Mathematical Notes, VoL 59, No. 1, 1996

C o e r c i v e n e s s o f F u n c t i o n a l - D i f f e r e n t i a l E q u a t i o n s

L. E. Rossovskli UDC 517

ABSTRACT. We consider functional-differential equations with the Dirichlet conditions and with contraction and dilatation of the arguments. Necessary and sufficient conditions are obtained under which a Gs type inequality holds. These results allow us to verify coerciveness by using a special "symbol" of the equation considered.

The coerciveness problem for quadratic forms generated by differential operators was studied by M. I. Vi- shik [1], L. Gs [2], S. Agmon [3], D. G. Figueiredo [4], and J. Necas in the book [5]. In particular, it was proved that the strong ellipticity of a system of differential operators with Dirichlet conditions on the boundary is a necessary and sufficient condition for this system to satisfy a Gs type inequality.

A. Skubachevskii [6] obtained necessary as well as sufficient conditions in algebraic form for difference- differential operators to satisfy a Gs type inequality. He also proved that these conditions coincide for almost all domains Q c R".

The present paper treats functional-differential equations with Dirichlet conditions and with contracted and expanded arguments. We present necessary and sufficient conditions for a Gs type inequality. With the equations under consideration we associate an algebraic expression (the symbol of the equation) whose positiveness is the desired condition. Theorems 1 and 2 contain the main results. Let us begin with some preliminary considerations.

w Let S =-1 be the unit sphere in R n. To each function g G C(S "-1) we assign a function G G L~(R ") by setting

c(=)=g (= e R" ) . (i)

The multiplication by G(x) is a bounded operator in L~(R ") k. It will be also denoted by G. Thus,

Gu(=) = G(=)u(=) = g ~,(=).

Obviously, the mapping g ~-. G is a homomorphism of the algebra C(S n-1 ) into the algebra of bounded operators B(L2(R")). Let us show that this is an isometry. On the one hand, we have

(fR)1,2 IIG,-,II,.,~R-) = . IG (= ) l~ lu (= ) l ~ d= < I IGI I~| = I Igl Iccs"- ' ) I I ' - ' I IL'~R"),

(1=-61 >~).

that is,

IlCll _< Ilgllccs.-,). On the other hand, suppose that Ig(6) l = Ilgllc(s-- ,) and

~,,(=) = 1 ( 1 = - 61 < ~), ~,,(=) = o

Then

II,-,,ll~,c~-) ~,_~oi< d= = la(~' ) l (1~- ~ol < ~),

lira [[IGu~ll'~ = IG(~o)l = Ig(~o)l.

(2)

(3)

Translated from Matematicheskie Zametki, Vol. 59, No. 1, pp. 103-113, January, 1996. Original article submitted June 28, 1994.

0001--4346/96/5912-0075512.50 C)1996 Plenum Publishing Corporation 75

By (2) and (3), we have I IGII = I lg l l ccs - - , ) - Thus, C(S "-~) is isometrically isomorphic to the closed subalgebra AG C B(L2(R")) of operators of

multiplication by functions G(z) of the form (1). Hence, first, we see that each complex homomorphism h of the algebra AG has the form

h(C) = g(~h) (~h e S"-~).

Second, since by the Stone--Weierstrass theorem [7, p. 137 of the Russian edition] the polynomials P ( ~ l , . . . , ~ n ) are dense in C(Sn-~), it follows that the polynomials P(G1,. . . , Gn) (where Gi is the operator of multiplication by the functions Gi(z) = xi/[x[) are dense in the algebra Aa.

w Consider the dilatation operator R0: L2(R n) --* L2(R n) given by the formula

Rou(=) =u(q-lx) (q > 1).

It is easy to see that R~u(x) = qnu(qx) = qnRolu(z ).

Since q-nl2R.o is a unitary operator, we have

~(a0) c {~ e C lIAI = q"/=} =q "/2T1.

Let us show that a(R0) = qnl2TX, i.e., the operator R0 - AI is not invertible if [A I = q,,12. construct a sequence IlukllL'(~-) --' or such that the sequence (Ro - AI)uj, is bounded.

Set ( M -1 i f q - i < [ z [ < q - J + l , j = l , . . . , k ,

uk(z) = 0 otherwise.

