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

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4.6. Second Order Partial Differential Operators

217

tinuously to all of L2(S2), and there exists a unique element v E L2(12) such

that

1\(u,w)-Bx(u,w)=(v,w) (wEH0'(E2)).

(4.6.15)

By definition,

v=A(/3)u

(4.6.16)

and it is clear that A(/3)u is linear. We show that

(XI - A(/3))D(A(/3)) = L2(E2).

(4.6.17)

To do this, let v E L2(E2). Since (D(w) = (v, w) is continuous in L2(E2)-thus

in H01(9)-we use Lemma 5.6.1 to deduce the existence of a u E Ho(I)

such that Ba(u, w) _ (v, w); in view of the definition of A (p), u e D(A(/3 )),

and (A I - A(0))u = v, thus proving (4.6.17). We have not shown yet that

D(A(/3 )) is dense in

L2(E2). Since H01(2) is dense in L2(S2) it is enough to

prove that D(A(/3)) is dense in Ho(S2). If this were not the case, applying

Lemma 4.6.1

to ff., we could find a nonzero w E Ho (S2) such that

B,(u, w) = 0 for all u E D(A(/3)). In view of (4.6.15), this would imply that

((Al - A(/3))u, w) = 0 for all such u, hence w = 0 by virtue of (4.6.17). It

only remains to be shown that A(/3) belongs to (2+(1, w) for to large enough.

This is obvious from (4.6.14) since

Re(A(/i)u, u) = XII uII2 -ReBx(u, u) '< XIIuII2 (u E D(A));

(4.6.18)

it suffices to set w = A.

We have completed the proof of:

4.6.2 Theorem.

Let 0 be an arbitrary domain in ff8 m and let the

coefficients of the differential operator A0 be in L°°(2) and satisfy (4.6.2);

assume that /3 is the Dirichlet boundary condition. Then the operator A(#)

defined by (4.6.15) and (4.6.16) belongs to (2+(1, w) for w sufficiently large.

The treatment of boundary conditions /3 of type (I) is slightly

different. Here the basic space is H'(S2). To surmise the form of the

functional B to be used now, we perform the following computation, where

I' is the boundary of 9:

fa{EEa,kD1uDkv-(EbjD-'u)v+(A-c)uv)dx

f(D"u)vdo+ f((XI-A1)u)vdx,

thus if /3 is to be satisfied, the functional must be

Bx,R(u, v) = f{Y_Ea,kD'uDkv - (Y_ bD'u)v+(A- c)uv) dx

-fryuvdo=B.(u,v)- fryuvdx,

(4.6.19)

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Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

and if the assumptions of Lemma 4.6.1 are to hold we need to postulate

that:

(B) I' is smooth enough to allow definition of the hyperarea differen-

tial da.

exists a positive constant a' < 1 such that

frym do

< a'IIq)IIxII

III (9), O E 6D)

(4.6.20)

for X large enough.

The functional Bx,, is defined by (4.6.19) (strictly speaking, first for

u, v E 6D due to the surface integral term and then extended to u, v E H'(12)

by continuity of Bx

and denseness of 6D: see Theorem 4.6.5). To obtain an

inequality of the type of (4.6.14) for B,,,, in H'(S2), it suffices to use (4.6.14)

in H'(S2) for Ba with a < 1- a'.

The construction of A(/3) runs very much along the same lines as in

the case of the Dirichlet boundary condition. Since B,,,,, satisfies the

assumptions of Lemma 4.6.1, we can define A($) by

X(u,w)-Bx,,,(u,w)=(A(/3)u,w), (4.6.21)

the domain of A($) defined as the set of all u e H'(2) such that w -->

Ba $(u,w) is continuous in the norm of L2(SZ). Denseness of D(A(,8)) and

equality (4.6.17) are proved exactly in the same way. Finally, to show that

A(#) E e (l, w), we only have to note that an obvious analogue of (4.6.18)

holds for A($) and B., . We have thus completed the proof of

4.63 Theorem. Let 9 be a domain in R' with boundary r satisfying

(B). Assume the coefficients of A0 are in L°°(1) and satisfy the uniform

ellipticity condition (A). Finally, let y be a locally integrable function such that

w large enough.

It is natural to ask in which sense, if any, functions in the domains of

the operators ^8) in Theorems 4.6.2 and 4.6.3 satisfy the boundary

condition P. The fact that every element of D(A(/3 )) (in fact, of H(2)) will

(almost) satisfy the Dirichlet boundary condition $ if adequate conditions

are imposed on I' follows from the next result, which is stated in much more

generality than needed now, with a view to future use.

