°=0
k)
+Qlu[
is
§ i v e n hY
= Fi(n ( 1 f c - 1 ) ,4 f e _ 1 ) ),
( M ) G QT,
(12.2.27)
350
ft
Elliptic and Parabolic
Equations
(fe)
% - - A«ife) + 0 ^ dt
= FM^Mt^),
(x,t) € QT,
(12.2.28)
(x,t)
QT,
(12.2.29)
(i,t)6QTl
(12.2.30)
u[k)(x,t)=u[k\x,t)=gi(x,t),
(x,t)edpQT,
(12.2.31)
u^OMHu^OM^^fa,*).
(x,t)edpQT.
(12.2.32)
dtik)
2- - A u f » + c 2 f = ^(Sf',^"1'), <9* (fe) du 2 ^ A M ^ + C ^ ^ ^ ^ , ^ - 1 ' ) ,
G
Lemma 12.2.7 For mixed quasimonotone ( / I , / ^ ) , *fee iwo sequences {(u(k),u{2k))}kLo and {(u[k),w2fe))}£L0 #wen 6y (12.2.27)-(12.2.32) with (u{°),u{°))
= {ui,u2)
and
(ufKuf)
= (% i ; u 2 )
possess the monotone property (12.2.15) for every k = 0,1,2, • • •. The proofs of these three lemmas are similar to those of Lemmas 12.2.112.2.3 and we leave them to the interested readers. 12.2.4
Existence
results
Lemmas 12.2.1 to 12.2.3 imply that for each of the three types of quasimonotone functions, the corresponding sequence obtained from (12.2.11)(12.2.14) and (12.2.16)-(12.2.21) converges monotonically to some limit function. The same is true for the sequences given by (12.2.23)-(12.2.26) and (12.2.27)-(12.2.32). Define lim u\ (x,t) =Tii(x,t),
lim u\ (x,t) = Ui(x,t),
fc—>oo fc—*oo
(x,t)eQT
(i = 1,2).
(12.2.33)
Following the same argument as in the proof of Theorem 12.1.1, we will show that under the conditions (12.2.9) and (12.2.10), Ui(x,i) = Ui(x,t) = Ui(x,t),
(x,t)eQT
(i = l,2)
and u = (ui,U2) is the unique solution of problem (12.2.1)-(12.2.4) for each of the three types of quasimonotone reaction functions. Theorem 12.2.1 Let (ui,'u2), (M,i>Zi,o) ^e ordered supersolution and subsolution of problem (12.2.1)-(12.2.4), and (/i,/2) be quasimonotone nondecreasing in (jj,u) and satisfy the conditions (12.2.9) and (12.2.10). Then
Monotone
351
Method
problem (12.2.1)-(12.2.4) has a unique solution u = (1x1,1x2) in Moreover, the sequences {(u[ ' ,u2 from (12.2.11)-(12.2.14) with ( u i 0 ) , ^ 0 ) ) = (ui,U2) converge monotonically to (ui,u2)
)}fclo
a^
an
d ihA
(JJ,U).
>^2 )}/£o> obtained
("i 0) ,w 2 0) ) = (% i ; ^ 2 ),
and satisfy the relation
(M,v&2) < (uifc),W2fc)) ^ («i»"2) < (uifc),u2fe)) < ( u i , u 2 )
on
QT (12.2.34)
/or every /c = 1, 2, • • •. Proof. Consider problem (12.2.11)—(12.2.14) where the sequence {u(fc)}£L0 represents either {u{k)}^=0 or {u(fc)}£L0. Since by Lemma 12.2.1 this sequence converges monotonically to some limit (1*1,112) as k —» 00, the continuity and monotonicity property of Fi imply that F(u[ ,u2 ) converges monotonically to Fi{u\,u2) f° r i = 1,2. Prom the regularity of solutions of heat equations, U(D e
ca'a'\QT),
ujfc) G C2+a'1+a/2(QT),
k = 2,3, • • •
and \ui
\2+a,l+a/2;QT
+ lul'C~1) W,QT + \u2k~l) lo;QT)>
& = 2, 3, • • • ,
where i = 1,2, and C» > 0 is a constant depending only on a, fi, T and /i but independent of k. From the monotone property (12.2.15), {«!fc)}EU(* = i . 2 ) is_ uniformly bounded in C 2 + a ' 1 + a / 2 ( Q T ) . Therefore, u e C2+a<1+a/2(QT) is a solution of problem (12.2.1)-(12.2.4). And (12.2.34) follows from the monotone property (12.2.15). Now we show that Ui(x,t) =Ui(x,t),
(x,t) <=QT (i = 1,2).
(12.2.35)
Let Wi(x,t) =ui(x,t)
-Ui(x,t),
(x,t)eQT
(i = 1,2).
Then, from the monotone property (12.2.15), Wi(x,t)<0,
(x,t) £QT
(i = 1,2).
(12.2.36)
352
Elliptic and Parabolic
Equations
By_(12.2.1)-(12.2.4) and the condition (12.2.10), Wi € C(QT) (i = 1,2) satisfies - ^ - Awi =fi(ul,u2)
-
C2
fi{ui,u2)
>Ci((«i -iLi) + (u2 -M2)) = -c%{wi + w2), Wi(x,t) =gi(x,t) -gi(x,t) = 0 ,
(x,t) € QT, (x,t) e dpQT.
Therefore, w\ + u>2 satisfies the relation &t
~
A
^
x +W2)
~ ~ (5i
+5
2 ) ( ^ i + ^2),
(x,t) e QT,
(WI + w2)(x, t) =0,
(x, t) G dpQT-
The maximum principle guarantees that wi(x,t) + W2(x,t) > 0,
(x,t)£QT.
This and (12.2.36) lead to (12.2.35). Prom (12.2.35), to show the uniqueness of the solution of problem (12.2.1)-(12.2.4) in (ji, u ) , it suffices to verify that any solution u* S (JJ, u) to problem (12.2.1)-(12.2.4) satisfies the relation u(x,t)
(x,t) € QT.
This may be proved by the same argument as in the proof of Theorem 12.1.1 ii) and we leave the details to the reader. • For the other two types of quasimonotone reaction functions, we may prove similarly the following theorems. Theorem 12.2.2 Let ( w i , ^ ) , (%j,^i2) be ordered supersolution and subsolution of problem (12.2.1)-(12.2.4), and (fi,fa) be quasimonotone nonincreasing in (JJ,U) and satisfy the conditions (12.2.9) and (12.2.10). Then problem (12.2.1)-(12.2.4) has a unique solution u = (ui,U2) in (jj,u). Moreover, the sequences {(u[ ',u2 from (12.2.11)-(12.2.14) with (w(i0),u20)) = ( u i , ^ 2 ) converge monotonically to (u\,u2)
)}^0 and
an
d (C^i »*4 )}fcio> obtained
(u^ ,u20)) =
(%vu2),
and satisfy the relation (12.2.34).
Theorem 12.2.3 Let (ui,U2), (%i;,%2) be ordered supersolution and subsolution of problem (12.2.1)-(12.2.4), and (/i,/^) be quasimonotone nonincreasing in (JJ,U) and satisfy the conditions (12.2.9) and (12.2.10). Then
Monotone
353
Method
problem (12.2.1)-(12.2.4) has a unique solution u — (1x1,112) in Moreover, the sequences {(u[ , u 2 )}feLo from (12.2.16)-(12.2.21) with (uf} ,u{2]) = (u!,u2)
(JJ,U).
an
d {(^1 '—2 )}j£=o> obtained
and {uf\u20))
= (&v&2),
an
converge monotonically to (ui,u2)
d satisfy the relation (12.2.34).
When the iteration processes (12.2.11)-(12.2.14) and (12.2.16)-(12.2.21) are replaced by (12.2.23)-(12.2.26) and (12.2.27)-(12.2.32), respectively, the results of Lemmas 12.2.5 to 12.2.7 imply that the corresponding sequences converge to some limit functions in the same fashion as in (12.2.33). It is easy to prove by an argument similar to the proof of Theorems 12.2.1 to 12.2.3 that these limits are also solutions of problem (12.2.1)-(12.2.4) in accordance with the quasimonotone property of (/i,/2). This observation leads to the following conclusion. Theorem 12.2.4 Under the hypothesis of Theorems 12.2.1-12.2.3, except that the iteration processes (12.2.11)-(12.2.14) and (12.2.16)-(12.2.21) are replaced by (12.2.23)-(12.2.26) and (12.2.27)-(12.2.32) respectively, all conclusions in the corresponding theorem remain true. 12.2.5
Extension
As the scalar equations, the monotone method used above may be applied to coupled uniformly parabolic systems of general form with more general reaction terms, such as the system n
r\
n
-W - E afffaQDjUn j,i=i
j=i
( 1
+ c - \x,t)u1
^-J2
= fi(x,t,ui,u2),
afi^D^
+ c{2)(x,t)u2
+
(x,t) G QT,
j^bfix^D^
j=i
3,1=1
ui(x,t)
Y/bf)(x,t)Dju1
+
= h{x,t,uuu2),
= gi(x,t),
u2(x, t) = g2(x, t),
(x,t) G QT, (x,t)edpQT, (x, t) G dpQT,
where fl C R n is a bounded domain with dCl G C2,a for some 0 < a < 1, /.
G C a,a/2,a ( Q T x R 2 ) | g.
g cS+a.l+a/a^), a « ) 6 « > c M g
Q^a/2^^
354
a
jl
Elliptic and Parabolic
= a
ij
an
Equations
d there exist positive constants A ^ , A ^ , such that
A(%|2 < £
«!?(*>*&& < A ^ i a 2 ,
V£ e R", (x,t) g Q T
for each i = 1,2. At the end of this section, we point out that the monotone method may be used to coupled elliptic systems and also to parabolic and elliptic systems with an arbitrary finite number of equations, see more details in [Pao (1992)].
Exercises 1. Prove Remarks 12.1.1 and 12.1.2. 2. Prove Lemmas 12.2.5-12.2.7. 3. Prove Theorems 12.2.2-12.2.4. 4. Apply the monotone method to general coupled uniformly parabolic systems. 5. Apply the monotone method to elliptic equations and coupled systems. 6. Establish the theory of monotone method for elliptic and parabolic systems with an arbitrary finite number of equations.
Chapter 13
Degenerate Equations
The last chapter of this book is devoted to elliptic and parabolic equations with degeneracy. We first consider linear equations and then discuss some kinds of quasilinear equations.
13.1
Linear Equations
Let ft c i n be a bounded domain. Consider the equation Lu = — aij(x)DijU + bi(x)DiU + c(x)u = f(x),
x G fl,
(13.1.1)
where a^ (i,j = 1, • • • , n), bi (i = 1,- • • , n), cand / are functions in Q with suitable regularity, a^- = a,ji and the matrix
(
a
n
Ojn\
•••
' ' '
air,
Qjir
is nonnegative definite on Q, denoted by A > 0, i.e. Ma:)&&>0,
V£ = (£i, •••,£«)€]&",
xGft.
