Fattorini H.O., Kerber A. The Cauchy Problem

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4.8. Construction of m-Dissipative Extensions in LP(S2) and C(S2)

247

ses of Theorem 4.8.13 since (D

u and fo, D' f 1 , ... , D fm are discontinuous

across xm = 0. However, the Fourier transform argument is easily seen to

extend to the present situation.

We prove Theorem 4.8.12 by means of a cover of Sl by interior and

boundary patches. The notations in the proof of Theorem 4.8.4 will be

freely used here.

(a) Interior patches.

At one of these the function

= X'u satisfies

Aou' = g', (4.8.53)

where

g'=X(.fo+FD'f)+(EY-ajkD'DkX'+Y_bD'X')u

+2y_EajkDJX'Dku

=X'.fo-(ED'X')f -(EEajkD'DkX'-EbjD'X')u

- (2EY, DkajkD'X') u + FD'(X`fj)+2EDk((Y- ajkDJX')

=go+EDjgj.

After the affine change of variables x - n = L(x - x) necessary to trans-

form the principal part of Ao at x into the Laplacian, we obtain

Aou'=0u`+Y.E(ajk(n)-ajk(0))D'Dkai +Y_ bjD'u'+cii'

=g0+LLlkJDkgJ,

where (lr !rjk) = L. Observing that

LL(ajk(11)-ajk(0))D'Dkut -ED'((ajk(n)-ajk(0))Dku')

-EE(D'ajk(71)-D'ajk(0))Dku'

D'ajk(0)Dkui

and that

F,bjDju' _ EDj(bju')-(FDjbj)u',

we see that, if V' is small enough,

IIU'Ilw

(4.8.54)

j=0

from which we obtain

(4.8.55)

(b) Boundary patches.

The treatment is much the same, the only

difference being that the change of coordinates x -i is now in general

248

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

nonlinear; the function go +EDJgj transforms then into go +EElkjDkgk,

{Ijk) the Jacobian matrix of ri(x). Wg write this last expression in the form

go+EDk(Eljkgk)-(EEDkljk)gk and proceed as above, applying this time

Corollary 4.8.14.

We shall need to define below what is meant by a solution of (4.8.45)

when u, fo, f 1,

... ,fm are no longer smooth. This will be done in the style of

Section 4.6: given fo, f1,...,fm E L'(52) (0 an arbitrary domain) a function

u E W',1(2i) is declared a solution of (4.8.45) satisfying the Dirichlet

boundary condition if and only if u E W01"(a) and

EajkDjuDkgi - (EbjDju)q - cugq) dx

fu(fogp

- E f

Dkp) dx

(q9 E 61 (SZ))13

(4.8.56)

4.8.15 Theorem.

Let 2 be a bounded domain of class C(k+2) A an

operator of class C(k+') with k > m/2 satisfying the uniform ellipticity

assumption (4.8.9), 14 Q a boundary condition of type (II). Let I< p < oo,

fo, f1,...,fm E LP(S2), A> 0 large enough. Then there exists a (unique) solu-

tion15 u E W"() of

(JAI - Ao) u = fo + Y_ Djf, (4.8.57)

Moreover, there exists a constant C > 0 independent of fo, f1,..

. ,fm such that

lull W1.o(tt) 5 C YL ll f IILP(12)-

(4.8.58)

j=0

The proof hinges upon the case p = 2. Here the solution is con-

structed by means of the sesquilinear functional B. in Section 4.6 in the

space

Ho (0)

= W01'2(S2). It is sufficient to observe that the linear functional

4(v)=I (fov-EfDjv)dx (4.8.59)

is continuous in H'(9) and apply Lemma 4.6.1 to B. in Ho(S2). We deduce

that the solution satisfies the inequality

IIuII W01.2(tt) s CF, II

fjIIL2(u). (4.8.60)

j=o

The next step is to extend this result top >_ 2. To do this, another particular

case of the Sobolev embedding theorem will be needed.

