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

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4.7. Regularity Theorems

227

u E D(A(/3)) (/3 the Dirichlet boundary condition), that is, let u E Ho( 3)

satisfy

BX(u,w) = (f,w) (w E Ho(S2))

(4.7.18)

for some f e L 2(2) (f = (XI - A(/3))u). Then U E H2(Q) and there exists a

constant C (depending only on 0, K,, ic, A) such that

IIUIIHZ(Iz) < C(IIf IIL2(U)+IIUIIHi(tt)).

(4.7.19)

The central idea of the proof consists in dividing 52 into interior and

boundary patches (in the sense of Section 4.5) and then using Theorem 4.7.1

for the interior patches and Corollary 4.7.3 (via a change of variables) for

the others. The effect of these changes of variables is examined in the

following result.

4.7.5 Lemma. Let V, V be bounded domains in R "', 11: V -> V' an

invertible map onto V such that the derivatives D",q (resp. D"q - 1) exist and

are continuous in V (resp. V') for Ial < k. Given U E Wk"p(V'), define

Nu(x) = u(,q(x)).

Then N is a bounded, invertible linear operator from Wk,p(V') onto Wk,p(V ),

1<p<co.

For a proof see Adams [1975: 1, p. 63].

Proof of Theorem 4.7.4. Let V,', V2 , ... , V,b, V21',... be a finite cover

of 12 by interior and boundary patches and let X i, X z,

' X

i' Xb.

be a

subordinated partition of unity in the sense of Section 4.5. We have

un. (4:7.20)

Xnu=Eu;,+Y

u=yXnu+y

Each of the functions u' satisfies

BX(u,,q) = (gn1 0

(TE'6l(Vn')),

(4.7.21)

where

gn=Xnf

u(EEDJ(aJfkD'X;,)+EbJDJX'

)

- 2 EE ajkD1 uDkX n.

(4.7.22)

Thus it follows from Theorem 4.7.1 that u E H2(Vn'). Accordingly, u;, E

H2(SZ) and?

IIuin IH2(sz) < C(IIgin IIL2(Ia)+IlufIIHI(Q))

< C'(Il f II L'(Q) +II uII HI(I))

(n=1,2 ....)

(4.7.23)

7 Naturally, all the interior patches can be lumped together. Also, multiplication by X'

(and the subsequent calculation) can be easily avoided.

228

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

The treatment of boundary patches is not as immediate. It is still true that

un satisfies (4.7.21) (where q) belongs now to HH(Vb r) St)) with g, given by

an expression similar to (4.7.21). If q is the map associated with V ,,b, then we

can show that un (which is un as a function of i 1, ... , satisfies

Bjun

(wEHo(S+)),

(4.7.24)

where the tilde in gn is interpreted in a similar way and h,\ is the functional

defined in terms of the transformed operator Ao in the same way BA was

defined using A0. Evidently Ao is of class C(')(S+) and it was already

shown in Section 4.5 that

ELajksjsk = Eajksjsk,

(4.7.25)

where = J', J the Jacobian matrix of rl with respect to x; thus the uniform

ellipticity condition (4.7.2) is satisfied (with a different constant K bounded

below by a multiple of K in each patch). Since rib E Ho'(S') and it vanishes

for IxI < p < 1, Lemma 4.7.3 applies to show that un E H2(S ) and

IIunhiH2(S;,<C(IlgnlIL2(Sn+IIunhlHI(S ))1

which implies, making use of Lemma 4.7.5, that un E H2(St) and that

IlunllH2(St) < C'(Ilg,'llL2(sz) +IIunIIH'(t ))

<C"(Il liL2(0)+IIullH'(9))

In = 1,2,...).

Combining these inequalities with (4.7.20) and (4.7.23), we deduce that

u E H2(SZ) and (4.7.19) holds, thus ending the proof of Theorem 4.7.4.