(13.1.2)
Here, as before, repeated indices imply a summation from 1 up to n. If A is positive definite, i.e. A > 0, then (13.1.1) is elliptic; otherwise, (13.1.1) is said to be degenerate. In case bn > 0 and /
an :
•••
ai(n-i)
\ > 0,
\ a ( n - l ) l ••• a ( n - l ) ( n - l ) / 355
ajn = anj = 0 (j = 1, • • • , n),
356
Elliptic and Parabolic
Equations
(13.1.1) is a parabolic equation; this means that, parabolic equations are degenerate elliptic equations. If bn > 0 and f an
•••
ai(„_i)
\
:
>0,
ajn =an}=0(j
= l,--- ,n),
\ 0 ( n - l ) l ••• a ( n - l ) ( n - l ) /
then (13.1.1) is a degenerate parabolic equation. If a n • • • o-im ^ : >0,
0 <m < n-2,
^ ^ m l ' ' ' Q"m7n /
a-ij = dji = 0 (i = m + I, • • • , n; j = 1, • • • , n ) ,
then (13.1.1) is called an ultraparabolic equation. In the extreme case dij = 0 (i, j = 1, • • • , n), (13.1.1) degenerates into a first order equation. Sometimes, it is convenient to write (13.1.1) in divergence form Lu = -Dj(aij(x)Diu)
+ /3i(x)DiU + c{x)u = f(x),
x GQ
(13.1.3)
where Pi(x) = bi(x) + Dj
Formulation
x e fi (i = l, • • • , « ) .
of the first boundary
value
problem
Different from elliptic equations without degeneracy, to pose the first boundary value problem for degenerate elliptic equations, in general, we are not permitted to prescribe the boundary value on the whole boundary Suppose that £ = dCl is piecewise smooth. Denote E° = {x £ H;aij(x)vi{x)vj(x)
= 0}
and (3(x) = j3i(x)ui(x),
x€E,
where V = (v\, • • • , un) is the unit normal vector inward to E; 0(x) is called the Fichera function. Divide S° as follows S° = E 0 U Si U E 2
Degenerate
357
Equations
and denote E 3 = S\E°, where S0 = { x e E 0 ; / 3 ( x ) = 0 } , Ei = {x£ Z°;P(x) < 0 } , £2 = {xeE°;/?(x)>0}. Then E 3 = {x G E;aij(o;)^(1)^(3;)
>0}UJE,
where £ is a possible subset of measure zero on E. Since E = 60, is piecewise smooth, there might be a subset of measure zero on E, at any point of which, no normal exists. The first boundary value problem for (13.1.1) or (13.1.3) is then formulated as follows Lu = / , u
i£(l,
(13.1.4)
=g,
(13.1.5)
s 2 us 3 where g is a given function. Let us observe some special examples. If (13.1.1) is elliptic, then E° is an empty set, E = E3 and we need to prescribe the boundary value on the whole boundary E. For equations of the form Lu = —Dj(a,ijDiu) + c(x)u = f(x),
x G
fi,
(13.1.6)
we have f3(x) = 0, Eo = E° and E2 is empty. Thus only the boundary value on E3 needs to be given. Now we consider the parabolic equation du — - a,ij(x,t)DijU + bi(x,t)DiU + c(x,t)u = (i,t)eQT =
f(x,t), ftx(0,T),
(13.1.7)
where oy (i,j = 1, • • • ,n), 6» (i — 1, • • • ,n), c and / are functions on QT with suitable regularity and atj = a^ satisfy the condition Oij&^>0,
£ = (£i,---,€n)€Rn,£^0,
(x,t)eQT.
(13.1.8)
358
Elliptic and Parabolic Equations
Denote t = xn+1, a{n+1)i = a i ( n + i) = 0(i = 1,- • • ,n + 1), bn+1 = 1. Then (13.1.7) can be expressed as (13.1.1) or (13.1.3) with i,j = 1, • • • ,n + 1, x = (xi, • • • ,xn,xn+i) and fi x (0,T) in place of Q. In the present case, £ = d Q r = ( f l x { « = xn+1 = 0}) U (n x {t = ar n+1 = T}) U (Sfi x (0,T)) and the Fichera function is n+l
P = ^PiVi
n
= Y^PM + "n+l-
»=1
t=l
n+l
On the lower bottom ft x {t = x n + i = 0}, we have V J a^ViVj = 0, /3 = 1 > 0 and hence Q x {i = x n + 1 = 0} C E2, where the boundary value needs to be given. However, on the upper bottom CI x {t = xn+i = T}, n+l
2_] CLijVii/j = 0, /? = —1 < 0, which means that Ct x {t = xn+i
= T} C S i ,
»,j=i
where we should not give the boundary value. Since oy satisfy (13.1.8), n+l
n
on the lateral boundary dVt x (0,T), Y^ a^ViVj = V J a^ViVj > 0, i.e. dfl x (0, T) = E3, where the boundary value needs to be given. Prescribing the boundary value on the lower bottom and the lateral boundary is just the usual formulation of the first boundary value problem for parabolic equations. It is natural to ask why we formulate the first boundary value problem for (13.1.1) in the above manner. The basic idea is to search such a boundary value condition which can ensure the uniqueness and existence of the solution. A proper condition should first ensure that the homogeneous equation has only zero solution satisfying the homogeneous boundary value condition. Let us first consider the special equation (13.1.6). Suppose that u £ C 2 (Q) is a solution of Lu=~Dj(aijDiu)
+ cu = 0,
(13.1.9)
satisfying the homogeneous boundary value condition. Multiply both sides of (13.1.9) by u and integrate over O. After integrating by parts we obtain 0 = / uLudx = — / uDj(a,ijDiu)dx Jn Jn
+ / cu2dx Jn
Degenerate
359
Equations
= / uaijDiUVjda + / aijDiuDjudx+ JT,
JQ
= I
uaijDiUUjda + j
JT,0
/ cu2dx JQ
uaijDiUVjda
JT3
[a,DiuDjudx+[cu>dX.
+
JQ
JQ
Since ay£»£j > 0 for all £ e R" and dijViVj = 0 on S°, i.e. ay£j£j achieves its minimum at f = (i/i, • • • , fn)> we have ajji/j = 0 (i = 1, • • • ,n) on S° and /
JT° So, if u
s3
udijDiUVjida = 0.
= 0, then /
uaijDiUVjda = 0
JT,3
and we are led to / dijDiiiDjudx
+ / cu2dx — 0,
JQ
JQ
from which we see that if (13.1.9) satisfies the structure condition c(x) > Co > 0,
x £ n,
(13.1.10)
then we finally derive u = 0. As we have seen in the study of elliptic equations, the condition (13.1.10) seems to be reasonable. Now we turn to the general equation (13.1.1) or (13.1.3). A similar derivation gives 0 = / uLudx — — J uDj(dijDiu)dx JQ
+ / u[liDiudx+
JQ
JQ
= I uaijDiUVjda + I JT,
j
cu2dx
JQ
dijDiuDjudx
JQ
- 1 j j3u2da - / ( 5 A A ~ c^u2dx. Since dijVj = 0 (i = 1, • • • , n) on E°, we have / UdijDiUVj da = 0 JT,
(13.1.11)
360
if
Elliptic and Parabolic
s3
= 0. Also, if u
divide /
Equations
2
s
= 0, then [/ (3u f3u2da da = fI /3u2da. It is natural to
3 Pu da into three parts
JY.
JYP
2
JT,0
/ pv?d(T = / f3u2dx + / /3u2dx + [ Pu2dx. Jn° JEo VSi Js2 By the definition of E 0 , £ 1 , £2, we have I
(3u2da = 0,
JSo
(3u2da < 0,
/
f
(3u2da > 0.
«/S2
JSI
Only - / (3u2da plays a negative role to our purpose. If * is2 / j3u2dcr = 0 and from (13.1.11) we obtain
s2
= 0, then
JT.2
\ aijDiuDjudx
- - /
(3u2da -
I (-Dif3i-c)u2dx
= 0,
(13.1.12)
from which we find that if the structure condition - (^Di0i(x)
- c(x)) > co > 0,
x e J2
(13.1.13)
is assumed, then (13.1.12) implies u = 0. Summing up, we arrive at the following conclusion: in order that the first boundary value problem for (13.1.1) admits only one (classical) solution, it suffices to prescribe the boundary value on £ 2 U £3, provided the structure condition (13.1.13) is assumed. In fact, from the above derivation, we may obtain the following conclusion: for any function u e C2(Cl) satisfying u
— 0, there holds E2US3
/ u2dx < — / uLudx, Jn °o Jn which implies / u2dx <— (Lu)2dx, Jn °o Jn provided condition (13.1.13) is assumed.
Degenerate
13.1.2
Solvability
361
Equations
of the problem
in a space similar
to H1
In what follows, we confine ourselves to the consideration of the problem for (13.1.1) with the homogeneous boundary value condition = 0.
(13.1.14)
E 2 US 3
Suppose that u G C2(U) is a solution of (13.1.1), (13.1.14). Multiply both sides of (13.1.3) by v £ W = \v £ C 1 ^ ) ^
= 0 ) and integrate
over fi. After integrating by parts, we obtain
L
vfdx = B(u,v),
(13.1.15)
where B{u,v) = / [aijDiuDjV — piuDiV — (Di/3i — c)uv\dx — I
(3uvda.
Conversely, it is not difficult to verify that if u G C 2 (0) (~l W satisfies (13.1.15), then u is a (classical) solution of (13.1.1), (13.1.14). Let (U,V)H = / (aijDiuDjV+
uv)dx — /
fiuvda
and denote by H the completion of W endowed with the norm |HI = ( V ) H • It is easy to verify that if is a Hilbert space. H is not equivalent to H1 unless 0 = 0 on S i . Definition 13.1.1 Let / € L 2 (0). A function u € H is said to be a weak solution of (13.1.1), (13.1.14) in H if (13.1.15) holds for any v £ W = iv £ C^U^v
=0).
Theorem 13.1.1 Under the condition (13.1.13), the boundary value problem (13.1.1), (13.1.14) admits a unique weak solution in H. Proof.
It is easy to see that, for any u, v € W, \B(u,v)\
+ v>)dx + ^
v*do-)1/2 \\U\\H-
Hence B(u, v) can be uniquely extended to H x W. Since for v € W, - / PivDivdx = - /
f3v2da + -
(3v2da +-
D^v^dx
362
Elliptic and Parabolic
(3v2da + i f
>]-[
Equations
DipiV2dx,
have B(v,v)>
/ aijDivDjvdxJa
/ y (-Di/3i-c)v2dx-Jn * '
2
/ pv2da. •/£!
Using (13.1.13), we obtain B(v,v)>S\\v\\2H,
VveW
for some constant 5 > 0, which means that B(u,v) is coercive. Thus, by a modified Lax-Milgram's theorem (§3.1.2), for any bounded linear functional F(v) in H, there exists a unique u £ H, such that F(v) = B{u,v),
VveW.
Clearly / fvdx is a bounded linear functional in H. Thus there exists a unique u G H such that (13.1.15) holds for any v € W. In other words, (13.1.1), (13.1.14) admits a unique weak solution in H. O 13.1.3
Solvability
of the problem
in
Lp(Ct)
Now we proceed to introduce another kind of weak solutions. Suppose u G C 2 (fi) is a solution of (13.1.1), (13.1.14). Multiply both sides of (13.1.13) by v G V = \v G C 2 (fi); v
= 0 j and integrate over
O. After integrating by parts twice, we are led to / vfdx = I uL*vdx, Jn Jn where L*v is the conjugate of Lu, namely, L*v = -Di(a,ijDjv)
- PiDiV + c*v,
c* = - A A +c,
(13.1.16)
x G 0.