13Here b1....,bn are the first order coefficients of the operator Ao in divergence form

(see (4.4.3)) and Wo.P(S2)=6 (S2) c

14The concept of operator of class C(k) was only defined (in Section 4.7) for operators

written in divergence form (4.4.2). Of course, the definition is the same for operators written in

the form (4.8.8).

15See footnote

12, p. 244.

4.8. Construction of m-Dissipative Extensions in LP(R) and C(SZ)

249

4.8.16 Theorem.

Let SZ c R mm be a bounded domain of class CO),

p >_ 1, k an integer >_ 1. (a) If kp < m, then every u E Wk'P(SZ) belongs to

Lq(S2) if

1<q<mp/(m-kp).

(b) If kp = m, then every u E belongs to Lq(1) for

1<q<oo.

In both cases there exists a constant C = C( p, q, SZ) such that

IIUIIL9(Q)<CIIuIIW'<P(Q)

(4.8.61)

For a proof see Adams [1975: 1, p. 97].16

We continue with the proof of Theorem 4.8.15. Let p > 2, fo, f,,

...,fm

elements of LP(l ). Since LP(2) C L2(SZ), there exists u E W01.2(2) satisfy-

ing (4.8.57) and (4.8.60). Consider m sequences of smooth

functions (say, in 6 (SZ)) such that

IIf(")-fI1LI(0)-0 (0<j<m)

(4.8.62)

as n -* no. Sincefo +>D'f. E 6D (SZ), it follows from Corollary 4.7.16 that the

solution of

(XI-A0)u(")= f(")+Y, DJf(")

W0 I,2(

actually belongs to C(2)(S2)f1 Cr(SZ) so that Theorem 4.8.12

applies. We use now inequality (4.8.60) for u(") - u(k) deducing that (u(")) is

a Cauchy sequence in Wl,2(SZ), hence, by Theorem 4.8.17, in LP(2) for all

p>_2ifm=2 or for

1<p<2m/(m-2)

(4.8.63)

if m >_ 3. Combining this with inequality (4.8.46) (for the operator A0 - XI),

we deduce that

(u(")) is

a Cauchy sequence in the space Wo'P(SZ). Taking

limits in (4.8.56), we obtain a solution of (4.8.57) in Wo'P([ ), but we still

must show that u satisfies (4.8.58) instead of the weaker inequality (4.8.46).

To this end we consider the "solution operator" of the equation (4.8.57),

S(fo, fl,...,fm) = U

mapping LP(S2)ii+1 into W1"P(SZ) in the range of p given by (4.8.63).

Obviously S is well defined (uniqueness of u for p>2 follows from

uniqueness for p = 2) and it follows immediately from (4.8.46) and (4.8.56)

that S is closed; hence we obtain from the closed graph theorem (Section 1)

that C is bounded, so that (4.8.58) holds. A completely similar argument

pivoting on p = 2 m /(m - 2) extends the result to the range p >_ 2 if m < 4

16 This result is proved there for arbitrary domains satisfying the cone property (Adams

[1975: 1, pp. 76 and 66]).

250 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

2<p<2m/(m-4)

if m >- 5; arguing repeatedly in the same way, we prove Theorem 4.8.16 in

the range p >, 2 for arbitrary values of m.

The case 1 < p < 2 (which, incidentally, is the one of interest for us)

is handled by means of a duality argument. Let fo, 11,... , fm be again

smooth functions so that the solution u E W0 2(SZ) of (4.8.57) belongs to

C(2)(SZ)n Ct.(I) and let g0, g1,...,g,, be arbitrary functions in LP'42)

(p'- 1 + p -1 = 1). Since p' >, 2, if X is large enough there exists a (unique)

v E Wo'P'(0) such that

(XI- A'

) v=go+Y_Djgj

(4.8.64)

