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

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3.3. Ordinary Differential Operators in a Closed Interval

137

having a single maximum at x = 0 and such that u(0) = 1, u"(0) = a. Then

0(u) = (6), 8 the Dirac measure, and (6, Aou) = a(0)a + yb(0)+ c(0), which

can be made positive by adequate choice of a.

We consider next the case E = LP(0, oo). Here D(A0(/3)) consists of

all infinitely differentiable functions with compact support in x >, 0 satisfy-

ing the corresponding boundary condition.

3.3.2 Lemma.

The operator A0(/3) is dissipative in LP (1 < p < oo)

a(x) >,0, c(x)+ P (a"(x)- b'(x)) < 0

(x>0) (3.3.6)

with boundary condition (II). In case a boundary condition of type (I) is used,

A0(/3) is dissipative if both inequalities (3.3.6) and

ya(0)- P (a'(0)-b(0)) > 0

(3.3.7)

hold. The first inequality (3.3.6) is necessary in all cases; the second, as well as

(3.3.7), are necessary when p = 1.

Proof. We consider first the case p > 1, using integration by parts as

in the proof of Lemma 3.2.2 but now taking boundary terms into account.

Keeping in mind the observations about the case 1 < p < 2 made in the

proof of Lemma 3.2.2, we have

IIuIIP-2Re(0(u),A0(/3)u)=Re f ((au')'+(b-a')u'+cu)IuIP-2ii

(ya(0)- p (a'(0)-b(0))}

I u(0)IP

- (p -2)(00aI uIP-4(Re(uu'))2 dx

- f

0aIuIP-2Iu'I2dx

+ 1 f (a"-b'+pc)IuIPdx<0

(3.3.8)

if a boundary condition of type (I) is used; if the boundary condition is of

type (II), the boundary term vanishes. The case p =1 is settled through a

limiting argument exactly as the one in the proof of Lemma 3.2.2.

The necessity of the first condition (3.3.6) for p >, 1, as well as the

necessity of the second for p = 1, follow as in the previous section; we only

have to repeat the arguments there using functions that vanish near zero and

thus satisfy any conceivable boundary condition. We show now that (3.3.7)

is necessary when p = 1. Assume ya(0)- a'(0)+ b(0) < 0 and let u be an

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Dissipative Operators and Applications

element of D(Ao(/3)), which is positive in an interval [0, a), a > 0, zero for

x >, a, and has norm 1. Then any u* E O(u) equals 1 in 0 < x < a and

<u*, A,($)u) = f ((au')'+(b - a')u'+ cu) dx

=-(ya(0)-a'(0)+b(0))u(0)+ fa(a"-b'+c)udx,

which clearly can be made positive by taking, say, u(0) large and u

sufficiently small off the origin.

3.3.3

Example.

The second inequality (3.3.6) is not necessary for dissipa-

tivity of A0(/3) for p > 1 regardless of the type of boundary condition used.

3.3.4. Example. Inequality (3.3.7) is not necessary for dissipativity of

A0(/3) for p > 1 when a boundary condition of type (I) is used.

We bring into play the adjoint A' defined as in the previous section.

The Lagrange identity takes here the following form: if u, v E 6D,

f (A(, u)vdx= f °°uA'0vdx-a(0)(u'(0)v(0)-u(0)v'(0))

+ (a'(0) - b (0)) u (0) v (0), (3.3.9)

where the functions u, v do not necessarily satisfy boundary conditions at

the origin. Formula (3.3.9) allows us to compute the adjoints of differential

operators when boundary conditions are included in the definition of the

domains. (The notion of dissipativity does not play a role here.) To this end,

we introduce the notion of adjoint boundary condition (with respect to A0),

assuming from now on that a(0) > 0. If the boundary condition /3 is of type

(II), the adjoint boundary condition /6' is /3 itself. If /3 is of type (I), then /3'

u'(0) = (y - a(0)-'(a'(0)-b(0)))u(0) = y'u(0). (3.3.10)

The motivation of this definition is, clearly, that the boundary terms in the

Lagrange formula (3.3.9) vanish when u satisfies the boundary condition /3

and v satisfies the boundary condition /3'; in other words,

(Ao(/3)u)vdx = f

00uA0'(/3')vdx

(u E D(A0(/3)), v E D(A' (/3'))).

f o

(3.3.11)

It is clear from the definition of /3' that if /3 is any boundary condition,

(#')'= /3 (note that in applying the definition of adjoint to /3' the coeffi-

cients a, b, c of the adjoint operator must be used: in other words, the

second prime indicates adjoint with respect to Ao).

