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

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4.3. Applications to Ordinary Differential Operators

187

for n large enough. This shows, in view of (4.2.21) that IIR(A,,Al <S-t By

virtue of Lemma 3.5, A E p(A). This ends the proof, since (4.2.24) is an

immediate consequence of (4.2.21).

4.2.5

Corollary.

Let A be a densely defined operator in E such that

Re(B(u), Au) < - SIIm(0(u), Au) I (u E D(A)), (4.2.25)

where 0: D(A) - E* is a duality map and 8 >, 0. Assume, moreover, that

(XI - A)D(A) = E (4.2.26)

for some A > 0. Then A E I p -) f1(2+(1, 0), where ro = arc tg S. In particular,

if (4.2.26) holds and Im(0(u), Au) = 0,

(0(u), Au) < 0

(u E D(A)), (4.2.27)

then A E l ((vr/2)-)fl (2 +(1,0).

Proof.

Inequality (4.2.25) clearly implies that r(0, A) contains the

sector 1+(((p+ it/2)-) with p = arctg S. We obtain immediately from

(4.2.24) that

IIR(A)II <1/Reae'P

in Re Ae'' > 0; likewise, if Re Ae-'' > 0,

IIR(A)II<1/ReAe

and the proof ends exactly like the first part of the proof of Theorem 4.2.1

by using inequalities (4.2.6) and (4.2.7).

We note that, under the hypotheses of Corollary 4.2.5, we can obtain

very precise estimates for

in the entire sector 2+(qq). In fact, using

(4.2.25) we deduce that

Re(0(u),e"4'Au) = (cos,Ji)Re(0(u), Au)-(sin )Im(0(u), Au) <0

(4.2.28)

if I tg < S. Accordingly, the operator e4A is dissipative if I'P I < (P = arc tg S.

Since e'4A generates the strongly continuous semigroup t -> S(e'a't) (t >, 0),

we obtain from Theorem 3.1.8 that

IISQ)II < l

(p).

(4.2.29)

We also note that the argument leading to (4.2.29) implies in particular that

has a strongly continuous extension to the closed sector E(q)).

4.3. APPLICATIONS TO ORDINARY

DIFFERENTIAL OPERATORS

We examine the differential operators

A(ao,Q,)u(x) = a(x)u"(x)+b(x)u'(x)+c(x)u(x) (4.3.1)

in a finite interval 0 < x < I in the light of the previous results on the class

d. As in Section 3.4, our standing assumptions are that a is twice continu-

188

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

ously differentiable, b is continuously differentiable, and c is continuous in

0 < x < 1. The boundary conditions at each endpoint are of type (I) or (II)

(see (3.4.1) and (3.4.2)). Also, we assume that

a(x)>O (0<x<1). (4.3.2)

The domain of A (/io, /3I) in L P(O,1) (1 < p < oo) consists of all u E W z' p (o, l )

that satisfy the boundary condition at each endpoint: in the space C[O,1]

(Co, C,, Co,1 depending on the boundary conditions used), D(A(/3o, /3,)) is

the set of all u E C(2)[0,1] satisfying the given boundary condition at each

endpoint with A(/3o, /3,)u in the space. When several spaces are at play, we

indicate by AP(/30, /3,) the operator in LP and by A(/3o, /3,) the operator in C

or subspaces thereof.

4.3.1

Theorem.

The operator Ap(/3o, /3,) (1 < p < oo) belongs to

C1(g7p - ), where

( 2 1/2

-1}

. (4.3.3)

l 1

For every p with 0 < 9) < qzp, there exists w = w(p, q)) such that

IIS Q)ll<ewIII

(4.3.4)

where /3,)).