4.6.4 Theorem.

Let 0 be a bounded domain of class C(k), k >_ 1,4

and let l < p < oo. Then (a) if

1< p<m/k, l<q< (m-1)p/(m-kp)

(4.6.22)

4Or, more generally, having the uniform C(k)-regularity property and a simple k-exten-

sion operator: see Adams [ 1975: 1], especially pp. 67 and 83. It follows from the definition in p.

67 that a bounded C(k) domain has the uniform C(k)-regularity property; on the other hand,

since r is bounded, the existence of the simple k-extension operator is automatic (loc. cit. p. 84)

4.6. Second Order Partial Differential Operators

219

there exists a constant C (depending only on SZ, p, q) such that

IIUIIt9(r)=

(fIuIda)

<Cjjujjwk.p(.)

(4.6.23)

for every u

E C(k)(SZ).

(b) If

p >_ m 1k,

(4.6.24)

then (4.6.23) holds for every q > 1.

The proof (of a more general result) can be seen in Adams [1975: 1,

p. 114].

The next result we need is on approximation of elements of

by functions of 61. (which was already used in the construction of B,,,, in

(4.6.21)).

4.6.5 Theorem.

Let SZ be a domain of class 00), 1 < p < oo. Then 61.

(or, rather, the set of restrictions of functions of 6D to SZ) is dense in Wk-P(SZ).

For a proof see Adams [ 1975: 1, p. 54], where the assumptions on the

domain 2 are less stringent.'

By means of Theorems 4.6.4 and 4.6.5 we can prove that elements of

H'(SZ) = W"2(SZ) "have boundary values" in a precise sense. Let u E H'(SZ)

and let be a sequence in 6D converging to u in H'(S2). Then it follows

from (4.6.23) that the restrictions of

to F (which we denote by the same

symbol) converge in L2(F) (actually in L2(m-')/(ni-2)(0)) to an element of

L2(I'). We can thus define the restriction or trace of u to F by

u=limu,

the limit understood in L2(I') (it is easy to see that it does not depend on

the particular approximating sequence C 6D used). It follows then that

elements of A($) c HH(SZ) do vanish on F in the sense outlined above since

the approximating sequence may (and must) be chosen in 6D(2).

Treatment of a boundary condition of type (4.6.6) is not as simple

since functions in H'(Sl) do not "have derivatives at the boundary," and a

more careful characterization of D(A($)) is imperative. This will be done in

the next section, under additional smoothness assumptions.

From now until the end of the chapter we place ourselves under

more comfortable hypotheses than those hitherto used. The domain SZ will

be bounded and at least of class C(') (although a piecewise C(') boundary,

like that of a parallelepipedon, is acceptable). The coefficients ask, b1 will be

supposed to be continuously differentiable in SZ, while c is continuous there;

also, y is continuous on F. We also assume that the uniform ellipticity

5The segment property suffices (see Adams [1975: 1, p. 54]).

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Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

assumption (4.6.2) holds everywhere in 12 for some [c > 0. In this framework,

hypotheses (B) and (C) on the boundary r are automatically satisfied. This

is obvious for (B); to prove (C) it is enough to apply Theorem 4.6.4 with

k =1, p = q =1, obtaining

fry9f 4, d a

But

<C 11'FOLi(r)<C119fOwI l(a).

II I'Ilw'.I(a)= II

III IIL-(sz)+II(D'vf)PIILI(sz)+E11'WOIIL'(0)

< 11p1IL2(c2)II

IIL'(SZ)

-1

+2m1/2K-1/2K-1/2)IIq

IIall ll2,,

(4.6.25)

where K is the constant in (4.6.12) and y), 0 are arbitrary functions in 6D.

Under the present assumptions, the operator Ao can be applied to

twice differentiable functions. Given a boundary condition /3 of any of the

forms (4.6.6) or (4.6.7), we denote by A0(/3) the operator A0 acting on the

space C(2)(SZ),3 of all functions u E C(2)(S2) which satisfy the boundary

condition /3. Since 6 (SZ) c C(2) (S2 )R, it is clear that D(Ao (/3 )) is dense in

L2(S2) for any boundary condition P.

Assume for the moment that $ is a boundary condition of type (I)

be the formal adjoint of A0,and let A'

u =

EEDJ(ajkDku)- EDJ(bj(x)u)+c(x)u, (4.6.26)

and /3' the adjoint boundary condition

D"u(x) _ (y(x)+b(x))u(x).