A simple computation shows that an operator A0(/3) satisfies the

assumptions in Lemma 3.3.2 in LP if and only if A' (/3') satisfies the same

assumptions in LP' (Co if p = 1).

3.3. Ordinary Differential Operators in a Closed Interval

139

In the following result no use is made of the dissipativity of AO;

however, we assume from now on that

a(x) > 0

(x >, 0). (3.3.12)

3.3.5

Lemma.

(a) Let 1 < p < oo and let f E L(, in x > 0. Assume

that u is a locally summable function defined in x >, 0 and such that

f uAo($')9pdx= f

(3.3.13)

Then u coincides with a function in W1j (0, oo) that satisfies the boundary

condition $ and

au"+bu'+cu=f (x>-0). (3.3.14)

If f is continuous, u is twice continuously differentiable in x > 0.

(b) Let v be a locally finite Borel measure, and µ a locally finite Borel

measure defined in x > 0 that satisfies

f A' (#')9)d,s = f

qdv ( E D(Ao(Q )))

(3.3.15)

Then u coincides with a continuous function in t > 0.

Proof. We can continue the coefficients a, b, c to x < 0 preserving

their smoothness and in such a way that the extended is positive; we

denote the extensions by the same symbols. To prove (a), assume that the

boundary condition is of type (I) and let 9'o be a fixed function in 6)1 with

q)0(0) = 0, gvo(0) = 1, and u _ a

smooth (say, CCzl) function in x < 0 with

compact support and such that u _(0) =1 and

u' (0) = yu _ (0) = y. (3.3.16)

We may and will assume in addition that

Todx= fgpodx=0.f

f° Aou

(3.3.17)

It follows from the first equality and the Lagrange identity (3.3.9) in x < 0

that

f°

u_A'

0.0990

Define now u1 = u for x > 0, u1 = au

in x < 0, where a is such that

f u1A'op0dx=0.

(3.3.18)

If i/i is an arbitrary function in 6D, we can clearly write

_ 40 + T

(3.3.19)

where p satisfies the boundary condition a'. It is a

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Dissipative Operators and Applications

FIGURE 33.2

consequence of (3.3.16) and of the analogue of (3.3.11) in x S 0 that

uiAoy?dx=

(Aout)q,dx.

(3.3.20)

We obtain, combining this equality with (3.3.19), (3.3.18) and (3.3.17), that

we have

00 -C

(3.3.21)

for all 4 E 6D, where ft = f for x >, 0, ft = Aout for x < 0. Making use of

Lemma 3.2.5 we see that u t E Wta (- oo, oo) and that au '' + bu ', + cu t = f

in - no < x < oo, from which (3.3.14) follows. That u satisfies the boundary

condition $ results from the fact that u_ does. To prove the statement

regarding f continuous, we need f, to be continuous. This can be achieved

by imposing on u

- the additional boundary condition Aou _(0) = ft(0). The

case of boundary conditions of type (II) is handled in an entirely similar

way and we omit the details.

The proof of (b) runs much along the same lines; the measure µ is

continued to x < 0 in such a way that the analogue of (3.3.21) holds for an

arbitrary test function 1JI, and then we apply the second part of Lemma

3.2.5.

We note that an immediate consequence of part (a) is that the

adjoint of the operator A0' (fl') in LP'(0, oo) (1 < p < oo) is the operator A(/3)

in LP, defined by A(/3)u = au"+ bu'+ cu, where D(A(/3)) is the set of all

u e W1(0, oo) r) L P (0, oo) such that A(fi) u E L"(0, oo) and u satisfies the

boundary condition ft.

We examine now the problem of finding m-dissipative extensions of

A0(/3). The treatment is exceedingly similar to that in the previous section

for (- oo, oo) so that we limit ourselves to stating the main result and

sketching the proof.

f l ftgpdx= f 1 ft"dx

3.3. Ordinary Differential Operators in a Closed Interval

141

3.3.6

Theorem.

(a) Let the coefficients of the operator A0 and the

boundary condition /3 satisfy

a(x)>O, c(x)<0 (x>,0) (3.3.22)

and

y >' 0 (3.3.23)

(the latter inequality if /3 is of type (I)), and assume A'(#) is dissipative in LP,

for some p > 1, which is the case if

c(x)+ p (a"(x)-b'(x)) <0 (x >,0) (3.3.24)

and

ya(0)- p (a'(0) - b(0)) >, 0 (3.3.25)

if /3 is of type (I). Let A(/3) be the operator defined by

A($)u(x) =a(x)u"(x)+b(x)u'(x)+c(x)u(x),

(3.3.26)

where D(A(/3)) is the set of all u E C(2) [0, oo)fl Co[0, oo) such that (3.3.26)

belongs to Co[O, oo) (if the boundary condition is of type (II), Co is replaced by

C0,o). Then A(#) is m-dissipative in Co (Co,o).