Proof. The result is a consequence of Corollary 4.2.5. To check this

we denote by 0 the only duality map in LP and perform an integration by

parts similar to that in (3.4.6). Assume for the moment that p >_ 2, that both

boundary conditions are of type (I), and take u twice continuously differen-

tiable and satisfying the boundary conditions. For any S we obtain

II uIl p-2(Re(O(u), AP(Qo, #I)u)± SIm(O(u), AP (/3o, PI) u))

Ref '((au')'+(b-a')u'+cu)IuI'-2udx

± SImf'((au')'+(b - a')u'+ cu)I

ulp-2udx

={,a(l)_(a'(l)-b(l))}Iu(l)v

Yoa(0) -

(a'(O)- b(O))} I u(O)I P

-(p-2) f'alulp-4(Re(uu'))2dx

+S(p-2) f'alul'-4Re(uu')Im(uu')dx

- fIaIuIp-2lu'I2dx± f

(b-a')lulp-2Im(uu')dx

+ p fI(a"- b'+ pc)lulpdx. (4.3.5)

4.3. Applications to Ordinary Differential Operators

189

We transform the sum of the first three integrals on the right-hand side

using the following result: given a real constant a > - 1,

Izlz + a((Rez)2 ± 6(Rez)(Im z)) , 0

(4.3.6)

for every z E C if and only if

(4.3.7)

To prove this we begin by observing that (4.3.6) is homogeneous in z, thus

we may assume that z = e`9', reducing the inequality to the trigonometric

relation 1 + a(cos2 (p ± 8cos psinq)), 0, or, equivalently (setting 4 =

2 + all + cos 4' ± S sin 4') , 0. Let a >, 0. Since the minimum of the function

g(i)=cos¢ f Ssin4i equals -(1+82)1/2, (4.3.6) will hold if and only if

2- a((1 + 62)1/2 -1), 0, which is (4.3.7).

On the other hand, the maximum of g is (1 + 62)1/2; if - I < a < 0

(4.3.6) will hold if and only if 2 - 1 a 1((1 + S

2) 1/2

+ 1) , 0, which is again

(4.3.7).

In view of the homogeneity of (4.3.6), it is obvious that if (4.3.7) is

strict, there exists T > 0 (depending on 8) such that, for every z E C,

Iz12 + a((Rez)2 ± S(Rez)(Im z)), TIz12.

(4.3.8)

We use (4.3.8) for z = uu':

-(p-2)f'alulp-4(Re(uu'))2dx+8(p-2) fIalulp-4 Re(uu')Im(uu')A

-fIaIuIp-21u'I2dx=- faluIP-4fdx,

(4.3.9)

where

f = I uu' I2 + (p - 2)((Re(uu'))2 ± S Re(uu')Im(uu'))

,TIuu'12=TIu121u'I2

(4.3.10)

for some T > 0 if

i/z

0<S<1(pp2)

-1J

(4.3.11)

We must now estimate the other terms on the right-hand side of

(4.3.5). To this end, consider a real-valued continuously differentiable

function p in 0<x<l. Since (pIulp)'=p'Iulp+ppIulp-2Re(uu'), we ob-

tain, for any e > 0,

IP(1)lu(l)IV -P(0)lu(0)IpI

p2z f IPI

lulp Zlu'IZdx+

fo(IP'I+p2 ZIPI)luIpdx, (4.3.12)

190

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

where we have applied the inequality

21ul lu'I

=2(e-'lul)(elu'I)<EZIu'I2+E-2Iul2.

(4.3.13)

We use (4.3.12) for any p such that

p(1)=yra(1)- p(a'(l)-b(1)),

p(0) =yoa(0)- p (a'(0)-b(0))

(4.3.14)

(we may take p linear) to estimate the first two terms on the right-hand side

of (4.3.5); for the fourth integral we use the inequality (4.3.13), in both cases

with E sufficiently small (in function of the constant T in (4.3.10)). We can

then bound the right-hand side of (4.3.5) by an expression of the type

-c

fIlulp-2lu'I2dx+w fIlulpdx

where w = w(S) and c = c(S) > 0. Upon dividing by Ilullp-2 we obtain

Re(O(u), Ap(/3o, #I) u) < ± SIm(O(u), Ap(/3o, /3,)u)+ wIIuII2.