(4.6.27)

We can apply the theory hitherto developed to A' and /3'; the bilinear

functional is now

BA 8,(u,v)= fU{EEajkDhiDkv+(Eb, DJu)v+(A-c +EDJbj)uv) dx

-f (y+b)uvda.

Assuming that u and v E 6D, we obtain applying the divergence theorem (see

(4.5.40)) that

(u, v) =B' Q v, u

(4.6.28)

4.6. Second Order Partial Differential Operators

221

for u, v E 6D and thus (by the usual approximation argument) for u, v E

H'(2). The operator A'(/3') is defined by

A(u,w)-B"

0,(u,w)= (A'(/3')u,w),

(4.6.29)

the domain of A'(/3') consisting of all those u E H'(1) that make Ba

L2()-continuous in HH(SZ). It follows easily from this and from (4.6.28)

that (A'(/3')u, w) = (u, A($)w) if u E D(A'(/3')) and w E D(A(/3)), so that

A'(/3') c A(/3)*. But A'(/3') belongs to C'+, while A(/3)*, being the adjoint of

an operator in C+, belongs itself to (2+ (Theorem 3.1.14). Accordingly both

R (X; A(#)*) and R (X; A'(/3')) exist for A large enough and we prove as in

the end of Theorem 2.2.1 that

A'(/3') = A(#)*. (4.6.30)

Equality (4.5.40) shows that Ao(/3) c A(#), A' (/3') c A'(/3'), thus we obtain

upon taking adjoints,

A0(18) c A($) c Ao(/3')*,

Ao($') c A'($') c Ao(/3)*. (4.6.31)

As we shall see in the next section (under additional assumptions), the two

double inclusion relations (4.6.31) are actually equalities (with A0(/3), A'(#')

replaced by their closures).

It remains to deal with Dirichlet boundary conditions. Here HI(a)

must be replaced by Ho(S2) and the adjoint boundary condition /3' is of

course /3 itself (see Section 4.5); we leave the details of the argument to the

reader.

We collect some of the preceding results.

4.6.6

Theorem. Let 2 be a bounded domain of class CM and let

ask, bj be continuously differentiable in S2, c continuous in S2; assume the

uniform ellipticity condition (4.6.2) is satisfied everywhere in SZ for some K > 0.

Then (i) If l3 is the Dirichlet boundary condition, Ao(/3) is densely defined and

possesses an extension A(/3) E e+(1, w) defined from the functional B;, for A

large enough by (4.6.15), (4.6.16). (ii) If /3 is an arbitrary boundary condition

of type (I) with y continuous on r then AO(/3), again densely defined, possesses

an extension A(/3) E 3+(l, w) defined as in (i) but using the functional BA,,8.

The duality relation (4.6.30) holds in all cases.

4.6.7 Remark.

All the developments in this section can be applied to the

case SZ = R ', where F is empty. The treatments for all boundary conditions

coalesce.

4.6.8 Remark.

The method used to construct the operator A(#) can be

made to function under much weaker assumptions. For instance, consider

the operator

Aou= E Y_Dj(ajk(x)Dku)+c(x)u, (4.6.32)

where the ask and c are merely measurable and locally integrable (but not

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Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

necessarily bounded) in an arbitrary domain 0 C R m; we require in addition

that c be essentially bounded above and that the symmetric matrix d(x) _

(ajk(x)) be positive definite, that is,

(dt,

(4.6.33)

a.e. in SZ (note that (4.6.33) does not prevent the quadratic form

from degenerating anywhere in S2, or even the ajk from vanishing at the

same time in parts of 0). Consider for simplicity the case where Q is the

Dirichlet boundary condition. Instead of Ho(1), the basic space is now

(S2), the completion of 6 (S2) in the norm derived from the scalar product

(q:, tP)x= f {Y-Eajk(x)DjrpDki+(A-c(x))gfp)dx (4.6.34)

with A >_ esssup c + 1. The functional Ba is nothing but Bju, v) = (u, v)x

(in which case Lemma 4.6.1 actually reduces to the representation theorem

for bounded linear functionals in a Hilbert space) and the construction of

the operator A($) proceeds without changes.

Addition of first-order terms bjDju to the operator A0 is possible,

but we must assume that, roughly speaking, "the bj vanish where

degenerates." Precisely, we postulate the existence of constants A > 0,

0<a<1 with

(Lbj(x)Sj)2 5 a2(A- c(x))EEajk(x)Sjsk

(5 E=-R')

(4.6.35)

a.e. in Q. This guarantees an inequality of the form (4.6.14), thus the

construction of A(/3) is carried out in the same way.