(where Ao is the formal adjoint of A0) satisfying the estimate

IlvIlwa"''(u)<C E IIgjIILP'(t)

(4.8.65)

j=0

An approximation argument shows that (4.8.56) can be used for functions

q) E CM(S2)n Cr(S2). Taking p = u, we obtain

f"(gou

-Yg;Dju) dx

= f

{EEajkDivDku+(Y_DJ(bjv)u)+(X

c)vu} dx

f(Y_Y_ajkDjvDku

-v(EbjDju)+(X - c)vu} dx

(fov - EfDJv) dx,

hence, using Holder's inequality,

f (gou+ YgjDju) dx

< E IIfIILP(e) llvllwo'o'fa>

j=0

\I M

<C(E

II fjII LP(Q)) (Y- IIgjIILP'(

)

-0

j=0

(4.8.66)

(4.8.67)

by (4.8.65). Since the g0,. .. , gm are arbitrary, inequality (4.8.58) is obtained

for u, fo,...,Jm.

Let now fo, 11,..

.'fm be arbitrary elements of LP(l ); for each j select

a sequence (f

(")) in 6D (SZ) such that f f"> - f in L P (S2) and let u f"> E

C(2)(SZ)f) Cr(S2) be the solution of (Al - A0)u(") = f f") +EDjf ("). Then it

follows from the previous considerations that u(") is Cauchy, hence conver-

4.8. Construction of m-Dissipative Extensions in LP(Q) and C(S2)

251

gent in Wo'P(S2) so that u = lim u(") is a solution of (4.8.57) satisfying

(4.8.58).

It remains to be shown that the solution of (4.8.57) is unique also for

I < p S 2. This is easiest seen as follows. If u E Wo P(S2) satisfies (4.8.57)

with fo = f,

fm = 0, then a simple integration by parts shows that

u E D(A0'(/3')*) and (XI - A' ($')*)u = 0, where A0'(/3') is thought of a an

operator in LP'(SZ) and /3'=/3 is the Dirichlet boundary condition. Since

AP(/3)- wI = Ao(/3')* - wI is m-dissipative in LP(Q) for w sufficiently

large, u = 0 and uniqueness follows.

We use the results obtained for an alternate characterization of the

domain of A,(#), the closure of AO(P) in L'(2).

4.8.17 Theorem.

Let C be a bounded domain of class C(k+2), AO an

operator of class C(k+) in 2 with k > m/2 satisfying the uniform ellipticity

assumption (4.8.9)" /3 the Dirichlet boundary condition. Then D(A,($))

consists of all u E L(2) such that A, (/3) u (understood in the sense of

distributions) belongs to L'(), i. e., such that there exists v (= A, (l3) u) in

L'(9) satisfying

fUuA0'(/3')wdx= favwdx

(wED(Ao(/3'))=C(2)(S2)r1Cr(S2)

(4.8.68)

Every u E D(A(/3)) belongs to Wo 9(S2) if q < m/(m - 1). For any such q

there exists a constant C = C(q) such that

(uED(A,(/3))), (4.8.69)

where A > 0 is independent of q.

The proof is based on the theory of the #-adjoint in Section 2.2

applied to the operator A' (/3') in Cr(S2). We begin with the identification

of Cr(S2)#=D (A' (/3')*)=D(Ao(/3')*)cCr(S2)*=Er(S2). Let v be an

arbitrary measure in E(S2) = C(S2)* (see Section 7). According to Theorem

4.7.16, if p > m every function u E W',P(S2) is (equivalent to) a continuous

function in S2 and the identity map from into C(S2) is bounded.

We can thus define a continuous linear functional 0 in W" °(S2) by means

of the formula

4)(u) = f u(x)v(dx).