(b) Let 1 < p < oo. Assume inequalities (3.3.22) and (3.3.23) hold and

suppose Ao is dissipative in LP' (for which (3.3.24) and (3.3.25) are sufficient

conditions). Let A(/3) be the operator defined by (3.3.26), where D(A(/3)) is

the set of all u c- WWP(0, oo)fl LP(0, oo) such that the right-hand side belongs

to LP. Then A(/3) is m-dissipative in LP.

a"(x)-b'(x)+c(x) <0,

(3.3.27)

ya(0)-a'(0)+b(0) >, 0.

(3.3.28)

Then the operator A(/3) defined as in (b) with p =1 is m-dissipative in L'.

To prove (b) let v E LP; we obtain, availing ourselves of the inequal-

(/3')qIIp- (9p E D(A0'(/3')) a u E LP such that IIuIIP

ity IImIIP, < A-'llXm

- A'

< X-'IIvIIP and

(XI - A' (/3')*)u= (XI-A(/3))u=v,

and the uniqueness of u follows from the fact that A0(/3) is dissipative in

Co,o or CO, which is in turn guaranteed by conditions (3.3.22) and (3.3.23).

Part (a) follows exactly in the same way as in Theorem 3.2.7. Finally, part

into c(x) < 0, -y'< 0 for the formal adjoint Ao and the adjoint boundary

condition /3'. On the other hand, inequalities (3.3.22) are a(x)> 0, a"(x)-

b'(x)+ c(x) < 0, and (3.3.23) becomes y'a(0)- a'(0)+ b(0) >_ 0. We obtain

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Dissipative Operators and Applications

the inequalities a"(x)- b(x)+2c(x) < 0 and 2y'a(0)- a'(0)+ b(0) >,0,

which imply the assumptions in (a) for A'(#'). We then extend A'(#') to a

m-dissipative operator and apply the #-adjoint theory; in the final identifi-

cation of the domain of A'(#')#, a decisive role is played by the dissipativ-

ity of AO(/3).

The comments after the proof of Theorem 3.2.8 have a counterpart

here; in particular, under the hypotheses in (c),

(/3')* =Ao($)

A($) = A'

for

1 < p < oo. We note finally that if we write Ao in variational form

(3.2.35), we only need to assume that a, b E 0l, c E Ct0), as in the case

considered in Section 3.2.

3.3.7 Example. If A0 is written in variational form, the dissipativity

condition (3.3.25) becomes

yd (0) - p

b (0) >_ 0.

(3.3.29)

3.3.8

Example.

Let /3 be an arbitrary boundary condition at x = 0, Av (/3 )

the operator defined in Theorem 3.3.6(b). Then

A',(/3')=Ap(/3)* (1<p<oo)

(3.3.30)

for p'-' + p -' =1, where A,.(/3') is the operator in LP' corresponding to

the formal adjoint Ao and the adjoint boundary condition /3'. (We adopt

here the hypotheses in (c)).

3.4. ORDINARY DIFFERENTIAL OPERATORS

IN A COMPACT INTERVAL

The problem of finding m-dissipative extensions of A0 is considerably

simpler in the case of a compact interval [0, 1] (at least when does not

vanish in 0 < x < 1) since we have at our disposition the theory of Green

functions of ordinary differential operators. We assume once again that a is

twice continuously differentiable, b continuously differentiable, c continu-

ous in 0 < x < 1. For the sake of simplicity we consider only separated

boundary conditions at the endpoints, that is, a boundary condition /3o at 0

and a boundary condition /3, at 1, where /3X indicates one of the following

two conditions:

(I)

u'(x)=yxu(x), (3.4.1)

(II)

u(x) = 0.