Then

It suffices to

k

2 = Z IAI ~<j-1) dx = ~ qnO-x)mes(q-j < I=1 < Ilu~IIL=cR-) q - i § -J <l=l<q-i+~ j = l j = l ' (-) =~_, q - z 1 mes( [=[< l ) - - , c r , k--.or

j = l an

Moreover, it is obvious that

{ _,~k-~ ifq-k < [z[ < q-k+l,

(Ro - ~_r)~k(=)= 1 if 1 < I=1 < q, 0 otherwise.

q2n -- 1 II(Ro - ,~I)u~ll~,(~.) = mes(Izl < 1) q ;

Thus, we obtain a(Ro) = q"/2T1. The operators P~ and /to commute with each other and with elements of AG. Hence, the closure AR,G (with respect to the operator norm) of the set of polynomials P(Ro,R~, Gi) is a commutative B*-algebra (as a subalgebra of the B*-algebra B(L2(R'~))). Obviously, AG is a closed subalgebra in AR,a.

The Gelfand-Neimark theorem [7, p. 311 of the Russian edition] asserts that the Gelfand transformation P ~* ff is an isometric isomorphism of the algebra AR,a onto the algebra C(AR,G), where AR,G is the space of maximal ideals of AR,G. The space AR,G is equipped with the Gelfand topology, which is Hausdorif and compact [7, p. 300 of the Russian edition].

"/6

w Let us show that, up to a homeomorphism, Aa,G is a compact set K C q"12T ~ x S ~ - 1 . To this end, let us consider the mapping ~: AR,a ~ C "+1 given by

~(~) = ( ~ ( ~ ) , ~, (~) , . . . , ~ , (~) ) .

By the definition of the Gelfand transformation, this mapping is continuous and the range of the first coordinate is a(Ro) = q' /2Ti. If h is a homomorphism of AR,~, then h is a homomorphism of Av, and SO

h(~) = ~(~) (~ �9 s " - ~ ) .

But then we have

( ~ ( ~ ) , . . . , ~ , (~) ) = ( ~ ( c , ) , . . . , ~ (~ , ) ) = ((r . . . , (r = ~,.

Thus, the range of the vector function ~(h) is a compact set K C q"/2T1 x S n - 1 .

Furthermore, let ~ (h l ) = ~(h2), that is, hl(Ro) = h2(Ro) and hl(G~) = h2(Gi) (i = 1 , . . . , n ) . Since the Gelfand transformation preserves involution, we have h~(R~) = h2(R~). Therefore, the homo- morphisms coincide on the polynomials P(Ro, R~, Gi), and since the polynomials are dense, we obtain hi = h2. Thus, ~ is one-to-one; but a continuous one-to-one mapping of a compact space into a Hausdorff space is necessarily a homeomorphism.

Finally, we obtain the isometric isomorphism

r r = ~B o ~-1 (PEAR,a ) ,

r ~) = ~ ( ~ - ' ( ~ , ~)) = ~, r162162162 (i = 1 , . . . , n ) .

Since r is a homomorphism, we have

r 2_, a**~,~,.., . . . v~ "+~" = ~ 2.,v-'a~'"~+~ M,l~ l=m I I,~1,1~1=,~ I

((A, ~) E K and a ~ z E C, where a = ( a l , . . . , a, ,) , B = ( i l l , . . . , fin), and l E Z ranges over a finite set).

In what follows, we shall use the notation

l I

The proof of the main result is based on a fact that follows from the isomorphism constructed above: if

rte ~ r . a ( ~ W § > 0 I~l,l~l='n

((A, ~) E q'/UT 1 x S ' -1) ,

then the operator

is positive definite, namely,

P = ~ (Ro~ + R*~)G ~§ I,,I,1~1=-,

where u E L2(R ") and c > 0 is independent of u. *-algebras containing this operator.)

Re(Pu, U)L2(~.) >_ c[[u[[~2(~.),

(The spectrum of an operator is the same in all

77

w In this section we shall study a functional-differential equation and prove sufficient conditions for the Ghrding inequality to hold.

Consider the equation

r-R~,(~) = ~_. v ~ = f(~) I~I,IPI_<~

(xEQ) (4)

with the boundary conditions v"~ulo~ =o (u=0,...,m-1),

where / ~ L'(Q), ~ ~ W~(R"), =(~) = 0 ( , e R" \ Q), and Q c R" is a bounded domain with smooth boundary; here

1 0 1 0 ~=~? ' "~"~" ' ~J-io~' and ~-iO~

is the normal derivative on the boundary. Here Wm(R ") is the Sobolev space equipped with the inner product

I~1<_=

The subspace of functions vanishing outside Q will be identified with I ~ ( Q ) . In I~'~(Q) we can introduce the equivalent inner product

=LL ( u, v ) ~ , . ( Q) Z)"ug'~v d=. I =

The following inequality for the operator s is called the GJrding inequality:

Re(/~Ru, u)t~,(Q) >_ call~il~w(Q) - c~ll~ll~,(Q), where u E 0~176 suppu C Q, and ca, c2 > 0 are independent of u. The expression

(5)

I~1,[~1=,-

will be called the symbol of Eq. (4). This brings up the question: How is inequality (5) related to the positiveness of the symbol?