(4.8.70)

It is plain that W',P(S2) can be linearly and isometrically embedded in the

Cartesian product LP(& )'"+' of LP(S2) with itself m + I times by means of

the assignation

u (u, -D u,...,-Dmu

17 See footnote 14, p. 248.

252 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

(see Section 7) and we can thus extend 1 by the Hahn-Banach Theorem 2.1

to LP(S2)m+' with the same norm; hence there exists an element f =

(fo, ft,

of LP'(2)m+' p'-' + p-' =1, such that

I/ ,P

IIf II LP'(t)"

F- II f IIiP'tu> CIIVIIE(t)

(4.8.71)

j=0

(where C does not depend on v and IIvIIE(sz) = total variation of v) and

(D(u) = ful fou Y_ f

Diu) dx (u E W' P(SZ)). (4.8.72)

Let p. E Er(S2) be an element of D(A0'(f3')*) = D((AI - A'(#'))*), so

that there exists v = (Al - A' (f3'))µ E Er(S2) with

fD(xI- A' (f3'))u(x)µ(dx)

= (u E C(2)(SZ)r1Cr(SZ))

(4.8.73)

We apply the argument above to the measure v. Once the functions

fo, fI,

,fm E LP'(S2) in (4.8.72) are manufactured, we use Theorem 4.8.15

to construct a solution u' E-= W011P'(&2) of (4.8.57). It follows from this

equation and (4.8.73) after an integration by parts that the measure a(dx) _

p.(dx)- u'(x) dx satisfies

fA'($'))u(x)o(dx)=0

(4.8.74)

We choose now A > w. (see (4.8.5)); since (XI - A0 '(f3'))(C(2)(St)o Cr(SZ))

is dense in Cr(S2), it follows instantly that a vanishes identically, so that

p(dx) = u'(x) dx and D(A0'(f3')*) c Wo'P'(S2), in particular, D(A0'(f3')*) c

L'(S2). Since any u E C(2) (U). belongs to D(A'

(#')*) and this space is

dense in L'(SZ), it follows that

Cr(S2)#=L'(S2),

and A'(/3')* is defined as the operator with domain consisting of all

(,8')* satisfying (4.8.68). Obviously,u e L'(SZ) such that there exists v = A'

A0(f3) c A' (,0')*' so that A1(f3) =Ao(R) c A' (f3')#; since both operators

00 00

belong to (2+,

Ai(Q) =

A' (Q')4,

(4.8.75)

and the proof of Theorem 4.8.17 for boundary conditions of type (II) is

complete (see the comments after (4.8.44) in relation to the second equality

(4.8.75)). Inequality (4.8.69) is an obvious consequence of (4.8.71) and

(4.8.58).

4.8.18 Remark. We show in the next section that the last vestige of

dissipativity requirements (namely, (4.8.4) on boundary conditions of type

4.8. Construction of m-Dissipative Extensions in LP(f) and C(S2)

253

(I)) can be removed. In the case E = LP(S2) the argument involves no more

than a sharpening of inequality (4.5.7) and shows that A(,6)- wI is m-

dissipative for co large enough (possibly w > wp) when the boundary condi-

tion /3 is of type (I) and (4.8.4) does not hold. As a bonus, we shall deduce

that AP (P) E d. (Theorem 4.9.1)

The cases E = L' (S2) and E = C(S2 ), Cr (12) are far more delicate

since (4.8.4) for p =1 is necessary for dissipativity of any operator A(/3) (in

particular, ofA(/3)- wI with co arbitrary). The same comment applies to the

cases E = C(S2), Cr() (see Lemma 4.5.3). However, a renorming of the

space will do the trick. (Theorem 4.9.3)

In Example 4.8.19 through Example 4.8.21 a proof of Theorem 4.8.8

is given. We assume that m >_ 3 in all examples except in Examples 4.8.22,

4.8.23, and 4.8.24.

4.8.19

Example. Let be a function in

(1 < p < oo) with

support in

I x I < M. Then there exists a function qq E W 2. P(R m) with sup-

port in

x < 2 M such that D mq) (.z, 0) _ ¢ (1,O) (boundary values and

derivatives understood as in Lemma 4.8.1) and

Ilrollw2'P(R,) < C01141IIWI.P(R,),

(4.8.76)

where Co depends only on p and M.