Conversely, if u G C2(Vt) and (13.1.16) holds for any v £V, then u is a solution of (13.1.1), (13.1.14). Definition 13.1.2
A function u G LP(Q) is said to be a weak solution
of (13.1.1), (13.1.14) in L p (ft), if (13.1.16) holds for any v G V = iv G
c2(ay,v
=o\.
Degenerate
363
Equations
It is easy to see that weak solutions in H are weak solutions in L2(£l), but not weak solutions in Lv{0) (p > 2). To prove the existence of weak solutions in LP(Q,), we need to establish some a priori estimates. Proposition 13.1.1 Suppose that c > 0, c* > 0 on fi. Then = 0 and p > 1, i) For any u £ C 2 (Q) with u s 2 us 3
"M»^^^min(^;(P-i)c)"Luii^; ii) For any v £ C2(Q.) with v
< i3 - u? >
= 0 and q>l, E1UE3
IMlL.(n) < • , J , 7^dL*v\\L,m. K v ' min(c+ (p — l)c*) ' Proof. 0),
(13.1.18)
We merely prove (13.1.17). Multiply Lu by (u2 + S^'^^u
(u2 + S)^-2^2uLu
= -(u2 +
(S >
S^-WuDjiayDiu)
+ (u2 + S^-W&DiU
+ (u2 +
5)^-2^'2cu2.
Substituting (u2 + S^WuDjfajDiu)
=Di ((u 2 + <5)(p~2)/2uay £ \ « ) - ((p - l)u 2 + 5){u2 +
S^WoijDiuDjU,
(u2 + S^-Wu&DiU
=DJ-(u2 + 5)*'2fr) - -{u2 + \p ) p into the above formula and integrating, we obtain / (u2 +
Sy^DiPi
6)ip~2)/2uLudx
JQ
= I (u2 +
+
S^-V^uaijDiUVjda
[ ((p - l)u 2 + <5)(u2 +
S^-V^aijDiiiDjudx
JQ
- - f(u2 + 5)P/2piPida - - f\u2 + + I(u2 + 5)P'2cu2dx. JQ
SY^DiPidx
364
Elliptic and Parabolic
Since a^Vj
Equations
= 0 (i = 1, • • • , n), u ^ = 0 , the first integral of the right
side vanishes. In addition, - f(u2 + SfVpiVido P V */E
= - - ( {u2 + 5)p'20du - - f r JTi\
5p>2pda.
P JS2UE3
Thus [ (u2 + Jn
6)(p-2)/2uLudx
= I ((p - l)u 2 + j ) (u 2 + J) ( p - 4 ) / 2 ayDiuDjudx - - f(u2 + 6)^2Dipidx+ P Jn
f(u2 + Jn
- - [ (u2 + 8)p'2j3da - - f
8)^-2V2cu2dx
8p'2pda,
from which, noting that the first and fourth terms of the right side are nonnegative, we obtain / (u 2 +
Jn
5){p-2)/2uLudx
>- [(u2 + 6)P/2(c*-c)dx+ P Jn
[(u2 + 5)(p-2V2ciL2dx Jn
6p'2f3do-.
-- f
P ^S2UE3 +
Letting 6 —> 0 then gives p - f (c* + (p - l)c)\u\ dx J P Jn ^
< - [ \u\p-lLudx PJn
<-
[ pJn
luF^lLuldx,
from which, (13.1.17) follows by using the assumption c* > 0, c > 0 and Holder's inequality. • Theorem 13.1.2 Suppose that c* > 0, c > 0 on fi. Then for any f G L P (Q) withp > 1, (13.1.1), (13.1.14) admits a weak solution in LP(Q). Proof.
Prom Holder's inequality and (13.1.18), we have
/ fvdx Jn where 1/p+l/q
< ||/||LP(n)ll«l|L«(n) < Kq\\L*v\\Lgm\\f\\LPm,
W G V,
= 1. So / fvdx is a bounded linear functional in {L*v; v G
in
V} C Lq(Cl). Let Lq{Q) be the completion of {L*v;v G V} in L«(ft). Then
Degenerate
365
Equations
we can first extend / fvdx to be a bounded linear functional in Lq(Q.) and then use Hahn-Banach's theorem to further extend it to be a bounded linear functional l(w) in Lq(Q,). Thus, by Riesz's representation theorem, it can be expressed as / uwdx with some u € Lp(Cl), in particular,
Ja
/ fvdx = l(L*v) = /
uL*vdx.
It is to be noted that, since the extension of / fvdx to Lq(Cl) is not
Ju
unique, we can not assert the uniqueness of the weak solution in Lp(Cl) from the proof of Theorem 13.1.2. 13.1.4
Method
of elliptic
regularization
A frequently applied approach in treating equations with degeneracy is the elliptic regularization. The basic idea is to consider the regularized equation Leu = —eAu + Lu = —eAu — aijDijU + biDiu + cu = f,
xGfl
(13.1.19)
with e > 0 and hope to obtain the required weak solution of (13.1.1) as the limit of the solution of (13.1.19). (13.1.19) is an elliptic equation; we can solve it by means of those methods and theories presented in previous chapters. Let ue be a solution of (13.1.19), (13.1.14). Under the condition c > Co > 0 on il, by the maximum principle, we have
l/l
\ue\ < sup — , CO
which implies the existence of a subsequence £fc —» 0 and a function u, such that u£k —^ u (£fc —» 0)
in L 2 (fi),
as k —> oo,
where —*• denotes the weak convergence. We hope that the function u thus obtained is a solution, at least a weak solution of (13.1.1), (13.1.14), i.e. for any v G V = \v € C2(U);v I
0 } , (13.1.16) holds.
S1US3
366
Elliptic and Parabolic
Equations
Prom (13.1.19), we have / fvdx = — e Au£vdx + / Ja Jn Jn =e /
-—^vda + e / DiU£Divdx + /
JE 0 UI; 2 VV
f =£ /
Luevdx
Jn
9uE , f dv , f -^zrvda — e / u£-—acr —£ /
JE0UE2 W
JE
ueL*vdx
Jn
W
A , ueAvdx
Jn
+ / ueL*vdx. (13.1.20) Jn By the uniform boundedness and the weak convergence of uEk, we may assert e / u£-—da —> 0,
£ / u£Au
/ ueL*vdx —> / uL*vdx Jn Jn
(£ = ££ —> 0).
If, in addition, we can prove £/
-^vdcr^O
JE0UE2
(e = e f c ->0),
(13.1.21)
""
then (13.1.16) follows from (13.1.20) by letting £ = ££ —> 0 and u really is a weak solution in L 2 (0) of (13.1.1), (13.1.14). In order (13.1.21) holds, it suffices to establish the following key estimate: \Dtue\ < M£~1/2,
Vx e S 0 U S 2
(i =
l,---,n).
For the proof, we refer to [Oleinik and Radkevic (1973)]. 13.1.5
Uniqueness ity
of weak solutions
in Lp(ft)
and
regular-
It has been proved that the weak solution in Lp(Cl) with p > 3 of (13.1.1), (13.1.14) is unique, but it is not the case when 1 < p < 3 (see [Oleinik and Radkevic (1973)]). Here we merely sketch the method of proof, which is based on Holmgren's idea.
Degenerate
367
Equations
What we have to do is to prove that, if u £ LP(Q.) satisfies the identity
J
uL*vdx = 0,
(weC2(fl);«
\/veV=
=o),
L EiUEs in Jn then u = 0 a.e. in 0 . If for any tp € C^{Q), the conjugate problem
L*v = pmQ,
v
(13.1.22)
>
=0
(13.1.23)
EiUE 3
had a solution v € C2(Cl), then we would have / uipdx = 0, Vv? e C^°(Q) in Jn and u = 0 a.e. in Q would follow immediately. However, it is difficult, even impossible, to prove the existence of classical solutions of the problem for the conjugate equation which is also degenerate. In view of this, instead of (13.1.23), one naturally considers its regularized problem -eAv + L*v = ip in Q,
v
=0
(e > 0).
(13.1.24)
If v G C 2 (fi) is a classical solution of (13.1.24), then, by assumption (13.1.22), we have / wpdx = — s / uAvdx + / uL*vdx = —e Jn Jn Jn Jn
uAvdx.
To our purpose, it suffices to prove e / uAvdx -» 0
(e -> 0).
(13.1.25)
Jn Under certain conditions, one can establish the estimate (13.1.26) / (Avfdx < ^£ Jnn with some constant M > 0 (see [Oleinik and Radkevic (1973)]). (13.1.25) is an immediate consequence of this estimate. To verify this fact, we express e / uAvdx as
Jn
e / uAvdx = e / (u — u')Avdx
Jn
Jn
+e \ Jn
u'Avdx
368
Elliptic and Parabolic
= e where u' G CQ°(Q,). u' G
CQ°(Q)
Jn
Equations
(u - u')Avdx + £
Jn
Au'vdx,
For any given 5 > 0, using (13.1.26), we may choose
such that
f (u - u')Avdx\
Jn
(u — u')2dx is so small to make
< e ( / (u - u') 2 rfo;) 1 / 2 ( /" (Au) 2 dz)
1/2
J
<2-
For fixed u', using the uniform boundedness of v in e (following from the maximum principle), we have s I Au'vdx <2' Jn when e > 0 is small enough. Another important problem for degenerate equations is the regularity of weak solutions. Many authors have studied the global regularity of weak solutions by means of elliptic regularization. Uniform estimates in Ck(Q) of solutions {ue} of the regularized problems have been established under certain conditions. Based on these estimates, weak solutions of the original problem are proved to be functions in Cfc(fi). Also uniform estimates in some Sobolev space have been established under certain conditions and hence weak solutions are proved to be functions in this space. The study of local regularity is related to the subellipticity of equations. A linear differential operator with C°° coefficients defined in il is said to be a subelliptic operator, if for any distribution u and any domain ff CC Q, Pu £ C°°(0') implies u G C°°(Q'). It has been proved that any subelliptic operator possesses nonnegative or nonpositive characteristic form. Various conditions have been discovered for linear degenerate elliptic equations of second order to be subelliptic.
13.2
A Class of Special Quasilinear Degenerate Parabolic Equations — Filtration Equations
In this section, we proceed to discuss quasilinear equations. As seen in the previous section, the study of linear degenerate elliptic equations is more difficult than that for equations without degeneracy. Much more difficulty would be caused by the quasilinearity of equations. We do not attempt to present the argument for general degenerate elliptic equations. We merely
Degenerate
Equations
369
introduce theory and methods for some special kinds of such equations. In this section, we are concerned with a certain kind of typical quasilinear degenerate parabolic equations, called filtration equations: ^
= AA(u),
(13.2.1)
where A(s) G C 1 [0,+00) satisfies A(0) = A'(0) = 0,
A'(s) > 0 for s > 0.