It is easy to prove that (4.7.19) implies

IIu1IH2(ti)<CII(AI-A(/3))UIIL2(st)

(4.7.26)

with C depending only on St, K1, K, A. To see this, we only have to observe

that

IIullHo(se)<C11fIIL2(u)=C'll(AI-A($)u)IIL2(ti)'

which is an immediate consequence of the first inequality (4.6.5) for B,,,

since

cllullHa(tt)IIullL2(u) < cllullH1(tl)IIullHo(u) < I BX(u, u)I = I (f, u)I

< IIf

llL2(t)IIullL2(t2)

(u E Ho(st)).

We consider now boundary conditions of type (I). The arguments are

very similar, and the only piece of fresh information needed is a result

analogous to Lemma 4.7.3 for these boundary conditions.

4.7.6

Local Boundary Regularity Lemma.

Let Ao be of class C(') in

S+ and uniformly elliptic there. Assume u E Ht(5+) is zero for I x I < p < 1

4.7. Regularity Theorems

229

and satisfies

Bx,s(u,fp)=Bx(u,ro)- f§

yugpdo=(f,,p)

(q'E6D)

(4.7.27)

where y is continuously differentiable in 90 - and f E L2(S ). Then u E H2(S )

and there exists a constant C (depending only on K tc, A, MI, where

Mk =

max

(I D«Y(x)I )

X (=- 9p

ford=(a,,...,am_1), IdI=a,+ +am_I<k) such that

IIUIIH2(S) C(IIflIL2($

)

(4.7.28)

Proof.

Let i =1, 2,..., m -1. We compute the left-hand side of

(4.7.27) with Shu in place of u. The expression for Bx(Ahu, q9) is exactly the

same as that in (4.7.9). On the other hand,

f YI

hugPdo=-

f u0'-h(YP)da=- f u(0`_hYT`hrP+YO`-h9P)da,

§o

thus, if B. is the functional in (4.7.9) and B,,,, is similarly defined with

respect to the homogeneous operator (4.7.10),

A\,#(Alhu,T) = - BX,#(u, S-hP)

(EEL)r_hajkT' hDk

cpDju) dx

+f OA< -h7T` hqpdo,

(4.7.29)

§o

where the hypersurface integral on the right-hand side of (4.7.29) is esti-

mated with the help of the Schwarz inequality and Theorem 4.6.4:

u12da f

IThI2do

(f_hYThdx)2

§o

S,-

We proceed now exactly as in the end of the proof of Theorem 4.7.1 to show

existence of, and to estimate the second derivatives in (4.7.16); the missing

derivative (Dm)2u is accounted for by means of (4.7.17). Details are

omitted.

4.7.7

Global Regularity Theorem. Let 0 be a bounded domain of

class C(2) and let A0 be of class C(')(S2) and uniformly elliptic in Q. Let /3 be a

boundary condition of type (I) with y continuously differentiable on the

230

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

boundary I'. Assume that u E H'() belongs to D(A(/3 )), that is,

Bx #(u,w) = (f,w)

(w E H'(SZ)) (4.7.30)

for some f E L2(9) (f = (Al - A(#)) u). Then u E H2([) and there exists a

constant C (depending only on S2, K1, K, A, M,) such that

I1u11H2(Q)<C(IIfIIL2(t)+IIuIIH2(l)).

(4.7.31)

The proof is essentially similar to that of Theorem 4.7.4, the only

slight difference appearing in the treatment of the boundary patches

V,b, Vb.... /and the functions un

= Xnu. It is easy to see that each un satisfies

Bx,s(un,q)=Bx(un,q))-J

ynuny9da=(gn,gq)

(ggE60),

rnVh

(4.7.32)

where gn is given by an expression of the type of (4.7.22) and

yn = DYXn + Xny.

(4.7.33)

If q is the map associated with the boundary patch Vb and -

indicates, as

before, that a function is considered to be a function of a

straightforward computation shows that

Ba,R(Tin,92)=Ba(u"n,gp)-f pYnUny,dUgn,92)

(4D

E=- 6D)

where again B,, s, B. are the functionals corresponding to AD and p is the

normalization factor in (4.5.28). We can then apply Lemma 4.7.6 and the

proof runs much like that of Theorem 4.7.5.