We prove this result with the help of the Neumann kernel K(z, xm)

_ (2/(2 - m)('0m)(I X 12 +

xm)-(m-2)/2 of R m, wm the hyperarea of the unit

sphere in R m. Using an obvious approximation argument, we may and will

assume that is infinitely differentiable. Define

0(x)=0(x,xm)=f

p(4,0)K(X-4,xm)d5 (4.8.77)

with

Obviously 0 is infinitely differentiable in xm > 0.

Observe that

e(x) f { f 0"D

S, xm - J dS dSm

+ f

(4.8.78)

hence, if 1 < k < m -2

Dk8(x)

(m 2)wm

l' Dm

lx-El

-(m-2) d

(4.8.79)

(m-2)wm

_ x.

254

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

and another differentiation produces the formula

D'Dke(x) = 2

D' p(S)Dx'D, Ix -

-(m-2) d

(m-2)wm

m-2 w

(4.8.80)

if j m; for j = m we have instead

D-Dk9(x)

(m 2) w Jm_

(m-2 wm

(Ix-SI2+X2 m-2)/2d4

(m-2)wm m°o

(4.8.81)

To deal with the two integrals in (4.8.80) or the first two integrals in (4.8.81)

we note that the Fourier transform of (the distribution) IxI

-(m-2) equals

- Cm I a I

-2 (C

a constant), thus the convolutions amount to multiplication

of Fourier transforms by Cmaj ak I a I - 2 and Cmakam 101 -2,

respectively, and

can be handled using Theorem 4.8.5.

To deal with the third integral, we integrate by parts, obtaining

fRm_ DkI (, x

) H( - x, xm) d4

(4.8.82)

where the expression H(x, xm) _ (2/(2- m)wm)Dm(1x1 2 +

xm)-(m-2)/2 =

(2/wm )xm (IxI 2 + xm) - m /2 is the Dirichlet kernel of R R. But it is a classical

result (see Courant-Hilbert [1962: 1, p. 268]) that

f m-,H(x, xm) dx =1

(xm > 0). (4.8.83)

Since H >, 0, the integral (4.8.82) can be estimated with the help of Young's

Theorem 8.1. A bound of the form IIDki(., xm)IILP(R--I) is obtained, thus

we only have to take the p-th power and integrate with respect to xm. The

missing derivative (Dm)2 is included observing that the function 0, being

defined by (4.8.77), is harmonic in R m so that - (D m) 2 = E(DJ)2 . Deriva-

tives of order < 2 are accounted for using (4.8.15) as in Theorem 4.8.6.

Finally, the fact that Dm0(x,0) = 4i(x,0) follows from an argument familiar

in the theory of mollifiers using (4.8.83) and the fact that the integral of

H(x, xm) on 191 >, 8 > 0 tends to zero as xm - 0. We have then shown that

0 satisfies all the properties required of 99 except for the location of the

support. To achieve this we simply define (p = X9, X a function in 6

with

support in

I x I < 2 M such that X (x) =1 for I x I < M.

4.9. Analyticity of Solution Operators 255

4.8.20 Example. Let U E W 2, n (R +) (1 < p < oo) with support in x I < M

and such that Dm u (z, 0) = 0. Then there exists a constant Co (depending

only on p, M) such that (4.8.16) holds in R +n.

The proof is essentially similar to that of Corollary 4.8.7: we use now

the even extensions u(.C, u(.X, - x), f (X, xm) = f (Ji, - x,,,) (x,,, < 0),

where f = Au. We apply then Theorem 4.8.6, generalized to functions in

W2, P(R ') through

an obvious approximation argument.

4.8.21 Example. Prove Theorem 4.8.8. (Hint: Let 99 be the function

constructed in Example 4.8.20. Apply Example 4.8.20 to u - q9.)