T h e o r e m 1. Let

Re Z rao(A)~a+O > 0 (A fi qn/2T1, 0 # ~ e Ra). I"1,181="

Then inequality (5) holds for Eq. (4).

Proof . First, suppose for the moment that the differential operator in (4) is homogeneous of degree 2m. Integrating by parts and taking into account the condition supp u C Q, we obtain

1 Re(/:Ru, U ) L 2 ( Q ) - -

lal,l/~l=m

Straightforward verification shows that (Rou)(~) = R~(~). By the Plancherel theorem, we have

78

1

I,~t,l~l=,,, I"I,1~1="

= ~ ~ ( R ~ + R:~)I~-- V ~ , ++VI~I l"'l,t.~l=m L~(~")

++ ( ,+,-..-+~ ) = y ] ( R ~ + J%~),+-m-~l~l ~, I~1 ' ~ i,.i,1~1__ m L~(m,,)

1 - ~ y ] ((R+# + R~)G~+#I~I'~, lSl'~)t,(m~ ).

I~l,l~l=,,,

The assumptions of the theorem imply

H,l~l=m

((+~, ~) e K C q"/2T~ x S"-~).

Using the results of w we obtain

m ~ 2

lald~l--m

The estimates for lower-order terms lead to inequality (5) in a standard way. The proof is complete. []

w In this section we show that, for a certain class of domains, the positiveness of the symbol is also necessary for inequality (5) to hold. Namely, we additionally assume that

-Q c qQ. (6)

We shall need a matrix representation of the operators Rao. Consider a bounded domain Q C R n such that

N

+=~ (?)

~k = q- (k-1)~ l (k = 1 , . . . , N ) .

Suppose that u E L2(~). Then the formula

(Uu)k(x) = q-"(k-1)/~u(ql-kx) (x �9 ~i, k = 1 , . . . , N) (8)

specifies a unitary operator U: L2(F~) --+ L2,N(~I). Indeed, by a change of variables, we obtain

II"ll~.cm = I.(~)1 ~ a~ = ~q-"ck-~) I.(q~-+~)l 2 d~ k - ~ l t, k = l l

N

- ~ II(u-)kll~,(m) = IIU-II~,N(m). k----1

Now let us put

We also define R: L2(fl) outside ft.

R = ~ a,Pd: L2(R ") -+ L2(R"). Itl<_3

--+ L2(F/), assuming the functions in L2(~) to be continued to R" by zero

79

We introduce the Toeplitz N x N matrices R with entries

{ qn(k-J)/2a~-j (Ik--j[~_ J),

PJ~ = 0 ( l k - j l > J).

Then we have

( U ' R B ) k ( ~ ) ~--- q--n(k-1)/2(R~)(ql--k X) -~ Z q-n(k-1)/2alu(ql-(k+i)X') Ifl_<J

-- Z q-n(J-a)/2qn(j-k)/2aJ -icu(ql-jx) IJ-kl<J N

= ~ p , , j ( t r , , l j ( k v , ) , , ( ~ , ) (~, e a~). j----1

We see that the operator R: L2(~) ---} L2(~) is unitarily equivalent to the operator of multiplication by the matrix R in the space L2,N(~x):

R = v - ' fiv. (9)

L e m m a 1. Let the matrices R + R* be positive det~nite uniformly ~ t h respect to N = 1, 2, . . . . Then the operator

R + R*: L2(R ") -~ L2(R ")

is positive definite.

Proof. We choose N and a domain fl that satisfies (7) and depends on N. Suppose that u E L2(R ") and u _= 0 outside Q. Then, using (9), we obtain

Re(Ru, B)L, (Rn) ---- Re(URu, UU)L,,N(~I) -~ R e ( k V ? ~ , UU)L,,N(Cll )

_> cl lu~ll~, , . (n,) = cll~,ll~,(~-),

where c > 0 is independent of N , ~ , and u. But for various N and ~ , the function u ranges over a dense subset in L2(Rn).

inequality

Re(Ru, ~)L , (Rn) ----. cllult~,(~.)