In some instances it is advantageous to replace inequality (4.7.31) by

one of the type of (4.7.26), that is, by

I1ullH2(g)<CII(AI-A($))UIIL2(a) (4.7.34)

with C depending only on SZ, K1, K, A, M,. The argument leading to (4.7.34)

is very much the same as that justifying (4.7.26) in the case of Dirichlet

boundary conditions.

An important consequence of the regularity results just proved is

that the domains of the operators A(,8) can be now identified in a much

more satisfactory way than hitherto possible.

4.7.8

Corollary. Let S2 be a bounded domain of class C(2), and let A0

be of class C0) and uniformly elliptic in SZ. Finally, let 8 be a boundary

condition of type (I) or (II) (in the first case, with y continuously differentiable

sln this, an important role is played by the invariance of the conormal vector (or

derivative) shown in (4.5.28).

4.7. Regularity Theorems

231

on r). Then D(A(,3)) consists of all u E H2(SZ) that satisfy the boundary

condition $ almost everywhere on F. When Q is the Dirichlet boundary

condition, we can also write

D(A($)) = H2(1)n Hot (0). (4.7.35)

Proof. We begin with the case of Dirichlet boundary conditions.

The inclusion D(A(fl3)) C H2(S2) results from Theorem 4.7.4 and the fact

that any u E D(A($)) c Ho(2) vanishes a.e. on IF has already been dis-

cussed in the previous section (see the comments after Theorem 4.6.5). The

converse is a consequence of the following result and of a simple application

of the divergence theorem:

4.7.9 Lemma.

Let 9 be a bounded domain of class C('), 1 < p < oo,

u an element of W ' , n (9) such that u = 0 almost everywhere on the boundary

r. Then u e W I.P(SZ) = closure of 6D(0) in W'"P(2).

The proof is based on Theorem 4.6.5 and on the method used in the

proof of Theorem 3.18 in Adams [1975: 1, p. 54]. We omit the details.

The case of boundary conditions of type (I) is handled as follows.

Let u be an arbitrary function in 6D. Making use of Theorem 4.6.4 for the

first derivatives of u, we obtain

IID°uIIL2(r) < CIIuIIHZ(u)

(4.7.36)

(the index 2 on the left-hand side can be improved to 2(m - 1)/(m - 4)).

We take u E H2(E2) and approximate it in the H2 norm by a sequence (un)

in D. Applying (4.7.36) to u - un,, we can then define

D°u = lim D°un,

(4.7.37)

the limit understood in the norm of L2(I') and (as easily seen) defined

independently of the sequence (un). Let $ be a boundary condition of type

(I) and u E D(A(/6)). We have already shown that u E H2(SZ) (Theorem

4.7.7), thus Dv is defined in the sense indicated above. Applying the

divergence theorem to each member u of an approximating sequence and

to a 99 E 6)1 we obtain

Bx,s(un,T)=Bx(u,,,'T) - frD°unTda=fa((JAI-A,)un)T dx.

(4.7.38)

Letting n --> oo, we use the fact9 that AIun -* Atu in L2(S2) while D''un

converges in L2(r). We take limits in (4.7.38) and conclude from the

9An abuse of notation is implicit here since AI was not originally defined in H2(0)

(see Section 4.5).

232

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

definition of A(#) through B,\,# that

f Yibpdo= f yugpdo

r r

for arbitrary 99 E 6D; hence

D"u = yu a.e. on IF (4.7.39)

as claimed. Conversely, let u E H2(1) such that (4.7.39) is satisfied a.e. on IF

and let be again an approximating sequence in 6D. Using the divergence

theorem and taking limits, we easily obtain that BA s(u, qp)

p) for

every T E 6D, where f = (Al - A0)u, thus u e D(A(fl)).

Stronger regularity results can be obtained under stronger assump-

tions on the coefficients of A0 and the domain Q. These results, which we

prove below, will be the basis of the treatment in LP spaces for p > 2 in the

following sections.

4.7.10 Interior Regularity Theorem. Let the hypotheses of Theorem

4.7.1 be satisfied with A0 of class C(k+D in SZ and f in Hk(S2). Then

u E H(k+2)(Q) and there exists a constant C (depending only on 0, S2', Kk, K,

A) such that

IIuIIHk+(&2.)<C(IIfIIHk(a)+IIuIIHt(St)).