4.8.22 Example. Prove the results in Examples 4.8.19 to 4.8.21 for m = 2.

(The only notable difference is that the Neumann kernel of Ris K(x,, x2)

_ (1/2ir)log(x2 + x2). The Dirichlet kernel of U82, is the same: H(xl, x2) _

D2K(xl, x2) = (1/1T)x2(x1 + x2)

= (2/w2)x2(x1 + x2)_

4.8.23 Example. Show how the proof of Theorem 4.8.13 must be mod-

ified for m = 2.

4.8.24 Example. Show that the characterizations of in Theo-

rem 4.8.11 and of D(A1(/i)) in Theorem 4.8.17 cannot be substantially

improved in the sense that, in general,

C(2)(S2)s and D(A1(p))

W2"(S2)13 when m > 2 (take, for instance, 2 = S= (x;

I x I < 1) and solve

Au = f, u = 0 on r = (x;

I x I =1) by means of the Green function G(x, y) of

the unit sphere. The second derivatives are expressed by singular integral

operators

D'Dku= f DjD'G(x,y)f(y)dy,

which do not map Cr() into Cr() or Ll(52) into L'(0)).

4.9. ANALYTICITY OF SOLUTION OPERATORS

We apply the theory in Sections 4.1 and 4.2 to the differential operators in

the last five sections. Although treatment of operators with measurable

coefficients in L2(S2) is possible, only "analyticity in the mean" is obtained

and we limit ourselves to the operators in Sections 4.7 and 4.8. Most of the

results are obvious generalizations of those for the one-dimensional case in

Section 4.3, thus some of the details will be omitted. The operator A0 will be

written in divergence form,'8

A,, = EEDJ(ajk(x)Dku)+Ebj (x) Diu + c(x)u.

(4.9.1)

"As in other places, we write bj instead of hj for simplicity of notation.

256

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

We shall show below that the operators Ap (,(3) = A o (,3 ), 1 < p < oo, defined

in the previous section belong to &; as an added bonus the dissipativity

assumptions (4.8.4) on the boundary condition can be abandoned.

4.9.1 Theorem.

Let I < p < oo, SZ a bounded domain of class C (2)

for 1 < p 5 2 (of class C(k12) with k > m /2 if p > 2), A 0 an operator of class

C(') in SI for 1 < p S 2) (of class C(k+q with k > m/2 if p > 2) satisfying the

uniform ellipticity assumption (4.8.9), $ a boundary condition of type (II) or of

type (I) with y E C(')(F) (y E C(k+I)(I') if p > 2). Then the operator A,(/3)

=Ao($) belongs to & ((pp -) in Lp(SZ), where

q)p=arctg{( p

) -1) (4.9.2)

111

For every T with 0 < p < p,, there exists to = w(p, qp) such that

IIS

e"13'1

T),

(4.9.3)

where

Proof. Let 9 be the only duality map in LP(SZ). We perform an

integration by parts similar to that in (4.5.7), but keeping track of imaginary

parts: for p >- 2 the calculation makes sense for arbitrary u in

C(2)

(2)0,

whereas for 1 < p < 2 we use the customary approximation argument.

Ilullp-2(Re(9(u), AP($)u)± 8Im(9(u), AP($)u))

Ref,(

bj D'u+cu)lulp-2udx

± S Im f,(FEDi(ajkDkU)+ EbjD'u + cu)I uIp-2udx

=fr(y+ pb)Iul"da

-(p -2) fu lulP-4( EEajkRe(iiD'u)Re(i Dku)) dx

+ 8(p -2) f I ulp-4( EEajkRe(uD'u)Im(uDku)) dx

ful ulp-2(YYajkD'uDku) dx ± SfI

ulp-2(

Y-bjIm(uD'u)) dx

+p f(pc-Y_ D'b,)luIPdx.

(4.9.4)