(4.3.15)

To extend (4.3.15) to any u E D(Ap(/3o, /3I)), we use an obvious approxima-

tion argument. Inequality (4.3.15) is obtained in the same way when /30, /3I

(or both) are of type (II). In the last case the use of the function p is of

course unnecessary.

When I < p < 2 we prove (4.3.15), say, for a polynomial satisfying the

boundary conditions and take limits as before. See the comments in the proof of

Lemma 3.2.2.

Inequality (4.3.15) is none other than (4.2.25) for the operator

Ap(/30, /3,)- wI, thus Corollary 4.2.5 applies. The estimate (4.3.4) is a direct

consequence of (4.3.15); in fact, if qq' < T

and I I < T', we have

Re(9(u), e`TAp(/3o, #I) u) = (cosp) Re(O(u), Ap(/3o, /3,)u)

- (sin p) Im(O (u ), Ap (/3o, /3,) u)

< w(cosg7)IIuII2

(u E D(A))

(4.3.16)

so that e'9Ap($o, /3I)- wcosqJ is dissipative.

A careful examination of the result just obtained reveals that the

restrictions on the domain of analyticity of

(which become progres-

sively worse when p - I or p -* oo since qqp - 0 in those two cases) are

actually restrictions on the validity of (4.3.4).

This makes clear the limitations of any analysis of operators in CT

based on Corollary 4.2.5; an operator A E & (qp) such that

satisfies

Cew111 (

E E(qp))

(4.3.17)

4.3. Applications to Ordinary Differential Operators

191

FIGURE 4.3.1

but not (4.3.4) in any sector may slip past Corollary 4.2.5 in the same way

an operator in 3+(C, w) may fail to be detected by Theorem 3.1.8. These

considerations are thrown into focus by the following

4.3.2 Example. Let A = Ap($o, f j), where the basic interval is 0 < x < ?r,

Au = u", and both boundary conditions are of type (II). The operator

can be explicitly computed by the separation-of-variables

techniques in Sections 1.1 and 1.3: if u -- Ea,, sin nx, then

Sq)u=

e-"z

ansinnx

n=1

1/2 f

e cX

(4.3.18)

where in the last formula u has been continued to - oo < x < oo in such a

way that the extended function is odd about = 0 and = ir. It is plain that

if Red' > 0, the operator is bounded in L P (0, 77), 1 < p < oo and in

C0,,j0,77 ]. We estimate its norm in the latter space:

I)-1/2( f

e-E2Rei'/41tI2dE)IIuIIC

(ICI/Red

)1/211uIIc

(4.3.19)

so that

(ICI/Red')'/2.

(4.3.20)

A corresponding bound in LP can be obtained by interpolation. But an

estimate of the type of (4.3.4) will not hold in any sector I q9 I < qDo if E = L'

or if E = C. This follows from the next example and a simple interpolation

argument.

It is of some interest to ask whether the angle q9p in Theorem 4.3.1 is

the best (i.e., the largest) such that (4.3.4) holds for every 9) < (pp. This is in

fact the case.

4.3.3 Example. Let 1 < p < oo, AP(/30, /3I) a general differential operator.

Then, if IroI > yip, the operator e'qAP(/3

, /3,)- wI is not dissipative for any

192 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

w, so that (4.3.4) does not hold for any such ip even though it may be the

case that A E 6T (T) with T'> TP. We sketch the proof for the boundary

conditions u'(0) = u'(1) = 0. Assume that

I/2

S>{((

`pp2) -11

Then we can find a complex number z of modulus 1 such that

Iz12+(p-2)((Rez)2-8 (Rez)(Imz))=-µ<0.

(4.3.21)

Let

be a smooth real-valued function such that

rl'(0) = 'q'(1) = 0.