(4.7.40)

Proof We only consider the case k =1 since extension to k > 1

involves no new ideas. Let 2" be a domain with U' c: 0", SZ" c 2. By

Theorem 4.7.1, u E H2(S2"); it is then easy to see that ui = D'u E H'(S2")

satisfies

BX(ui,T)= (fi,T)

(4.7.41)

for p E 6D(9"), where

fi =

+Y_Y_ D'aI'kDjDku+Y_ D'(b, +Y_ Dkajk)Diu+(D'c)u.

(4.7.42)

Clearly we have

IlfIIL2(a..)

If IIH'([Z-,)+CIIuIIH2(12,,),

where C depends only on 2, 0", K21 K, A. Inequality (4.7.40) then follows

immediately from Theorem 4.7.1.

4.7.11

Local Boundary Regularity Lemma.

Let the hypotheses of

Lemma 4.7.3 be satisfied with A0 of class C(k+1) in S+, and f E Hk(S+)

Then u E Hk+2(S+) and there exists a constant C (depending only on Kk, K,

A) such that

IIuIIHk+z(S;) , C(IIf II Hk(s+) +II uII HI(§+) )'

(4.7.43)

Proof.

Again we only consider the case k = 1. Since u E H 2 (S + ),

ui = D'u E H'(S+) for any i. Moreover, if i =1,2,...,m - 1, we check easily

4.7. Regularity Theorems

233

that u; = 0 a.e. on So ; thus, using Lemma 4.7.9, we see that u, = Du E

H(S) for i =1, 2,..., m -1 and each u, satisfies (4.7.41) with f, given by

(4.7.42) and vanishes in

I x I < p. Hence Lemma 4.7.3 shows that all third

generalized derivatives D'DjDku exist for 1 < i < m - 1,

1 < j, k < m and

their L2 norm is bounded by the right-hand side of (4.7.43). The only

derivative absent from the argument is (D' )3u. However, we can express it

by means of the others in the style of (4.7.4) keeping in mind that

EEajkD'Dk(Dmu)+F,(bj +EDkajk)Di(Dmu)+ c(Dmu) = ADtu - fm.

This ends the proof.

4.7.12 Global Regularity Theorem.

Let the assumptions of Theorem

4.7.7 be satisfied with 9 of class C(k+2), A0 of class C(k+q in SZ and

f = (XI - A()(3))u in Hk(SZ). Then u E Hk+2(S2) and there exists a constant

C (depending only on 9, Kk, Ic, A) such that

IIuIIHk+2(t) <C(II If IIHk(Q)+IIuIIHl(O))

(4.7.44)

The proof follows closely that of Theorem 4.7.4 and is thus omitted;

in the boundary patches we use of course Lemma 4.7.11 in lieu of Lemma

4.7.3.

If A is large enough, inequality (4.7.44) implies

IIuIIHk+2(U) < CII(XI - A(Q)) ull

(4.7.45)

with C depending only on 0, Kk, Ic, A; to see this, we argue as in the proof

of (4.7.26).

4.7.13

Local Boundary Regularity Lemma.

Let the hypotheses of

Lemma 4.7.6 be satisfied with A0 of class C(k+') in S"' Y E C(k+ 1)(90M),

f E Hk(S+). Then u E Hk+2(S+) and there exists a constant C (depending

only on Kk, K, A, Mk) such that

IIuIIHk+2(S,)<C(IIfiIHk(§m)+IIuIIH'(S,)).

(4.7.46)

Proof. Once again we only consider the case k = 1, and set u, = D'u,

1 < i < m - 1. The function u, belongs to H'(S+) and satisfies

Ba.,(ut,q)=Bx(u,,gq)- f§'- 7Ujggdo=(fr,p)+ f§

h,(p do

(4.7.47)

for qD E 6 (1), where f, is given by (4.7.42) and

h; _ (D'y)u+(Y_D'ajDiu.