(13.2.2)
Equation (13.2.1) is parabolic when u > 0. However, it degenerates when u = 0. If we do not restrict ourselves to the study of nonnegative solutions, then, instead, we assume that A(s) E C1(—00, +00) satisfies A(0) = A'(0) = 0,
A'(s) > 0 for s ^ 0.
An important example of (13.2.1) is the Newtonian filtration equation ^
- Aum
(13.2.3)
with m > 1, which corresponds to the slow diffusion. If we do not restrict ourselves to the study of nonnegative solutions, then (13.2.3) should be written as
^ = Aw-1!*). In this section, we merely consider the Cauchy problem for (13.2.1) with the initial condition U(X,0)
= UQ(X),
x£Rn,
(13.2.4)
where 110(2;) is a nonnegative and locally integrable function. 13.2.1
Definition
of weak
solutions
Let QT = Mn x (0, T) and G be a subdomain of QT. Definition 13.2.1 A nonnegative function u is said to be a weak solution of (13.2.1) in G, iiu,A(u) G Lloc(G) and u satisfies
[J (u^ + A(u)A
370
Elliptic and Parabolic
Equations
Definition 13.2.2 A nonnegative function u is said to be a weak solution of (13.2.1), (13.2.4) in QT, if u, A(u) € L\oc{QT) and u satisfies (u-£+A(u)&
//
uo(x)(p(x,0)dx
=0
(13.2.5)
for any (p e C°°(QT), which vanishes when |x| is large enough or t = T. Remark 13.2.1
If
dA(u) v J
6 L\oc(QT) (i = 1, • • • ,n), then (13.2.5) can
OXi
be transformed to the form //
(u-£-VA(u)-V
JJQT
^
°^
uo(x)
(13.2.6)
JVLn
'
8A(u) If both u and —r (i = 1, • • • , n) are bounded, then, by approximation, it OXi
is easy to see that (13.2.6) holds for any
W1,00(QT),
vanishing when
Remark 13.2.2 Ifu is a weak solution of (13.2.1), (13.2.4) «n QT, then, for any r S (0, T), there holds \\
( U 7 T "*" A(u)Aip)dxdt + /
u0(x)ip(x,0)dx
for any
—/
u(x,r)
=0
(13.2.7)
vanishing when \x\ is large enough, where QT —
To prove, we choose <pr]e as a test function in (13.2.5), where rje G C°°[0,T] such that rjs(t) = 1 for t € [0,r - e], rjE(t) = 0 for t e [T,T], (j
—
|J?E(0I
If
—
• Then we have
r
)e(u^+A(u)A
(J
u(pt]'edxdt + J
uo(x)tp(x, 0)dx — 0,
from which, letting e —* 0 and noticing that // JJQT
//
u(prj'£dxdt — /
u(x,T)(fi(x,T)dx
JRn
^eWf^C^iOvC 1 )*)
—
u(x,T)ip(x,T))dxdt
Degenerate
<— /
/
Equations
371
(u(x,t)
Jr-e JRn —0 (e -> 0), £
we obtain (13.2.7). Conversely, if for any r £ (0,T) and any
Uniqueness equations
of weak solutions
for one
dimensional
We first discuss the uniqueness of weak solutions of the Cauchy problem for (13.2.1) in one spatial dimension:
xeRn,
u(x,0) =u0(x),
(13.2.9)
where A(s) £ C ^ + o o ) satisfies (13.2.2) and 0 < u0{x) £ Lloc(R). Theorem 13.2.1
The Cauchy problem (13.2.8), (13.2.9) has at most one dA(u) weak solution u which is bounded together with the weak derivative — - — . Proof.
Let ui,u2
be weak solutions of (13.2.8), (13.2.9), which are
bounded together with —
, — C/tJu
. Then u%,U2 satisfy (13.2.6) and
CJJu
hence
IL £<- - - ^ - 1 ( ^ - ^ ) " 1
for any (p £ W '°°(QT) If
-'
which vanishes when |x| is large enough or t = T.
- A{u2(x,T)fjdT
(13.2.11)
could be chosen as a test function, then from (13.2.8) we would have //
(A(ui) - A{u2)) (ui -
u2)dxdt
JJQT =
ff J* /dA(Ul(x,r)) JJQTJT ^ dx
_
dA(u2(x,r))\dT dx )
372
Elliptic and Parabolic
/dA(ui(x,t)) V dx
Equations
_
dA(u2(x,t))\ dxdt dx /
-i£(jC(^-^H*
= ak{x) / (A(ui) -
A(u2))dxd,T,
where ak(x) is a smooth function such that ak(x) = 1 when \x\ < k — 1, ak{x) = 0 when \x\ > k, 0 < ak(x) < 1 when k — 1 < \x\ < k and a'k{x) is bounded uniformly in k. Substituting
ak{x)(A{u\)
- A(u 2 ))(wi - u2)dxdt
s//8,oiw(/; w»,) -
w x ^ - 5^)**
=J2fc>
(13.2.12)
where Q r = {(a:,i) e Q T ; A ; - l < |x|
Degenerate
dA(u) Since Ui, —
373
Equations
(i = 1,2) are bounded and o/fe(x) is bounded uniformly
in k, lik is bounded, so is I\k- By definition, ak{x) increases with k, so does I\k- Thus lim I\k exists and we have fc-»oo
lim Jlfe = / /
(A(ui) -
A(u2))(ui-U2)dxdt.
IQT
It is easy to verify that lim hk = 0.
(13.2.13)
k—»oo
In fact, from the boundedness of ut,—^ox boundedness of a'k(x), we have
\hk\
(i = 1,2) and the uniform
\A{Ul) - A(u2)\dxdt
< C( ff
(A(Ul) - A{u2)fdxdtjl
2
< C ( (J
(A(«i) - A(u 2 ))(ui - u2)dxdt}
,
where C is a constant independent of k. The finiteness of the integral //
(A(u\) — A(u2))(ui
— u2)dxdt implies
JJQT
lim / / (A(ui) - A{u2))(u\ =-"» J Jot
-u2)dxdt
= 0
and hence (13.2.13) holds. Combining (13.2.13) with (13.2.12) we finally obtain // JJQT
(A(ui) - A(u2))(ui
- u2)dxdt = lim hk = 0, fc^°°
which implies that u\ = u2 a.e. in QT13.2.3
Existence equations
of weak
solutions
D for
one
dimensional
Now we proceed to discuss the existence of weak solutions of the Cauchy problem (13.2.8), (13.2.9).
374
Elliptic and Parabolic
Equations
Theorem 13.2.2 Assume that UQ(X) > 0 is a continuous and bounded function in R, A(uo(x)) satisfies the Lipschitz condition, A(s) is appropriately smooth and lim A(s) = +oo. Then for any T > 0, the Cauchy s—»+oo
problem (13.2.8), (13.2.9) admits a continuous, nonnegative and bounded dA(u) weak solution u in QT such that —^—^ is bounded. Moreover, the solution ox u is classical in {(x,t) € Qr;u(x,t) > 0}. Proof. Denote vo = A(uo) and choose a sequence of smooth functions {v0k(x)} converging to vo(x) uniformly as k —* oo and satisfying 0
-r-v0k{x)
xeR
(fc = l,2,---)
with some constants M and KQ. Construct a sequence of smooth functions {u>k(x)} such that Wk{x) =
v
ok(x),
when |x| < k — 2,
M,
when |x| > k — 1
and 0<wk(x)<M,
—wk(x)
= max{K0,M},
Denote v = A(u), $(i>) = A~l(v).
x G R (k = 1,2, • • •).
Then (13.2.8) becomes (13.2.14)
Consider the initial-boundary value problem for (13.2.14) with conditions v(x,0) = wk(x),
xe(-k,k),
v(±k,t)=M,
t € (0,T).
(13.2.15)
Since the initial-boundary value is positive, we may apply the standard theory of parabolic equations to assert that (13.2.14), (13.2.15) admits a classical solution vk(x,t) in Gk = (—k,k) x (0,T). The maximum principle for classical solutions shows that 0 < vnxnwk{x) < vk(x,t)
< M.
(13.2.16)
A crucial step is to prove dvk < N, dx
in Gfc.
(13.2.17)
Degenerate
375
Equations
For this purpose, we first use the maximum principle to Pk = -7-— which ox satisfies *'(«fc)
dPk -dT
d2Pk 1 d .,. ,dPk -dxT--¥{^)-b-x^{vk)-dx-'
=
mGk
and then obtain max|Pfe| = max Gk
Gk
dvk dx
ovk dx
< max|Pifc| = max
where Tk = dpGk is the parabolic boundary of Gk- Since dvk
dx
t=o
=\w'k(x)\
xe[-k,k],
to prove (13.2.17), it suffices to show dvk dx
x=±k
(13.2.18)
te[0,T}.
Notice that vk achieves its maximum M on the lateral {k} x [0,T]. Hence dvk dx
>o,
te[o,T].
(13.2.19)
x=k
Consider the auxiliary function zk(x, t) = vk(x, t) - M(x - k + 1),
(x, t) e Gk,
which satisfies
$>fe)
ll = ^'
M € Dk = {k
~ 1,fc) X ( °' T) '
Zk(x,0) = wk(x) - M(x - k + l) = M(k - x), zk(k,t)=0,
zk{k - l,t) = vk(k - l,t) > 0,
x e (k - l,k), ie(0,T).
Since zk achieves its minimum minzk on {k} x [0, T], we have Dk
dzk = dv^ dx x=k dx
x=k
M
ie[o,r],
which combined with (13.2.19) yields 0<
dvk dx
<M
t£[0,T}.
x=k
Another part of (13.2.18) can be proved similarly.
376
Elliptic and Parabolic
Equations
The estimate (13.2.17) implies the uniform Lipschitz continuity of vk in x: for any (x,t), (y,t) £ QT and sufficiently large k such that (x,t), (y,t) € \vk{x,t)-vk(y,t)\
(13.2.20)
1
Denote uk = A" ^). The existence of the inverse function A~1(s) for s e [0, +oo) follows from the assumptions A'(s) > 0 and lim A(s) = +oo. 3—*+00
Prom (13.2.14) we have duk
d2Vk
< 13 - 2 - 21 >
~i = -d-
For any (x, t), (y, s) € QT, choose k large enough such that (x, t), (y, s) € Gk,x+ \At\l/2 £ [-k,k] with At = t-s. Integrating (13.2.21) over (a;,x + |At| 1 /2) x (s,t) yields /
(uk(z,t)
=
-uk(z,s)Jdz
-^^dzdT
L L
Hence from (13.2.17) we obtain /
(uk(z, t) - uk(z, s))dz\ < 2N\At\.
Jx
Using the mean value theorem for integrals, we see that there exists x* £ [z.x + IA^I1/2] such that 2
fx+\At\^
/
.
(uk(z,t)-Uk(z,s))dz=
[uk(x*,t)-Uk(x*,s)J\At\1/2.