(3.4.2)

3.4. Ordinary Differential Operators in a Compact Interval

143

In the treatment of the regular case (a > 0) we can dispense entirely with the

operators A0 of the past two sections. Given two boundary conditions fl o, /3I,

we define an operator A(/30, /3!) as follows. In the case E = Lp(0,1)

(1<p<oo),

A(/3o,/3r)u(x)=a(x)u"(x)+b(x)u'(x)+c(x)u(x),

(3.4.3)

where the domain of D(A(/3o, /3,)) consists of all u E W2,p(0,1) that satisfy

the corresponding boundary condition at each endpoint. When E is a space

of continuous functions, the definition of the space itself depends on the

type of boundary conditions used. If both boundary conditions are of type

(I), then E = C[O,1] and A(/30, /3,) is defined by (3.4.3) with D(A(/30, /3,))

consisting of all u E C(2)[0, 1] that satisfy the boundary conditions at each

endpoint. If one of the boundary conditions (say, /io) is of type (II),

E = Co[0,1 ] (we use here the notation in Section 7)3 consisting of all

u e C[O,1] with u(O) = 0; D(A(/30, l3,)) is defined in a way similar to the

previous case, where now A(/30, /3,)u must belong to Co. Similar restrictions

of the space C are taken if /3,, or if both boundary conditions are of type

(II); the corresponding spaces will be denoted C, [0, 1], C0,1 [0,1 ].

The dissipativity results for the operators A(/3o, /3,) in the different

spaces considered can be proved essentially as in the case of one boundary

condition (which was considered in the previous section) and will be for the

most part left to the reader.

3.4.1

Lemma. The operator A(/30, /3I) is dissipative in C[0,1 ] (or in

CO, C,, C0,! depending on the boundary conditions used) if and only if

a(x)>O, c(x)<0 (0<x<1) (3.4.4)

and

Yo>0, Y,<0

(3.4.5)

when boundary conditions of type (I) are used at either endpoint.

The case E = LP is dealt with by integration by parts

for

1 < p < oo and by means of the usual limiting argument when p =1.

As pointed out in the two previous sections, there are difficulties with the

differentiability of 0(u) when I < p < 2, which can be avoided by performing the

integration by parts for functions u in q@(2) (or simply polynomials) that satisfy

the boundary conditions and then approximating any u E D(A(/30, /3,)) by a se-

quence

in 6J'C?'1z> in such a way that u,, -> u, u;, --> u' uniformly in a < x < b,

u;; - u" in Lp(0,1).

3See footnote 2, p. 135

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Dissipative Operators and Applications

The analogue of (3.3.8) is

IIuIIP-2ReKO(u), A(Qo, #I)

=Re f'((au')'+(b-a')u'+cu)IuI'-2i dx

{ya(l)-P(a,(l)-b(l))}Iu(1)1°

-{yoa(0)- p(a'(0)-b(0))}lu(0)Ip

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

-fIaIul'-2Iu'I2dx+p

(3.4.6)

The boundary terms vanish when boundary conditions of type (II) are used.

The preceding inequality plainly implies the following result.

3.4.2 Lemma.

The operator A(/3o, $i) is dissipative in the space L'

(1<p<oc)if

a(x)>,0, c(x)+ I (a"(x)-b'(x))<0 (0<x<1) (3.4.7)

with boundary conditions (II). If boundary conditions of type (I) are used

instead at either endpoint, A(/3o, #,) is dissipative if (3.4.7) holds and

y0a(0)-

I (a'(0)-b(0))>0, yra(1)- I (a'(l)-b(l))<0

(3.4.8)

at the corresponding endpoint. The first inequality (3.4.7) is necessary in all

cases; the second, as well as both inequalities (3.4.8), is necessary when p = 1.

The necessity proof for p = I is entirely similar to that in the previous

section for the case of the half-line.

We turn now to the problem of solving the equation (I - A(/3o, /3I))u

= v in the different spaces under consideration. To this end we use the

theory of Green functions for ordinary differential operators (an account of

this theory, together with the results to be used below, can be found for

instance in Coddington-Levinson [1955: 1].) Let a(x) > 0 in 0 < x < 1, uo a

nontrivial solution of the equation

au"+bu'+(c-1)u=0 (3.4.9)

in 0 < x < 1 that satisfies the boundary condition $o, and let u1 be a

nontrivial solution of (3.4.9) in 0 < x < 1 that satisfies the boundary condi-

tion /3,. The functions uo and u1 are linearly independent if A(/3o, /3,) is

dissipative in any of the spaces L', I < p < no or C. In fact, if they were not,

3.4. Ordinary Differential Operators in a Compact Interval

145

uo would be a nontrivial solution of (3.4.9) in 0 < x < 1 satisfying the

boundary conditions at both endpoints. Then uo would belong to

D(A(/3o, //,))

in the space under consideration and (3.4.9) would be

A(/3o, /3,)u = u, an equation that can only have the solution u = 0 in view of

the dissipativity of A(/3o, /3,).