Strictly speaking, Lemma 4.7.6 is not directly applicable here due to the

presence of the hypersurface integral on the right-hand side of (4.7.47).

However, this additional term can be estimated with the help of Theorem

234 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

4.6.4:

If h;cpda

<C(M2IIuiIH'(S'")+mK211uliH2(S n)IIroIIH1(§m)

Using now (4.7.26) for the second term between parenthesis, we apply the

argument in Lemma 4.7.6 to the difference quotients Ohu,, 1 < j < m - 1

and prove that each u; belongs to H2(S2) and that the derivatives D'DJD k u,

1 < i < m - 1, 1 < j, k < m satisfy an estimate of the type of (4.7.73). The

derivative (D )3U is captured as in the end of Lemma 4.7.9.

4.7.14 Global Regularity Theorem.

Let the assumptions of Theorem

4.7.7 be satisfied with 2 of Class C(k+2), A0 of class C(k+') in g, .y E C(k+')(I'),

and f = (Al - A($))u in Hk(1). Then U E Hk+2(Q) and there exists a

constant C (depending only on 9, Kk, ic, A, Mk) such that

II U I I

Hk+2(u) < C(II If II Hk(g) +11u I I H'(9)).

(4.7.48)

The proof follows that of Theorem 4.7.12 and is thus left to the

reader.

As in the case of Dirichlet boundary conditions we can show that

(4.7.48) implies

IIUIIHk+z(t)<CII(AI-A($))uiIHk(U) (4.7.49)

for A large enough with C depending only on 2, Kk, K, A, Mk.

With the help of the following result, a particular case of the Sobolev

imbedding theorem, we can show that u must in fact have a certain number

of derivatives (in the classical sense) if k is large enough.

4.7.15

Theorem.

Let 9 be a domain of class C(') and let uE

Wk,p(1) with I < p < oo and (k - j)p > m for some j > 0. Then u E C(')(S2)

(after eventual modification in a null set) and

Il ull cu)(Si) < CII ull wk.P(u),

(4.7.50)

where the constant C does not depend on u.'0

The result is a particular case of Theorem 5.4 in Adams [1975: 1, pp.

97-98] and yields the following

4.7.16

Corollary.

Let the assumptions of Theorems 4.7.12 or 4.7.14

be satisfied with k +2- j > m/2 for some j > 0. Then u E C(')(S2) and

Ilul1c(J(5)<C(IIfIIHk(SZ)+IIuiiH'(0)),

(4.7.51)

'oThis result holds under the sole assumption that fl possesses the strong local

Lipschitz property (Adams [1975: 1, p. 66]), which holds for any domain of class CO).

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

235

where C does not depend on u. If the assumptions are satisfied for all k, then

u E C(-)(S2).

4.8.

CONSTRUCTION OF m-DISSIPATIVE

EXTENSIONS IN Lp(S2) AND C(0)

The results in the previous section are sufficient to assemble suitable

extensions in LP(2); however, a precise characterization of the domains can

only be obtained using a deep a priori estimate (Theorem 4.8.4). We assume

again throughout this section (unless otherwise stated) that 0 is a bounded

domain of class C(2) and that A0 is an operator of class 0 in 9, although

we shall occasionally be forced to work with smoother domains and

operators in several results. The uniform ellipticity assumption (4.7.2) is

assumed to hold. Since A0 will be written both in the form (4.4.1) and in

variational form, we shall denote by bi in this section the first order

coefficients in (4.4.2).

4.8.1 Lemma Let /3 be a boundary condition of type (II) or of type

(I) with y continuous on r, 1 < p < oo, W2'p(E2)a the space of all u E W2'p(SZ)

that satisfy /3 a.e on r, C(2)( 9),8 the space of all u E C(2)(S2) that satisfy /3 (in

the classical sense) everywhere on F. Then C(2)(C2)p is dense in W2"(Q)S in

the norm of

We note that the boundary values and the boundary derivatives of a

function u E

are defined in the same way as for p = 2 in Corollary

4.7.8:

u=limu inLP(F),

(4.8.1)

D°u=limD°u inLP(F), (4.8.2)

where is an arbitrary sequence in 6 converging to u in W2 , p( a );

existence of the limit follows from Theorem 4.6.4, and existence of the

sequence from Theorem 4.6.5. The exponent p in the two limit relations

(4.8.1) and (4.8.2) can be improved except when p =1 (see again Theorem

4.6.4). The proof of Lemma 4.8.1 is fairly elementary and we omit it.