Thus \uk(x*,t)-uk(x*,s)\<2N\At\1/2 and hence for some constant C \vk(x*,t)-vk(x*,s)\=\A(uk(x*,t))-A(uk(x*,s))\
Degenerate
377
Equations
=\A'(Zk)\\uk(x*,t)-uk(x*,s)\
-vk(y,s)\
<\vk(x,t) -vk{x*,t)\ + \vk{x*,s)
+ \vk(x*,t) -
vk(x*,s)\
-vk(y,s)\
L ("*£ " ^
• D—+ £^0M-(«W)* - o,
from which it follows by letting k —* oo and noticing that for large k, wk(x) = vok(x) converges to VQ = A(UQ) uniformly in any finite interval, that u satisfies (13.2.6), i.e. u is a weak solution of (13.2.8), (13.2.9). Finally, we prove that u is classical in {(x,t) £ Qx;u(x,t) > 0}. Let (xo,to) G QT, u(xo,to) > 0. Then in some neighborhood U C QT of (xo,to), we have u(x,t) > ao > 0 with some constant ao. Hence for any (x,t) G U and large k,
uk(x,t)>^->0.
378
Elliptic and Parabolic
Equations
This means that for large k, Uk satisfies duk
d (
,sduk\
.
.
TT
with a(x, t) = A'(uk), which is uniformly parabolic in U. Prom the standard theory of parabolic equations, it follows that for large k, Uk is uniformly bounded and equicontinuous in C2(U). Thus u s C2(U) and u satisfies (13.2.8) in the classical sense. • 13.2.4
Uniqueness equations
of weak solutions for higher
dimensional
Now we turn to the higher dimensional case. We first study the uniqueness of weak solutions. The same method as the proof of Theorem 13.2.1 can be used to prove that (13.2.1), (13.2.4) admits at most one weak solution u which is bounded together with S7A(u). However, it is difficult to prove the existence of such weak solutions in higher dimensional case; one can not obtain such solutions even under rather restrictive conditions on A(u) and
UQ(X).
In what follows, we present a uniqueness theorem for equation (13.2.3), which is valid also for equations (13.2.1) satisfying (13.2.2). Theorem 13.2.3 Suppose 0
Proof.
From Remark 13.2.1, we have /
Ui(x,T)ip(x,T)dx
SJ
— /
dip
Ui— +u™Atp\dxdt
uo(x,T)(p(x,0)dx (i = 1,2)
for any r G (0,T) and
u(x,r)
JJc
(u-£
+ (uf -
i(^+aAip\dxdt, Ml
uf)Aip)dxdt (13.2.22)
Degenerate
379
Equations
where ' a(x,t)
u?(x,t)-u2n(x,t) , ui(x,t) -u2(x,t)
ifm(x,t)^u2(x,t),
mur[l~1(x,t),
if ui(x,t)
(x,t)€QT.
= U2(x,t),
If for any function g £ Co°(R"), the problem dip
+ aA
dt
(x,t)eQT,
(13.2.23)
xeM.n
had a solution
u(x,r)g(x)dx
= 0,
(13.2.24)
which would imply u(x, r ) = 0 a.e. for x £ R" and this is just what we want to prove. However, since the coefficient a in (13.2.23) is merely a locally integrable function, (13.2.23) does not admit any smooth solution in general and even if (13.2.23) does admit, the solution can not have compact support in x in general. In view of this, we replace a by ak(x,t)
= pk(x,t)*a(x,t)
+ T,
(x,t) € QT,
K
where pk is a mollifier in R n + 1 , and consider the boundary value problem ( dip
-^-+akA
= 0,
\x\ < R, 0 < t < T,
\x\ = R, 0 < t < T,
ip(x,r)
\x\ < R,
=g(x),
where R > RQ + 1 such that supp<7(:r) C B^ choose pk such that
fj Jo
= {x £ R"; \x\ < Ro}. We
a — pk* a)2dxdt < —^. BR
(13.2.25)
(13.2.26)
*
JE
Let ipk be a solution of (13.2.25) and extend it to the whole QT by setting
380
Elliptic and Parabolic Equations
£R G Cg°(R") with the following properties to "cut-off"
(13.2.27)
Here and below, as we did before, we use C to denote a universal constant independent of R and k, which may take different values on different occasions. Choosing
u(x, T)g(x)£R(x)dx
= if
JR"
«
-
• Vy>fe +
<)(2V6J
JJQ-r
+ / / u£R(a - ak)A
(13.2.28)
Now we are ready to estimate Ik and Jk. Multiplying the equation in (13.2.25) by A
\V
^ J BR
ak{A
I
I
Jt
J BR
^J
\Vg\2dx, BR
from which it follows, in particular /
\Vipk\2dxds
/
JO
(13.2.29)
JBR
f f ak(Aipk)2dxds Jo JBR
< C.
(13.2.30)
Using (13.2.27), (13.2.29) and noticing that ut e L°°{QT) (i = 1,2) and
i4i
(u^+odv^i+i)dxdt JBR\BR-!
f
(u1+u2)dxdt.
Jo
(13.2.31)
JBR\BR-I
Using (13.2.30) and noticing that Uj £ L°°(QT){i formly bounded, we obtain \Jk\
f
{a
ak)2 a
dxdt)l/\f
= 1,2) and
j
ak{Aykfdxdt)
Degenerate Equations
381
^^dxdt)1/2.
f
W o JBR
O-k
'
Using (13.2.26), we further obtain \Jk\
I
(a - pk * a - ^ ) dxdt)
< -=.
(13.2.32)
Combining (13.2.31), (13.2.32) with (13.2.28), we finally arrive at /
u(x,T)g(x)£R{x)dx
< \Ik\ + \Jk\
I
(u1+u2)dxdt+-=.
JBUXBR-!
(13.2. 33) \k
Let k —> oo and then R —> oo. Since Ui S LX(QT) {i = 1,2), the right side of (13.2.33) tends to zero and hence (13.2.24) holds. • Remark 13.2.3 It is to be noted that requiring weak solutions to belong to L}{QT) n L°°(QT) is too restrictive, which means that they must be "small" at infinity and thus even the nonzero constant solutions are excluded. Fortunately, those weak solutions determined by initial data with compact support satisfy such condition. 13.2.5
Existence equations
of weak solutions
for higher
dimensional
Now we discuss the existence of weak solutions of the Cauchy problem (13.2.3), (13.2.4). Theorem 13.2.4 Assume that u0 e L^M") n L°°(QT) and u0(x) > 0. Then the Cauchy problem (13.2.3), (13.2-4) admits a weak solution u € L\QT)f\L™{QT). Proof. As we did in the proof of Theorem 13.2.2, the basic idea is to regularize the initial value and then to establish some estimates for the approximate solutions to obtain the desired compactness. First we choose a sequence of positive numbers Rk and rjk such that Rk -> +oo,
r)kRl -> 0,
as k -» oo,
(13.2.34)
and then construct w0fc S Cg0 (£?#,.) such that l|wofc||L~(M») < ||uo||z/»(R™)>
(13.2.35)
382
Elliptic and Parabolic
\\uok - '"olU1(R") -» 0,
Equations
as k —> oo.
To this purpose, we may define Vok = UQ for \x\ < BRk-i, and then mollify it. Consider the initial-boundary value problem ^
= Auf,
(x,t) e BRk x
u(x, t)=t)k,
(13.2.36) vok = 0 elsewhere
(0,T),
(x, t) e dBRk x (0,T),
u(x,0) =uok(x)
+ r)k,
(13.2.37)
x£BRk.
According to the classical theory, (13.2.37) admits a smooth solution uk. The maximum principle and (13.2.35) imply that Vk
|KI|L°°(R«) + r)k.
(13.2.38)
Multiplying the equation in (13.2.37) by pup~ (1 < p < +oo) gives
pdiv(ul-1Vukn)-pVur1-Vukn
=
= p d i v K - V O - mp(p - l ) ^ + p - 3 | V u f e | 2 . Integrating both sides of the above equality over BRk x (0, t) (0 < t < T) and noticing that ^ < 0 ,
ondBRx(0,T),
where [i is the normal vector outward to dBRk x (0,T), we deduce /
upk(x,t)dx-
{uok(x)+-qk)pdx
JBRk
=P f
JBRk
I
JO JdBRk
<-mp(p-l)
up-l^dadT-mp{p-l)
f
"V
h
u^+p~^uk\2dxdr
f JBRk
u^+p~3\Vuk\2dxdT
f [ Jo JBRk
or /
up(x,t)dx
+ mp(p-l)
JBRIC
<[ JB„,
f JO
(uok(x)+rik)pdx,
f JBRk
0
u™+p-*\Vuk\2dxdT
Degenerate
383
Equations
Using (13.2.34), (13.2.36), we see that the right side of the above inequality tends to /
u*dx as k^
oo, which implies that for any fixed p
€
[l,oo),
the left side is bounded. This combined with (13.2.38) gives, in particular /
uk(x, t)dx
0
(13.2.39)
ul{m~l)\Vuk\2dxdt
(13.2.40)
and f JO
[
\VuZ\dxdt
< m2 f
JBRk
f
JO
JBRk
Multiply the equation in (13.2.37) by m « ™ " ' y ,
/d4 m + 1 ) / V
4m
,. (dvp_
m\
19._
m|2
and integrate the resulting equality over Bjik x (t, T) (0 < t < T). Noticing that ^ = 0 ,
ondBRkx(0,T),
we obtain fT
Am
/d4m+1)/2N2
f
dxdTj
wn?Jt LS-^-) \VuT(x,T)\2dx+\
= - \ i
f
jB
<\
Rk
\Vu%(x,t)\2dx,
!
\VuT(x,t)fdx
Js
*k
0
Integrating with respect to t over (0, T) we further obtain fT
Am
Ml
<7i I 1
Jo
f
|V JBR,
f
Jdu^+l),\2,
\Vuf(x,t)\2dxdt.
M
384
Elliptic and Parabolic
Equations
Using this and (13.2.40) and the uniform boundedness of uk, we finally derive
< C.
(13.2.41)
The estimates (13.2.39)-(13.2.41) and Kolmogrow's theorem imply that for any S G (0,T), R > 0, {u%} is strongly compact in L2(BR x (5,T)) and hence there exists a subsequence of {u™}, supposed to be {u™} itself, which converges almost everywhere to a certain function v in QT u™ —> v a.e. in
as k —* oo,
QT,
i.e. Uk —> u = u 1 '" 1 a.e. in
QT,
as fc —>
OO.
(13.2.38), (13.2.39) imply that u G LX{QT) n L ° ° ( Q T ) Given any y> G Cco(QT), vanishing when \x\ is large enough or t = T. We have, for large k, I
(Uk~m
+ u
™Av)dxdt
+ /
(uok{x) + r)k)ip(x,Q)dx = 0.
Here we regard uk as zero outside BRk x [0,T]. Letting k —> oo, we see that u is a weak solution of problem (13.2.3), (13.2.4). • Remark 13.2.4 Different from the one dimensional case, the above existence theorem does not provide a continuous solution. 13.3
General Quasilinear Degenerate Parabolic Equations
The most general quasilinear degenerate parabolic equations, written in divergence form, are as follows
where ay = a,ji and
385
Degenerate Equations
Here, as before, repeated indices imply a summation from 1 up to n. However, in this section, we merely consider the one dimensional case and, for simplicity, we will present the arguments merely for equations of the form du _ d2A(u)
dB(u)
where A(s), B(s) G C ^ R ) and A(s) satisfies A(0) = 0,
A'(s) > 0 for s G R.