Then u(x) = exp(zq(x)) E D(AP(flo, X13,))

and we have u x u'(x) =

z rl'(x) exp(2(Re z) rl (x)) = z 4' (x) with J real. Accordingly, if f is the func-

tion in (4.3.10),

Estimating the other terms in the same way as in Theorem 4.3.1, we obtain

the inequality

Re(O(u), AP(/30, #1) u) - SIm(6(u), A,(Qo, #I) u)

%cIluII2-P f

lulP-21u'I2dx-CIIu112,

(4.3.22)

where c > 0. If (4.3.15) holds for some w, we deduce that there exists a

constant C' such that

f1lulP_2Iu'l2dx<C'

f1IulPdx,

an inequality that can be easily disproved.

Theorem 4.3.1 can be used to obtain sharp bounds on the norm of

certain multiplier operators. For instance, let 1= ir, Au = u", Qo, $ boundary

conditions of type (I) with yo = y, = 0. Then X13,)) is given

Sq)u=

e-"2

to"cosnx

n=0

for u - 7,a"cosnx. Consider

in the space LP,

1 < p < oo. Since

Examining (4.3.5), taking into account that

a = 1, b = c = 0, and making use of (4.3.6) and (4.3.9), we see that

e"PAP ($0, X13,) is dissipative if IT I < (pp so that

q,P.

(4.3.23)

On the other hand, it follows from Example 4.3.3 that the inequality does

not extend to

I arg

I > q), even in the weakened form (4.3.4).

Since Theorem 4.3.1 does not provide any information on A, or A,

we give later a sort of substitute that falls short of proving that these

4.3. Applications to Ordinary Differential Operators

193

operators belong to ll (Theorem 4.3.5); this result hinges upon the case

p = 2, where the sector of analyticity in Theorem 4.3.1 is optimal (see

Remark 4.1.7). As a preliminary step we prove that A, and A E C'+, which

was only shown in Section 3.4 under dissipativity assumptions on the

boundary conditions.

4.3.4 Theorem. (a) The operator A, (/30, /3,) belongs to (2+ in L1(0, 1).

(b) The operator A(/30, /3,) belongs to (2+ in C[0,1 ] (Co, C,, C0,, depending on

the boundary conditions used).

Proof.

Assume for the moment that both boundary conditions are

of type (I). If the dissipativity conditions (3.4.8) are not satisfied for p =1

there is obviously no hope to prove that A,(/3o,/3,)-wI is dissipative for

any w since they are necessary for dissipativity of any operator, A(/io, /3,)

using these boundary conditions in L'. The same comment applies to

conditions (3.4.5) in C. However, a renorming of the space will remove this

obstacle.

Let 1 < p < oo, p a continuous positive function in 0 < x < 1. Con-

sider the norm

IIUIIp =

(ux)PPx)dx)I/p

(4.3.24)

in LP(0, 1). Clearly II'IIp is equivalent to the original norm of LP; we write

LP(0,1)p to indicate that LP is equipped with II'IIp rather than with its

original norm. The dual space LP(0,1)* can be identified with LP'(0,1),

p'-' + p - ' = 1, endowed with its usual norm, an element u* E LP'(0,1) acting

on LP(0,1)p through the formula

(u*, u)p = f Iu*up dx.

(4.3.25)

If p > 1, there exists only one duality map 6p: LP -* LP' given by Bp(u) =

0(pu), 0 the duality map corresponding to the case p = 1

(see Example

3.1.6). For p =1 the duality set of an element u E L' (0, I) p coincides with

the duality set of up as an element of L'(0,1) (see again Example 3.1.6). We

take now u twice continuously differentiable and perform the customary

integrations by parts, assuming that p is twice continuously differentiable as

well:

IIupIIP-2 Re(ep(u),

AP (Qo, #I) u)p

= II upll

P-2Re(pO(pu), AP($0,QA)u)