Boundary conditions of type (I) are handled through assumptions

(B) and (C); here we require the constant a' in (4.6.20) to satisfy a' < 1- a,

a the constant in (4.6.35). Of course, the question of in what sense the

boundary conditions are satisfied admits an even less clean cut answer in

this level of generality, even for the Dirichlet boundary condition: roughly

speaking, the boundary condition u = 0 will be satisfied in the part of the

boundary F where EEa; is nondegenerate (see the discussion in

Section 4.5 on assignation of boundary conditions).

4.7. REGULARITY THEOREMS

We place ourselves here under (some of) the assumptions in the second half

of Section 4.6. To begin with, we shall assume throughout the rest of the

chapter, unless otherwise stated, that S2 is of class C(2). For the sake of

simplicity, we also assume that 0 is bounded, although this is not essential in

4.7. Regularity Theorems

223

some results. The operator A0 will be written in divergence form,

m m m

Aou= Y, E Dj(ajkDku)+ E bjDju+cu,

(4.7.1)

j=1 k=1 j=1

and to simplify future statements we introduce the following definition: A0

is of class C(k), k >-I in 0 (in symbols, A0 E C(k)(S2)) if and only if

ajk E C(k)(C2), bj, c E C(k- 1)(SZ). We shall assume throughout that A0 is

uniformly elliptic:

(4.7.2)

for some K > 0 and write

X E= 02

for 1 < j, k < m and I a I < n, I it I < n - 1; clearly the definition makes sense

if A0EC(k)(F2)and 1<n<k.

Roughly speaking, the results in this section state that the solution of

(XI - A (/3 )) u = f is "two degrees smoother" than f in the L2 sense, at least

if the coefficients of A0 are smooth enough. In all that follows, k > 0 will be

a sufficiently large parameter, this expression meaning that all the argu-

ments in Section 4.6 function, in particular, the second inequality (4.6.3) is

satisfied by B. in the case of Dirichlet boundary conditions or by B,,,, for

boundary conditions of type (I).

Our first result holds with no assumptions on Q.

4.7.1 Interior Regularity Theorem.

Let 9 be an arbitrary bounded

domain in Rm and let A0 be of class C(')(11) and uniformly elliptic. Assume

u E H1(52) satisfies

Bx(u, (P) = (f, 9)) = f fpdx (T E 6D(l)) (4.7.3)

for some f E L2(12), and let 2' be a domain with FTC Q. Then u E H2(1) and

there exists a constant C (depending only on 0, SZ', K1, K, k) such that

IIuIIH2(s1-) <C(II f IIL2(,Q)+IIulIHI(SZ)).

(4.7.4)

Proof.

Let X be a function in 6 (0) such that X(x) =1 in 2' and let

v = X u. Plainly v E Ho (S2) and a simple computation shows that v satisfies

Bjv, 9q) = (g, p)

(9' E 1 (52)), (4.7.5)

where

g = Xf -

u(Y_Y_Dj(ajkDkX)+ F,bjDjX)-2EEajkDjuDkX.

(4.7.6)

Let now h be a nonzero real number, i an integer (1 < i < m), e; =

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Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

(0,..., 0, 1, 0,..., 0) E iR m (1 in the i-th place). Given an arbitrary function

w = w(x), we define

TTW(x) = w(x + he;), L'hw(x) = (T w(x)- w(x))/h.

Obviously A'hw will approximate D'w for h small. The precise statement we

need is:

4.7.2 Lemma.

Let 1 < p < oo, 0 an arbitrary domain in p ', u a

function in LP(l ). Assume there exists a constant C such that

IIO'hWIILP(1') < C

for h small enough, where S2' is any domain with K2' c R. Then the derivative

D'u exists in the sense of distributions in 0, belongs to LP(S2) and

IID'wII

LP(0) <

C. (4.7.7)

For a proof, see Gilbarg-Trudinger [1977: 1, p. 1621.