We will illustrate the theory and methods by discussing the first initialboundary value problem for (13.3.1), i.e. the problem with conditions A(u(0,t))
=A(u(l,t))
=0,
u(x,0) =u0(x),
t G (0,T),
(13.3.2)
ZG(0,1).
(13.3.3)
Equation (13.3.1) degenerates whenever u = 0. If the set £ = { s e R; A'(s) = 0} does not contain any interior point, then (13.3.1) is said to be weakly degenerate; otherwise, (13.3.1) is said to be strongly degenerate. 13.3.1
Uniqueness equations
of weak solutions
for weakly
degenerate
Denote QT = (0,1) x (0,T). Definition 13.3.1 A function u G LX{QT) is said to be a weak solution of the first initial-boundary value problem (13.3.1)-(13.3.3), if A(u), B(u) G L^iQr) and the integral equality
// (U^
+ A(U)
I ? ~ B(u)%)dxdt
+ [ MxMx,0)dx = 0
holds for any function ip G C°°(QT) with tp(0,t) =
x=0,l
strict monotonicity of A(s), which follows from the assumption that the set E = {s G R; A'(s) = 0} contains no interior point.
386
Elliptic and Parabolic
Equations
Theorem 13.3.1 Assume that u0(x) G L°°(0,1), A(s), B(s) G C ^ R ) and (13.3.1) is weakly degenerate. Then the first initial-boundary value problem (13.3.1)-(13.3.3) admits at most one weak solution in L°°(QT). We will prove the theorem by means of Holmgren's approach. A crucial step is to establish the L 1 -estimate for the derivatives of the solutions of the adjoint equation. The proof of our theorem will be completed by using this estimate together with some L 2 -type estimates for the same solutions. Let ui,u2 G L°°(QT) be weak solutions of (13.3.1)-(13.3.3). By definition, we have
.<«-*>&+*S-*£)**=«
JJo
(Ul-U2)(^+A-.
for any
= f A'(9Ul + (1 - e)u2)d6, Jo
B =B(Ul, ua) = / B'{0ui + (1 - 6)u2)d6, Jo If for any / G CQ°(QT), the problem
'
(x,t) G QT, (x,
te(o,T),
t)eQT-
( 13 - 3 - 4 )
iG(0,l)
had a solution tp in C°°(QT), then we would have
I
(wi — u2)fdxdt
=0
QT
and ui — u2 a.e. in QT would follow from the arbitrariness of / . However, since A and B are merely bounded and measurable, it is difficult to discuss the solvability of problem (13.3.4). Even if we have established the existence of solutions of the problem, the solutions are not smooth in general. In view of this situation, we consider some approximation of (13.3.4). For sufficiently small 77 > 0,5 > 0, let g_
((v + Ay^B,
if|ui-u2|>«,
"
lo,
if|ui-u2|<&
Degenerate
387
Equations
Since A(s) is strictly increasing and u\,U2 € L°°(QT), there must be constants L(6) > 0, K(5) > 0 depending on 5 but independent of rj, such that A=A(Ul)-A(u2)^
|A$| < * ( * ) ,
if | « x - « 2 l > J.
Let J4 £ and A^£ be C°° approximations of A and A^ respectively, such that lim As = A, e—0
Ae
lim A* = A*, e->0 n'e
a.e. in
QT,
v
|A*,e| < A-(5),
inQT,
where C is a constant independent of e. Denote
S ^ A ^ + I,)1/2. For given / G
CQ°(QT),
consider the approximate problem of (13.3.4)
l + ( , + ^-^&.
(i,t)€Qr,
(13.3.5)
V(0Jt)=¥j(l,t)=0,
te(o,T),
(13.3.6)
^(i,T)=0,
ie(o,i).
(13.3.7)
= /,
The existence of solutions in C°°(QT) follows from the standard theory of parabolic equations. 13.3.1
The solution if of (13.3.5)-(13.3.7)
satisfies
sup \ip{x,t)\
(13.3.8)
QT
ff
(r, + Ae)(^)2dxdt
(13.3.9)
ffjg)'****™-1-
(13.3.10)
Here and in the sequel, we use C to denote a universal constant, independent of 5, rj and e, and K(5) a constant depending only on 5, which may take different values on different occasions. Proof.
(13.3.8) follows from the maximum principle. To prove (13.3.9)
and (13.3.10), we multiply (13.3.5) by -^-j and integrate over QT- Inte-
388
Elliptic and Parabolic Equations
grating by parts and using (13.3.6), (13.3.7) yield
Using Young's inequality and noticing that |A* | < K(S), we obtain
+ K(S)
JJQT U J
dxdt + Cr]-1.
(13.3.11)
Integrating by parts and using (13.3.6) and Young's inequality give rdxdt '<<*[[
{ri + Ae){-^\
dxdt + Ca-1^-1.
(13.3.12)
for any a > 0. Substituting this into (13.3.11) and choosing a > 0 (depending only on 5) small enough, we derive (13.3.9). (13.3.10) follows from (13.3.9) and (13.3.12). • L e m m a 13.3.2
Proof.
The solution ip of (13.3.5)-(13.3.7)
f1 sup )Jo te(o,T). For small (3 > 0, let
sgn^s = <
dip(x,t) dx
s
0'
I0(s) = / Jo
if s > /?, if \s\ < p, if s < -0, sgn0ed6.
satisfies (13.3.13)
Degenerate
389
Equations
Differentiate (13.3.5) with respect to x, multiply the resulting equality by sgn^-^- and integrate over St = (0,1) x (t, T) (0 < t < T). Then we obtain
z^M
sgn^
(tops dxdr \dx)
Hence, integrating by parts and using (13.3.7) yield
-//<" +i -»(S) 2 ^(l) l^t•£-*(£)**/
+ is) ay
>
dxdr
g-~>(%)%**
B& 9
fdipw*^1
W^)[>
(13 3 14)
--
dx 2 The first term of the right side is nonnegative. The third term is bounded. In addition, using (13.3.5), (13.3.6) and the fact / 6 CQ°(QT), we see that
(e^>0-*!.SK(DC-('-|Wl)C-* which shows that the last term of the right side of (13.3.14) is equal to zero. Thus we obtain
[1lJ^-)dX
dx + C, d
\
ox
/
//
xe[o,i];
jt
dip dx
*»}
Proof of Theorem 13.3.1. of (13.3.5)-(13.3.7). Then
Jixe[o,
\
<0}
as /3 —> 0.
• Given / € Cg°(QT)- Let ip be a solution
/I" (ttl - u 2 ) / ^ = // (Ul - U2)(f + („ + * ) § - BH,.g)«fe*.
390
Elliptic and Parabolic
Equations
By the definition of weak solutions, we have
1<--»>(! + 4S-*S)**-°Hence //
(ui
—
u2)fdxdt
JJQT
= //
(ui - u2)r]-Q^dxdt + / / - If
(it! - u 2 ) ( i £ -
A)-^dxdt
(ui - "2)(£*, e - B)^-dxdt
(13.3.15)
Now we proceed to estimate each term of the right side of (13.3.15). First, from Lemma 13.3.1, //
(ui - W2)(A£ -
A)-^dxdt
1/2
2
1/2
IQT
,V2
(Ae - Afdxdt)
Hence rj2
lim / / (ui - u 2 )(A £ - A)—^dxdt
= 0.
(13.3.16)
Denote Gs={{x,t)eQT;\ui-u2\
<6},
F5={(x,t)eQT;\u1-u2\
>6}.
Using Lemma 13.3.1 and Lemma 13.3.2, we have
JJ <
I
fa-mXB^-B^dxdt
II (ui - u2)B*iE-£dxdt +
// JJGS
(ui
—
dip
u2)B^-dxdt
^JL^^niL^^^L
9X
dip
dx
dxdt
Degenerate
(XU2dxdt)
( jj
391
Equations
V2
+ C5.
Since lim A*£ = 0 a.e. in Gs, it follows that dtp (ui-u2)(B£e-.B)gdzdt
jj
lim
£—0
Using Lemma 13.3.1, we have (Ul - u2)(BsVt£ -
jj
B)^dxdt 1/2
Bfdxdt)l'\
{Ble -
from which, noticing that lim Bt,
SSj^<
= B in F$, we infer
ui - u 2 )(5* >£ - B)^-dxdt
= 0.
Therefore lim
ff
£-*0
JJQT
(m-UiHB^-B&dxdt
(13.3.17)
It remains to treat the first term of the right side of (13.3.15). For any > 0 , 7
jj
}2,
(ui-u2)-^dxdt
(U1 U2)
-#J " &
dxdt
u2)
JJG-,
KCsupA-1'2 II A11/2 9V2 dxdt + 7 9a;
dxdt
dx2
dxdt
2
dxdt {A^-A1'2ddx)
C s u p A - 1 / 2 / / Al'2
i/2
//,
dx2
2 A
d2v d2
2
«v (/i w - "^y'VL (S) H
1/2
392
Elliptic and Parabolic Equations
(A\'2 - A^fdxdt)
+ CL[n)-V2K{6)V2rrV2
+
V2
C-yKtf)1'2*-1,
where we have used (13.3.9) again and the fact that A > L(j) > 0 in F 7 . Since lim Ae = A a.e. in QT, letting e —> 0 in the last inequality gives £—>0
lim
II
E-»0
d2 («i u )^dxdt (ui -u22)-
< CL^)-1'2^)1'2^2
+
CjKiS)1^-1.
(
IQT
Now we choose 7 = 5K{5)~1/2. Then we are led to 32
lim jj
£-•0
(ui - u2)r]-—^dxdt
(13.3.18)
Combining (13.3.16)-(13.3.18) with (13.3.15) we finally obtain II
< CLiSKiSy^y^KiS)1/2^2
(«i - u2)fdxdt
+ CS,
JJQT
from which, it follows by first letting 77 —> 0 and then letting 5 —> 0, that //
("l — u2)fdxdt = 0.
>Q
The proof of our theorem is complete.
Degenerate Equations
13.3.2
Existence equations
of weak solutions
393
for weakly
degenerate
Theorem 13.3.2 Assume that UQ is Lipschitz continuous in [0,1] with uo(0) = wo(l) = 0, A(s),B(s) are appropriately smooth, lim A(s) = ±00 s—>±oo
and (13.3.1) is weakly degenerate. Then the first initial-boundary value problem (13.3.1)-(13.3.3) admits a continuous weak solution. To prove the theorem, we consider the following regularized problem duE d2A£(ue) dB{u£) = + -df —dxl-&r> ue(0,t) =u£(l,t) = 0,
(rrt^n (x,t)£QT, te(0,T),
M „ 1 0 l (13.3.19) (13.3.20)
u e (a;,0) =ti 0e (a;),
x G (0,1),
(13.3.21)
where Ae{s) =es + A{s),
s£R
(e > 0)
and uoe is a smooth function approximating UQ uniformly with "0,(0) = u0E(l) = 4 , ( 0 ) - «o e (l) = <{0)
= 1^(1) = 0
and |u 0e | uniformly bounded. Let ue be a smooth solution of this problem, whose existence follows from the classical theory of parabolic equations. We need some estimates to ensure the compactness of {ue}. First, the maximum principle implies that sup\uE{x,t)\
<M
(13.3.22)
QT
with constant M independent of e. Next, we have Lemma 13.3.3 Then
Let u€ be a solution of problem
dx
V
'
;
te[0,T}
(13.3.19)-(13.3.21).