The linear independence of uo and u, implies of course that their

Wronskian W(x) = W(uo, u,, x) does not vanish at any point of the interval

0 < x < 1. We define then

G(x, )=

(uo(x)u,(f)/a(f)W(f)

(3.4.10)

It is an elementary matter to verify that if f E LP(0,1), 1 < p < oo, then

u(x)= (0<x<1) (3.4.11)

belongs to D(A(/3o, /3,)) and satisfies A(/3o, /3,)u - u = f. The same observa-

tion applies to the space C[0,1] (or CO, C,, C0,, according to which boundary

conditions are used). We have then proved the following

3.4.3 Theorem. Assume that the operator A(/3o, /3,) is dissipative in

L P (0, 1) (1 < p < oo) or in C [0,1 ] (Co, C,, Co,, depending on the boundary

conditions) and that

a(x)>0 (0<x<1). (3.4.12)

Then A(/3o, /3,) is m-dissipative.

Observations similar to those in the last two sections apply to

operators written in variational form (3.2.35).

3.4.4

Remark. The results in the last three sections can be extended in an

obvious way by replacing if necessary the operators A0, AO(P) or AO(flo, #I)

by A0 - wI, A0(/3)- wI or Ao(/3o, /3,)- wI for w sufficiently large; in this

way, the second condition (3.2.28) can be weakened to

c(x)<w (-oo<x<oo) (3.4.13)

and (3.2.29) becomes

c(x)+ I (a"(x)-b'(x))<w (-oo<x<oo). (3.4.14)

The conclusion of Theorem 3.2.8 is in this case that A - wI is dissipative in

the various spaces considered there, so that A E e+(1, w). The same observa-

tions apply to the operators A(/3) in Theorem 3.3.9 and to the operators

A(/3o, /3,) in a compact interval [0, 1] in Theorem 3.4.3, and are especially

interesting here: in fact, (3.4.13) and (3.4.14) will always be satisfied by to

large enough and it turns out that the only essential restriction on the

coefficients (besides smoothness) is (3.4.12); if it holds it will always be true

that A E e+.

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Dissipative Operators and Applications

3.4.5

Example.

Let /30 (resp. /3,) be a boundary condition at x = 0 (resp.

at x = 1), and let AP(130, /3I) (1 < p < oo) be the operator in LP defined by

(3.4.3): the corresponding operator in C (Co, C,, Co,1) is denoted by A(/3o,13,).

The operators defined with respect to the formal adjoint and the adjoint

boundary conditions are A,(13o, /3'): we only assume that a(x) > 0 in

0 < x < I (see Remark 3.4.4) and conditions (3.4.8). Then

A',(/3 ,Q;)=AP(Qo,ar)* 0<P<00),

(3.4.15)

where p'-' + p =1. For p =1 we have

A'(/3 ,/3)=AI(Qo,Q,)#

(3.4.16)

and

Ai(iao, $;) = A(Qo, #I)

(3.4.17)

3.4.6 Example. Denote by A0 the operators defined in Section 3.2 in

LP(- oo, oo) with domain 6 , A

(/3) the corresponding operators in L P (0, oo ).

Using (3.4.15) and assuming that a(x) > 0, we can prove that if 1 < p < cc,

then (a) Ao is dissipative in LP'(- oo, oo) if and only if Ao is dissipative in

LP(- oo, oo) and (b) A0'(13') is dissipative in LP'(0, oo) if and only if Ao(/3)

is dissipative in LP(0, oo). The general case a(x) >, 0 can be handled using

the previous result for a(x)+e, e> 0 and letting e- 0.

3.5. SYMMETRIC HYPERBOLIC SYSTEMS

IN THE WHOLE SPACE

Consider the differential operator

Lu= E Ak(x)Dku+B(x)u.

k = I

Here x = is a point in m-dimensional Euclidean space OBm,

u(x) = (uI(x),..., u, (x)) a v-dimensional vector function of x, Dk = a/axk,

and the coefficients Ak(x) = {ak(x)}, B(x) = {b`J(x)} are v x v matrix func-

tions defined in R-. Since no additional complications are caused by

allowing the coefficients Ak, B to be complex-valued, we shall do so,

assuming A, ( ), ... , A

to be self-adjoint and continuously differentiable

in IF m and B(.) to be continuous there. We shall work only in the complex

space L2(lRm)' (see Section 7); in view of the result in Example 1.6.1, we

scarcely need to apologize for lack of generality on this score. (We will

consider in Chapter 5 the operator L in certain Sobolev spaces; see Section

1.6 for the constant coefficient case.)

Our goal in this section is to find a domain for L that makes it

m-dissipative under adequate restrictions on its coefficients. This will be

accomplished by defining L = Coo in a domain consisting of very smooth