We note the following formal rules of computation with the opera-

tors Th and L)h:

DjI'h=O'hDj,

)zdx=- f0w(z _hz)dx,

0`h(wz) 0'hw)Thz + w0'hz (4.7.8)

(the second holds if w or z vanish near the boundary of 2 and It

sufficiently small). Using these rules, we take qp E 6 (SZ) and compute:

(lhv'T)=fu{,Y,ajk(0`hDjv)Dkg7+X(0'hv)q))dx

_ -ff(Y,Y,''_h(ajkDkp)Djv+Xv0'-hp)dx

=-8l(v,s-hT)

f.(LLA'_hajkT hDkWDjv) dx, (4.7.9)

where we denote by B. the functional corresponding to the homogeneous

operator

m m

Aou= E E Dj(ajkDku). (4.7.10)

j=lk=1

Let g be the function in (4.7.5). Define

g=g+37,bjDjv+cv

4.7. Regularity Theorems

225

We use now (4.7.9) keeping in mind that Ba(v, z'

_ h P) =

(g, 0'_ h97) and that

both inequalities (4.6.3) hold for fix (where c, C only depend on ic, A, K,) in

the norm II -IIA or (with different constants) in the norm of Ho(2). Note first

that, since A'hv E H01(2) for h sufficiently small, there exists q) E 6D(2)

(depending on h) such that

IISht)-9)IIHo(t)<

are, for the moment, the constants in (4.6.3). Hence

IBX(O'hv,9')I

SIB&(Shv,Shv)I -IBx(O'hv,9)-O'hv)I

c11O'hvllHO(a)-CIIO'hvIIHo(at)IIT-O'hVIIHo(st)%

2I1AhVIIHo(ct)

(4.7.11)

By virtue of (4.7.9) and (4.7.10),

IBx(O'hv,T)1 <C'(1181IL2(g)IIO'-h9911L2(a)+IIvIIHo(Q)IITIIHp(&2))I

(4.7.12)

where C does not depend on h. Noticing that

IIo'-h9'IIL'(t) < IITIIHo(t) < C"IIo'hvIIH,,(t)'

where C" is likewise independent of h, we obtain combining (4.7.11) and

(4.7.12) that

IIo'hVIIH,(a)<C(IIg11L2(a)+IIVIIHO(SZ))

IIO'hD'vIIL2(si) <

C(IISIIL2(g) +II VIIHl(J))

for arbitrary i, j, thus the theorem results from Lemma 4.7.2 and the fact

that u = v in S3'.

Under the assumptions of theorem 4.6.6, we obtain the important

result that A ($) (,0 any boundary condition) acts on elements of its domain

"in the classical sense"; in other words, if u E D( A (/3 )),

A(P)u=EF, aj,DjDku+Y_ (bj +Y, Dkajk)Dju+cu, (4.7.13)

where all the derivatives of u involved exist6 and belong to L 10C2

( Q). To see

this, let f be the right-hand side of (4.7.13); an integration by parts shows

that Bx(u, T) = (Au - f, qq) for qq E 6 (SZ), thus by uniqueness, A(/3)u is

given by (4.7.13).

Theorem 4.7.1 contains no information on the global behavior of u,

precisely, on whether u belongs to H2(SZ). To obtain results of this type, we

begin with a local result for Dirichlet boundary conditions.

60f course, in the sense of distributions.

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Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

FIGURE 4.7.1

4.7.3 Local Boundary Regularity Lemma.

Let Ao be of class 0) in

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

and satisfies

BA(u,(p)=(f,T)

(pE6D(§+)),

(4.7.14)

where f E L2(S ). Then u E H2(S ) and there exists a constant C depending

only on K,, K, A such that

IIUIIH2(§,,5 <c(IIfiiL2(S )+IIuIIHi(s ))

(4.7.15)

The proof is a rather immediate adaptation of that of Theorem 4.7.1.

Since u is already zero for I x I < p < 1, multiplication by X is unnecessary. A

moment's reflection shows that all the arguments of Theorem 4.7.1 apply as

long as the operators Th, Oh act on directions parallel to the hyperplane

xm = 0, that is, if I < i < m - 1. Accordingly, we can prove existence of

D`Dju (1<i<m-1, 1< j<m)

(4.7.16)

in L2(S') and establish for each an estimate of the required type. The only

second derivative not included in (4.7.16) is (Dm ) 2 u; however, by virtue of

Theorem 4.7.1, (Dm)2u exists in the sense of distributions in S

and is

locally square integrable there: in view of (4.7.13),

(Dm)2u = -

amm(Y-Y-ajkDJDku + Y_ (bj + Y_ Dkajk)Diu + cu)

+ am',(Xu - f ), (4.7.17)

where the double sum is extended to indices (j, k) * (m, m). Since

K-' by virtue of the uniform ellipticity assumption (4.7.2), the

corresponding estimate for (Dm)2, hence (4.7.15), follows.

We piece together these results.

4.7.4 Global Regularity Theorem.

Let Q be a bounded domain of

class C(2), and let Ao be of class C(') in 2 and uniformly elliptic. Let