(13.3.23)
1=0,1
with the constant C independent of e. Proof.
Let we(x,t)=
rX' Jo
X€(s)ds,
(x,t)£QT,
(13.3.24)
394
Elliptic and Parabolic
where
Ae(s) =
Equations
SGR
^f
(13325)
'-
and 6(s) is an auxiliary function of the form 6(s) = a + s with an arbitrary constant a greater than M, the constant in (13.3.22). For example, we may choose a = M + 1. Then dwE
...
sdu€
,„.
.
dAJur)
- ^ =^(« e )-^/fl(« e ) = 9W£
w/
% ^U£ /„,
-,
~Qf =A'e(Ue)-gf/6(Ue)
,„.
.
^
e
dy>/6(u€), 9j4£(ue) ... .
(x,t) e QT, ,
= —^-/0(ue),
.
„
(X,t) € QT-
Using (13.3.19), one can easily check that we satisfies
in which we have a term I -j—^ j ; as will be seen bellow, this term plays dA (u ) an important role in our proof. If we simply set w£ = —|——, then in the ox equation for we, this term disappears. This is just why we introduce the auxiliary function 6{x). Define an operator H as follows: rrr
H
. [ U
dw , ]
S
... _ - ^
.d2w
( U B
/dw\2
) _ - ( _ )
,
.dw
.
-B'(US)-,
_,
(X,t)GQr.
Then H[w£] = 0. Let ve(x, t) = K(x — 1) — we(x,t),
(x,t)eQT
with constant K to be determined. By a simple calculation, using (13.3.22), we see that, for sufficiently large K > 0,
K2 <-—-KB'(ue)<0,
inQT.
From this it follows that ve can not achieve its maximum at any point inside QT. In addition, since from (13.3.20), (13.3.21) and the uniform boundedness of u'0e, we have, for large K, ve(0,t) =-K
< 0,
ve(l,t)
= 0,
te[0,T],
Degenerate
dv£ dx t=o
395
Equations
= y_^yx))y)>
ox t=o
x
6(u0£{x))
we can assert that the maximum of u£ must be zero and must achieve at i = l. Thus dv£ dx
>o,
x-1
te[o,T]
and hence dA£(u£) dx
= {a +
u£)-^-
x=l
OX
<(a + u£) x=l
K
te[0,T].
x=l
Similarly, we can prove that dA£(ue) dx
>-C,
t€[0,T\.
x=l
Therefore the conclusion (13.3.23) for x = 1 is proved. Similarly, we can prove another part of (13.3.23). • L e m m a 13.3.4 Then
Let u£ be a solution of problem
dA£{u£) (x,t)
(13.3.19)-(13.3.21).
(13.3.26)
{x,t)GQT
Proof. Define w£ and X£(s) as in (13.3.24) and (13.3.25) with 6(s) to be determined. The first requirement is that 6(s) has positive lower bound on \s\ < M (M is the constant in (13.3.22)). Then from (13.3.19) we see that wF satisfies dwe dt
A
d2w£ dx2
e
and vE =
dx
dve dt
e„ , ,
2
- " ^' M ^ - '^(^ty -B>^=°. eju£)
s/dw£\
„,,
(«.«)e QT
sdw£
satisfies
*wf£-(*<**+ * « + *(-.)£)£ 6"(u£)
3
B"(u£)
o
.
n
.
n
Multiplying the above equality by ve gives 1 dv2 2 dt
...
.
d2v£ dx2
1/„.,,
.
„,,
.
.„.
2 du £ e\\ dv sdu dx
396
Elliptic and Parabolic
Equations
-Sty"=-Sr». 3 =°. <*.«>s«- <13-3-27> If u 2 achieves its maximum at some point on the parabolic boundary, then, by Lemma 13.3.3, (13.3.26) holds clearly. Suppose that v2e achieves the maximum at some point (xo,£o) n o t o n t n e parabolic boundary. Then at (zo,£o), the sum of the first three terms of the left side of (13.3.27) is nonnegative and hence 6"(uE) \E\uE)
4
B"(uE) A £ (u e )
3
.
n
.
^
namely -6"{uE)v2£ - B"(uE)vE < 0,
(x, t) e QT,
from which it follows by using Young's inequality that for 5 > 0, -e"(ue)v2£
< 6vl + 1 ( B > £ ) ) 2 ,
(X,t) G Q r ,
( - 0 > e ) - <5K2 < ^ ( B " K ) ) 2 ,
(i,t) e Q r .
namely
If 6(s) is chosen such that 8"(s) has negative upper bound on \s\ < M, then we can choose S > 0 so small that
u 2 (z,t)
(x,t)eQT
with constant C independent of e. This inequality implies (13.3.26), if #(s) is required to have positive lower bound on \s\ < M. The choice of such functions 6(s) is quite free, for example, we may choose 0(s) = l + (M-s)(M
+ s),
s£R.
This completes the proof of our lemma. L e m m a 13.3.5 Let u£ be a solution of problem Then for any (x, t), (y, s) e QT,
• (13.3.19)-(13.3.21).
\AE(uE(x, t)) - AE(ue(y, s))\ < C(\x -y\ + \t- s| 1 / 2 ) where the constant C is independent of e.
(13.3.28)
Degenerate
Proof.
397
Equations
Since Lemma 13.3.4 implies that
\A£(u£{x,t))-A£(u£(y,t))\
(x,t),(y,t)
e QT,
(13.3.29)
it remains to further prove |A£(n£(x,0)-^(w£(x,s))|
{x,t),(x,s)£QT.
(13.3.30)
Suppose, for example, At = t — s > 0. Given a £ (0,1) arbitrarily and denote d = (At)a. We may choose d < 1/2; otherwise, (13.3.30) follows immediately from the uniform boundedness of {u£}. In case x+d < 1, we integrate (13.3.19) over (x, x+d) x (s, t). Integrating by parts then gives rx+d
J
/"' /3A
(ue(U)-ue(Z,s))d£
=J
(11 \
\ \*+d
(?-J±-±L + B(ue))\x
dt.
(13.3.31)
Using the mean value theorem for integrals, we see that (u£(t,t)-u£(£,s))dZ
/
=
d(u£(x*,t)-u£{x*,s))
J X
for some x* G [x,x + d]. Combining this with (13.3.31) and using (13.3.22) and Lemma 13.3.4, we obtain \ue(x*,t) - u£{x*, s)\ <
C(At)1-01.
This, together with (13.3.29) gives \A£{u£{x,t))-A£(u£(y,s))\ <\A£(u£(x,t))-A£(u£(x*,t))\ + a
=C(2{At)a
+
\A£(u£(x*,t))-A£(u£(x*,s))\
\A£(u£(x*,s))-A£(u£{x,s))\ + C ( A i ) 1 _ a + C{At)a +
(At)1-0),
which implies (13.3.30), if we take a = 1/2. If x + d > 1, then since d < 1/2, we have x > 1 — d > 1/2 and can obtain the same conclusion by integrating (13.3.19) over (x — d, x) x (s,t). The proof is complete. • Proof of Theorem
13.3.2.
Denote
w£(x, t) = A£(u£(x, t)),
(x, t) e QT.
398
Elliptic and Parabolic
Equations
Lemma 13.3.5 and (13.3.22) imply the uniform boundedness and equicontinuity of {we} in QT- Hence there exists a subsequence, still denoted by {w£}, and a function w G Cl'1/2(QT), such that lim we(x, t) = w(x, t),
uniformly in QT-
e—>0
Let tp(s) be the inverse function of A(s), whose existence for s G R follows from the strict monotonicity of A(s) and the assumption lim A(s) = ±oo. s—»±oo
Then u(x, t) = lim ue(x, t) = lim ip(w€(x, t) — eu£(x, t)),
(x, t) G QT
exists and u G C{QT)To prove that u is a weak solution of problem (13.3.1)-(13.3.3), notice that, from (13.3.19)-(13.3.21), for any
{lis
r\
//
S3
P1
(u£—+WE-Q-^-B(ue)—)dxdt+
uOs(x)ip(x,0)dx
=0
and hence, by letting e —> 0, I
[u-^
+ w — ^ - B{u)-^Pjdxdt
+ J u0(x)tp(x,0)dx
= 0.
Since w = A(u), by definition, u is a weak solution of (13.3.1)—(13.3.3). Theorem 13.3.2 is proved. Theorem 13.3.3 suppose that
If in addition to the assumptions of Theorem 13.3.2, \A(si) - A(s2)\ > \\si - s2\m
(13.3.32)
for some constants m > 1 and A > 0, then the weak solution u of problem (13.3.1)-(13.3.3) given in Theorem 13.3.2 is Holder continuous, precisely, ueCl/m,l/(m+l)(QT)_
Proof. In the proof of Theorem 13.3.2, in fact, we have reached A{u{x,t)) e C 1 , 1 / 2 ( Q T ) , which follows from (13.3.28) by letting e -> 0. Thus, using the assumption (13.3.32), we obtain \u{x,t)-u{y,s)\<\-llm\A{u{x,t))-A{u{y,s))\llm +
\t-s\1'2)llm
< C{\x - y\l/m
+ \t- s\^2^),
(x, t), (y, s) G QT,
Degenerate
399
Equations
i.e. u G Cl/m'l/{2m){QT). We further prove that u G C 1 / m - 1 /( m + 1 )(Q T ). First, using (13.3.32) and Lemma 13.3.5 gives \u£(x,t)~u£(y,t)\<X^m\A(ue(x,t))-A(u£(y,t))\1/m A£(ue(y,t))\1/m Ce1/m\u£(x,t)-u£(y,t)\1/m
+ Ce1/m.
(13.3.33)
Next, for any given a G (0,1), by an argument similar to the proof of Lemma 13.3.5, we can assert that for any x G (0,1), there exists x* G (0,1) with \x — x*\ < d = (At)a (suppose At = t — s > 0), such that \u£(x*,t)-u£(x*,s)\
<
C{Atf~a.
Combining this with (13.3.33) gives \us(x,t) - ue(x,s)\
<\u£(x,t) — u e (x*,t)| + \u£(x*,t) -u£(x*,s)\ a/m
a/m
+e
+ \u£(x*,s)
l a
+ (At) - ).
-ue(x,s)\ (13.3.34)
TYl
Letting e —» 0 in (13.3.33), (13.3.34) and choosing a = to \u(x, t) - u(y, t)\
(x, t), (y, t) G QT,
1/{m+1
- , we are led
\
(x,t),(x,s)£QT,
which imply that u G C 1 / m ' 1 / ( m + 1 ) ( Q r ) 13.3.3
A remark on quasilinear strong degeneracy
n
parabolic
equations
with
Now we turn to the strong degenerate equation (13.3.1), i.e. equation (13.3.1) with A'(s) > 0 and E = {s G R;A'(s) = 0} containing interior points. Problems for such equation are much more difficult to study than those for equation with weak degeneracy. The root of difficulty is that the solutions of such equation might be discontinuous. This can be exposed in the following consideration. Suppose E D [a,b] (a
dB{u) dx
(13.3.35)
400
Elliptic and Parabolic
Equations
whose solutions, as is well known, might have discontinuity, even if the initial-boundary value is smooth enough. The first problem is how to define solutions with discontinuity for (13.3.1). Motivated by the theory of shock waves, a meaningful discontinuous solution u of (13.3.35) should satisfy the so-called entropy condition (u-k)jt<(B(u)-B(k))ix,
VfceR
(13.3.36)
at the points of discontinuity in addition to the integral identity fj
(u^-B(u)^dxdt
= 0,
V^eCo°°(QT).