4.8.2

Corollary.

Under the assumptions of Theorem 4.6.6, we have

A0(/3)= A($) = (A0($'))*,

(4.8.3)

the closure and adjoint understood in L2(S2).

Proof. Combining (4.7.13) with (4.7.26) for the Dirichlet boundary

condition (or (4.7.34) for other boundary conditions), we obtain that the H2

norm and the graph norm I J(A I - A (fl)) u I I (A large enough) are comparable

in D(A($)). The first equality (4.8.3) then results immediately from Lemma

236

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

4.8.1 for p = 2, the second from the first applied to the formal adjoint

A'(,8') and from (4.6.30).

We can now construct the desired extensions in LP(S2).

4.8.3 Theorem.

Let I < p < oo, 9 a bounded domain of class C(2) if

1 < p < 2 (of class C(1+2) with k > m/2 if 2 < p < cc), A0 an operator of

class CO) if I < p < 2 (of class C(k+l) if 2 < p < oo) satisfying the uniform

ellipticity assumption (4.7.2),,8 a boundary condition of type (II) or of type (I)

with y E C(')(F) (y E C(k+')(F) if 2 < p < oo) satisfying

(x)+ p Y_ bj (x) vj <0 (x e F),

(4.8.4)

l - I

where v = (vi,..., vm) is the outer normal vector at x (if p = oo, this inequality

is y(x) < 0). Finally, let bb E 01w(S2) and

8 5)

= max (c(x)-

P )

DJb(x))

. .

J=l

(wp = max c (x) ifp = oo ). If 1 <p <oo, the closure AP(fl)=A0(fl) in LP(2)

belongs to C3+(1, wp ). Moreover, if 1 < p < oo, we have

AP(13) = Ao($) = A'

($')* = (Ao ($l) )*,

(4.8.6)

where A'($') is thought of as an operator in LP'(SZ) (p'-' + p

' =1). Finally,

if p = oo, the closure A,,, ($) =Ao(/3) in C(S2) (Cy(st) if $ is of type (II))

belongs to (2+(1,

Proof We note that if w,, w. are defined by (4.8.5) in relation to

the formal adjoint A', then we check easily that wp = wp. for 1 < p < oo, and

that w = w 1. Obviously we may assume that wp = 0; otherwise we simply

replace Ao($) by A0($)-wPI. The same observation holds for w... We

begin with the case E = LP(R), I < p < 2. It follows from Corollary 4.8.2

that if A > 0 is

large enough, (XI - A0(/3))D(A0($)) is

dense in

(XI - A($))D(A($)) = L2(s2). We see that (JAI - A0($))D(A0(/3)) is dense

in L P(Q), 1 < p < 2, and by virtue of Lemma 3.1.11, Ao ($) is m-dissipative

in LP(2) (recall that Ao($) is dissipative in view of Lemma 4.5.2). The case

p > 2 is slightly more complex to handle. Let f E H"(SZ)fl LP(2) = Hk(s2)

and let u E D(A($)) be the solution of (XI - A(/3))u = f for sufficiently

large A > 0. In view of Theorem 4.7.14 (or Theorem 4.7.12 in the case of

Dirichlet boundary conditions), u E Hk+2(2), hence u E 02)(0) by Theo-

rem 4.7.15. It follows from Corollary 4.7.8 that u satisfies # almost every-

where (thus everywhere) on r, hence u E C(2)(st)s = D(A0(/3)). Since f is

arbitrary we deduce that (XI - A0($))D(A0($)) 2 Hk(st), which is dense

in LP. As we already know that A0($) is dissipative in LP (Lemma 4.5.2),

we obtain again from Lemma 3.1.11 and Remark 3.1.12 that A00) is