= Ref

I((au')'+(b-a')u'+cu)luIP-2upPdx

_{Via(l)-

P(a

(1)-b(1))}lu(l)IPp(l)P

194

Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

yoa(0)-

(a'(0)-b(0))}I u(0)Ipp(0)p

-(p-2)

f'aIUlp-4(Re(uu'))2ppdx

- f

IalUlp-2Iu'l2p°dx

- pf

1alUlp-2Re(uu')pp-1p'dx

If1(a"-b'+pc)lulpppdx+ f1(a'-b)lulppp-p'dx

= l(71

P(1))a(1)- p(a'(1)-b(1))IIu(I)IpP(1)p

(yo

(0))a(0)-P(a'(0)-b(0))}IU(0)I°P(0)p

-(p -2)f 'a I

p-4(Re( 7U,))2ppdx

- f /al

ulp-21 u'l2ppdx

+f{(app-ip')'+

(a"-b'+pc)pp+(a'-b)pp-p'}lulpdx.

(4.3.26)

Since p'(0) and p'(1) can be chosen at will, we do so in such a way

that the quantities between curly brackets in the first two terms on the

right-hand side of (4.3.26) are nonpositive, say, for 1 < p < 2. As the first

two integrals together contribute a nonpositive amount, we can bound

(4.3.26) by an expression of the form w'I I u I I p < w I I u I I

where w does not

depend on p. Consider now the space L'(0, l)P. Again under the identifica-

tion (4.3.25), the duality set E ),(u) of an element u consists of all u* E L°°(0,1)

with u*(x) = Il ull PI u(x)I -'u(x) = Ilupll Iu(x)l -'u(x) where u(x) * 0 and

Iu*(x)I < IIuIIP elsewhere. We can then take limits in (4.3.26) asp - 1 in the

same way as in Lemma 3.2.2 and obtain an inequality of the form

Re(u*, AI($0, #I)u)P < wIIU112

(4.3.27)

in V. The inequality is extended to arbitrary u E D(A 1($o, Q1)) by means of

the usual approximation argument. Now that A1($0, /3,)- wI has been

shown to be dissipative, m-dissipativity is established using Green functions

as in the end of Section 3.4. The case where one (or both) of the boundary

conditions are of type (II) is treated in an entirely similar way; naturally, the

use of the weight function is unnecessary in the latter case.

To prove a similar result for the operator A we renorm the space C

or the corresponding subspace by means of

IIuIIP = max I u(x)I p(x), (4.3.28)

0<x<1

4.3. Applications to Ordinary Differential Operators

195

where p is a positive, twice continuously differentiable function in 0 < x < 1.

The use of the weight function p is again unnecessary when both boundary

conditions are of type (II): we treat below in detail the case where RO and $,

are of type (I), the "mixed" case being essentially similar. Choose p in such

a way that

p'(0)+Y0p(0)>0

(resp.p'(1)+Y1p(1)<0) (4.3.29)

if yo < 0 (resp. y, > 0). The dual of C[0,1] equipped with II' II, can again be

identified with E[O,1], an element ft E E[0,1] acting on functions u E C[O,1]

through the formula

<µ, u> = f lu(x)p(x)tt(dx). (4.3.30)

Accordingly, the norm of a measure it E E as an element of C* is still

I I I I I = Jo l i

l (dx) and the identification of the duality sets OP (u) can be easily

adapted from Section 3.1; OP(u) consists of all µ E E with support in

mP(u) = (x; lu(x)Ip(x)I = IIuIIP) and such that uptt (or uµ) is a positive

measure in m(u) with I I µ I I = IIuIIP The same comments apply of course to

the spaces CO, C1, C011, where the corresponding measures are required to

vanish at 0, 1, 0 and 1.

We now show that A(ao, a,)- wI is m-dissipative for to large enough.