(13.3.37)
Here u = —{u+ +u~), u^ are the approximate limits of u at the points of discontinuity and (74,7x) is the unit normal vector to the line of discontinuity. It is not difficult to verify that (13.3.36) and (13.3.37) imply the following integral inequality jj
sgn(« - *)((« ~k)^-
(B(u) - B{k))^)dxdt > 0, V0 <
(13.3.38)
In fact, at least for piecewise continuous functions u, (13.3.36), (13.3.37) are equivalent to (13.3.38). It was Kruzhkov who first denned weak solutions of (13.3.35) in this way and proved the existence and uniqueness of weak solutions of the Cauchy problem for (13.3.35). Inspired by Kruzhkov's idea, Vol'pert and Hudjaev denned weak solutions of strongly degenerate equation (13.3.1) as follows. Definition 13.3.2 of (13.3.1), if ^
£
A function u €
^ € Llc(QT)
sgn(„ - *,((. - » , |
L°°(QT)
is said to be a weak solution
and
- <^>. fe - <*(., - *(*,)!)** > 0, VO < ^ e C H Q T ) , Vfc e R.
The existence and uniqueness of the weak solution thus defined has been proved in BV(QT) for both the initial-boundary value problem and the Cauchy problem. By BV(QT), it is meant the set of all functions of locally bounded variation, i.e. a subset of L\OC{QT), in which the weak derivatives of each function are Radon measures on QT-
Degenerate
Equations
401
The existence can be proved by means of parabolic regularization; the basic idea is the same as the proof of Theorem 13.3.2 but it is different in techniques. The proof of uniqueness is rather difficult, which is based on a deep study of functions in BV(QT) and a complicated derivation (see [Wu, Zhao, Yin and Li (2001)]).
Bibliography
Adams R. A., Sobolev Spaces, Academic Press, New York-London, 1975. Chen Yazhe, Parabolic Equations of Second Order, Beijing University Press, China, 2003. Chen Yazhe and Wu Lancheng, Elliptic Equations and Systems of Second Order, Science Press, China, 1997. Cui Zhiyong, Jin Dejun and Lu Xiguan, Introduction to Linear Partial Differential Equations, Jilin University Press, 1991. Evans L. C , Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74, American Mathematical Society, Providence, RI, 1990. Friedman A., Partial Differential Equations of Parabolic Type, Pentice-Hall, Inc., 1964. Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Heidelberg-New York, 1977. Gu Liankun, Parabolic Equations of Second Order, Xiamen University Press, 1995. Jiang Zejian and Sun Shanli, Functional Analysis, Higher Education Press, 1994. Ladyzenskaja O. A., Solonnikov V. A. and Ural'ceva N. N., Linear and Quasilinear Equations of Parabolic Type, TransL Math. Mono., 23, American Mathematical Society, Providence, RI, 1968. Ladyzenskaja O. A. and Ural'ceva N. N., Linear and Quasilinear Elliptic Equations, English TransL, Academic Press, New York, 1968. Lieberman G. M., Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Maz'ja V. G., Sobolev Spaces, English TransL, Springer-Verlag, BerlinHeidelberg, 1985. Oleinik O. A. and Radkevic E. V., Second Order Differential Equations with Nonnegative Characteristic Form, American Mathematical Society, Rhode Island and Plenum Press, New York, 1973. Pao C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. Saks S., Theory of the Integral, English TransL, Dover Publications, Inc., New York, 1964. 403
404
Elliptic and Parabolic Equations
Wu Zhuoqun, Zhao Junning, Yin Jingxue and Li Huilai, Nonlinear Diffusion Equations, World Scientific Publishing Co., Singapore, 2001. Zhong Chengkui, Fan Xianling and Chen Wenyuan, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, 1998.
Index
ck(n), 2
barrier function, 291, 293 barrier function technique, 242, 282 Bernstein approach, 282, 296, 297 bilinear form, 64 boundary gradient estimate, 282, 287, 289 boundary Holder's estimate, 287, 289 boundary Holder's estimate for gradients, 308
Cff(n), 2 Ck'a(Q),
7
ck
Caccioppoli's inequality, 168, 200, 205, 208 Campanato space, 160, 198, 256 Cauchy problem, 369 Cauchy's inequality, 1 Cauchy's inequality with e, 2 classical Harnack's inequality, 133 coercive, 64 compact continuous field, 316 compact continuous mapping, 315 compact embedding theorem, 20 comparison principle, 233, 235, 238 completely continuous field, 316 completely continuous mapping, 315 conjugate operator, 48 continuity module, 261 contraction mapping principle, 246 coupled elliptic system, 354 coupled parabolic system, 336 critical point, 314 critical value, 314 cut-off function, 5
24
W2k'k{QT), 24 Wk*(p), 15 ^ • p ( n ) , 15 BMO(Qo), 259 W2pk'k(QT),
24
W?'"{QT),
24
t-anisotropic t-anisotropic 28, 204 t-anisotropic t-anisotropic
Campanato space, 197 Poincare's inequality, Sobolev space, 24 embedding theorem, 26
relatively strongly compact, 16 a modified Lax-Milgram's theorem, 73,94 Aleksandrov's maximum principle, 264, 273 Arzela-Ascoli's theorem, 241 405
406
Elliptic and Parabolic Equations
De Giorgi iteration, 105, 110, 115, 282 degenerate equation, 355 diagonal process, 244 difference operator, 47 Dirichlet problem, 39 discontinuous solution, 400 domain additivity, 315, 317 domain of (A)-type, 160 Ehrling-Nirenberg-Gagliardo's interpolation inequality, 17, 260 elliptic regularization, 365 embedding theorem, 19, 204 energy method, 71, 94 entropy condition, 400 estimate near the boundary, 122 existence and uniqueness of the classical solution, 240, 248-251 existence and uniqueness of the strong solution, 264, 272 existence and uniqueness of the weak solution, 46, 62, 75 existence of classical solutions, 233 exterior ball property, 242 Fichera function, 356 filtration equation, 368 fixed point method, 277 Fredholm's alternative theorem, 67 Galerkin's method, 85, 101 general elliptic equations, 60, 187 general linear elliptic equations, 248, 260 general linear parabolic equations, 251, 271 general parabolic equations, 94, 231 general quasilinear degenerate parabolic equations, 384 global estimate, 193 global gradient estimate, 282, 296 global Holder's estimate for gradients, 310 global regularity, 56, 92 Holder space, 7
Holder's estimate, 143, 155, 282, 301 Holder's estimate for gradients, 282, 307 Holder's inequality, 2 Harnack's inequality, 131, 141, 145, 156, 282, 284 heat equation, 71, 111, 123, 199, 250, 266 heat equation with strong nonlinear source, 317 Hilbert-Schmidt's theorem, 85 homogeneous heat equation, 111, 123, 145 in the sense of distributions, 41 integral characteristic of Holder continuous functions, 161, 198 interior estimate, 178, 200 interior Holder's estimate, 155, 284 interior Holder's estimate for gradients, 307 interior regularity, 50 interpolation inequality, 17, 200 invariance of compact homotopy, 317 invariance of homotopy, 315 inverse Holder's inequality, 125 inverse Poincare's inequality, 124 iteration lemma, 177, 205, 207, 213, 301 Kronecker's existence theorem, 315, 317 Laplace's equation, 105, 118, 131 Lax-Milgram's theorem, 64 Leray-Schauder degree, 315 Leray-Schauder's fixed point theorem, 277 linear elliptic equation, 240, 255, 264 linear parabolic equation, 249, 272 Lipschitz space, 8 local boundedness estimate, 116, 118, 120, 121, 123, 126 local flatting, 6 lower sequence, 325
Bibliography maximum estimate, 282 maximum principle, 233, 237, 241, 282 mean value formula, 131 method of continuity, 246 method of solidifying coefficients, 188, 263 Minkowski's inequality, 2 mixed quasimonotone, 337 mollifier, 4 monotone method, 323 monotone sequence, 339 more general quasilinear elliptic equations, 310 more general quasilinear equations, 310 more general quasilinear parabolic equations, 311 Morrey's theorem, 282, 302 Moser iteration, 105, 121, 123, 137, 282, 285 near bottom estimate, 211 near boundary estimate, 181, 191 near lateral boundary estimate, 219 nonhomogeneous heat equation, 112, 126 normality, 314, 316 one-sided Lipschitz condition, 324 ordered supersolution and subsolution, 324, 338 partition of unity, 6 Poincare's inequality, 21, 28 Poisson's equation, 39, 47, 107, 120, 122, 178, 181, 240, 255, 287 property of segment, 16 property of uniform inner cone, 17 quasilinear degenerate parabolic equation, 368 quasilinear elliptic equation, 277 quasilinear parabolic equation, 280 quasimonotone, 337 quasimonotone nondecreasing, 337
407
quasimonotone nonincreasing, 337 regular point, 313 regular value, 314 regularity near the boundary, 53 regularity of weak solutions, 47, 50, 89 relatively weakly compact, 16 rescaling, 23, 204 Riesz's representation theorem, 41, 61 Rothe's method, 79, 96 Schauder's estimate, 159, 187, 197, 199, 231, 233, 264 Schwarz's inequality, 2 sector, 324, 339 semi-difference method, 79 shock wave, 400 sign rule, 297 smoothing operator, 3, 58 Sobolev conjugate exponent, 20 Stampacchia's interpolation theorem, 259 strong solution, 255, 263, 266, 272 strongly compact, 16 strongly degenerate, 385 subelliptic operator, 368 subellipticity, 368 subsolution, 324, 338 supersolution, 324, 338 support, 3 test function, 51, 54, 72, 73, 370-372, 377 the first boundary value problem, 356 the first initial-boundary value problem, 71 topological degree, 313 topological degree method, 313 trace of functions in H1 (fi), 29 uniform exterior ball property, 242 uniform parabolicity, 94, 231 uniformly elliptic, 60 upper sequence, 325
408
Elliptic and Parabolic Equations
weak derivative, 14 weak Harnack's inequality, 154, 284 weak maximum principle, 105, 107, 111, 112 weak solution, 40, 60, 361, 362, 369, 370, 385 weak subsolution, 116, 123
weak supersolution, 116, 123 weakly compact, 16 weakly degenerate, 385 Young's inequality, 1 Young's inequality with e, 2
,^i.
.•
Elliptic & Parabolic Equations This' book provides an introduction to elliptic and parabolic equations. While there are numerous monographs focusing separately on each kind of equations, there are very few books treating these two kinds of equations in combination. This book presents the related basic theories and methods to enable readers to appreciate the commonalities between these two kinds of equations as well as contrast the similarities and differences between them.
ISBN 981-270-025-0
YEARS OF
PUBLISHING
9 "789812' 700254
www.worldscientific.co