Observe first that if u'(0) = you(0), u'(1) = y,u(1), then up satisfies the

boundary conditions

(up)'(0)=Y0,p(up)(0), (up)'(1)=Y,,p(up)(1),

(4.3.31)

where

Yo,P=Yo+p'(0)p(0)-'>0, Yr,P=Yr+p'(1)p(1)-'<0.

(4.3.32)

An argument already employed in Section 3.3 shows that for any

u E D(A(flo, a,)), u - 0 the set mp(u) does not contain either endpoint of

the interval [0,1] if both yo, p, y,

> 0, so that

(I up12)'(x) = 0, (Iup12)"(x) < 0

(x E mp(u)). (4.3.33)

On the -other hand, if either yo,p or y, P vanish, m( u) may contain

the corresponding endpoint but arguing again as in Section 3.3, we see

that (4.3.33) holds. Writing q = p2, we have (IupI2)'= 2(u,uc + u2U' )ri

+(u +

(lupl2)" = 2(u,u',' + u2u' )Q + 2(u12 +

u22),q

4(u,u', + u2u2)r1'+(u1 + U2 2)71". Hence, it follows from (4.3.33) that

Re(u 'u')=llull 2(u,ui+u2u'2)=-zllull-2(u +u2)r1

,71

zrl

71',

(4.3.34)

Re(u-'u")=IIu11-2(u,ui'+u2u2)<-Ilull-2(ui2+u' )

+I1Ull-2(ui +

u2)ri-2n'2 - illull

2(u2

-Ilull-21u,12

71-2,nr2

- zr

(4.3.35)

196 Abstract Parabolic Equations: Applications to Second Order Parabolic Equations

in m

p(u ). Accordingly, if It E 8(u), we have

Re(p,A(/3o,/3,)u)= f

Re(u 'A(/3o,#I)u)pudµ

mv(u)

wIIuIIP

(4.3.36)

for some constant w, which shows that A(/3o, /3,)- wI is dissipative. That

(Al - A(/3o, 8/)) u = v has a solution u for all v is once again shown by

means of Green functions. This ends the proof of Theorem 4.3.4.

Theorem 4.3.1 only states that u (t) = SP (t) u, the solution (or gen-

eralized solution) of u'(t) = Ak(/io, /3,)u(t) can be extended to a function

"analytic in the LP mean." However, more than this is true,

since u(t, x) = u(t)(x) can be shown to be pointwise analytic. This result

extends to p = 1.

4.3.5

Theorem. Let u E LP (1 < p < oo ). Then

u (t, x) _

(SS(t)u)(x), after eventual modification in an x-null set for each t, can be

extended to a function x) analytic in 2+(or/2-). Moreover,

E C

(Co, C,, Co, I depending on the boundary conditions) for all E 2+(7r/2 -) and,

for every q) with 0 < q) < 7r/2 and every e > 0 there exist constants C = C(cp, e),

w = w ((p, e) such that

1llull (4.3.37)

where the norm on the right-hand side is the LP norm.

Proof.

Let p, q be such that 1 < p < q < oo. Since AP (/30, /3,)

Aq(/Jo, /3,) (both operators thought of as acting in Lq) we also have

R(A; Ap(/3o, /3,)) 2 R(X; Aq($o, /3,)) and it follows from formula (2.1.27)

that the same inclusion relation holds for the solution operators SP; pre-

cisely, if u E Lq(0,1), we have

Sp(t)u=Sq(t)u (4.3.38)

for t >, 0. Consider now a function u in

l) for some p >, 1. We have

u(x)-u(x')= fxu'(l) dE (4.3.39)

for almost all x, x', so that it follows from Holder's inequality that, after

eventual modification in a null set, u is continuous in 0 < x < 1. Moreover,

setting (say) x' = 0 in (4.3.39) we obtain an inequality of the form

I u(x)I

< Iu(0)I + l'-"IIu1ILp,

where the expression on the right-hand side is easily seen to be a norm

equivalent to the original norm of W',p. It follows that

W'°p(0, l) c C[0,1] (4.3.40)

with bounded inclusion.