Therefore, the

holds for any u E L2(R'~). The proof is complete. []

Note that positive definiteness of the operator R + R* : L2(R n) ~ L2(R n) means that the expression

Re Z atA'-Rer(A) Ill__J

()~ E qn/2T1)

is positive definite.

Theorem 2. Let the domain Q satisfy cond/tion (6) and let estimate (5) hold for Eq. (4). Then

Re Z ra~(A)~+$ > 0 I~I,[~I =m

80

Proof . Integrating (5) by parts and estimating the lower-order terms of the operator Ee , we obtain

I'~1,1~1 ='~

We introduce the notation

al = Q \ q-~-Q, gb, = q-(k-~)$2~ (k=~,.. . ,N), N

$ 2 = U$2k. k = l

Since Q satisfies (6), it follows that $2 satisfies (7). We take some u E C~176 with suppu e $2 and set

Ha(x) = qm(k-1)(Uu)k(x) (x e $21 ; k = 1 , . . . , N). (11)

For Is[---- rn we have

(vv',,)k(~) = q - - ( k - 1 ) / ~ v , ~ , ( y ) l , _ _ , , _ , , = q,,(',-%-.(*-~)/~v.(u(q~-~,)) = q'~(~-~)v'(~ru)k(~) = V'Hk(~),

which implies that

By using (9), (11), and (12), we obtain

up 'u = ~ 'H. (12)

( R,.,fl :D's u , :Da U ) L', ( Q ) = ( Ral~ ~ ' s u , :Z)"U ) L 2 ( i] )

= (UR~,~o%, Y~"U)L',"(nl) = (-a~,~UZ)~u, U~%)L,,~(n~)

= ( ,Raz~H, T) H)L',N(n,), N

II,.llf~.-(,~) = ~ [Iv'-[Ib(.)= ~ ~ l l ( uv ' - ) , ~ l l L ( . . ) i~l=,- Isl=-, k=1

N

IIv H~II~,(.~)- ~ ' V ' H " = H " , L',"(n.) II I tWm.N(. , ) , , fm- - , . .d , ,~ , - - . , d

I.fl=,., k= l I~1=,,, N N

I1'-,11~,,(~) = ~ II(V',.,),,ll~:(~i) = ~ q-2"V'-l)llHkll~,,(~). k----1 k = l

In this case, inequality (10) acquires the form

--callHllL~.~,(n,), I,',l,lPl=m

(13)

where c3 and c4 are independent of H and N , and H E C':r is an arbitrary vector function. Let z ~ E $21, and let B6(x ~ be the ball of radius 6 centered at x ~ . If 6 is sufficiently small, then inequality (13) holds for all H E C'~176176 We shall derive a similar inequality for the function F e 6 ~176 (B~(0)), where t = 7/6 > 1. Let us perform the change of variable y = t(x - x ~ and denote f (y ) = H(z(y)) . Inequality (13) then becomes

2m-- . . 2 --n 2 t~ ' - "Re Z (R'~:D~F'9"F)L='"(s,(o)) > cst [[FIIw-,~(B,(o))-c4~ [IFI[L,,N(S,(O)), I('1,1~1=',,

81

o r

_ - c4IIFIIL~.~(B=(o)). ( 14 ) c311FIIw,.,(e,(o)) V'I ,i#l = m

The const=ts c~ and c, ~ e independent of N , "~, = d F . We now ~et F = , , r , where ~ E C~176 but Y ~ C ~v . By applying the Fourier transformation to (14), we obtain

1

M,I~I==

> c3/R-I~l'=lgi21W(~)12 d~ -c4/~ . Iri2lW(~)l 2 ~. (15)

Here (-,-) is the inner product in C N . Since the set C~ for all 7 is dense in L~(Rn), it follows from (15) that

lal,l~l=m

If I~l ~ > 2 c , / c 3 , then

( ~_, (R,,, +-~*a)~'~+aY, Y) > c31~I"IYI 2. i,fl , l~l=,n

However, since the last inequality is homogeneous in ~, we see that it holds for all 0 # ~ E R '~ �9 Thus, the matrices

are positive definite uniformly with respect to N. Lemma 1 implies that the operator

(n~ + n:~)~ ~+~

is positive definite and

Re ~ ro~(~)~+~>0.

The proof is complete. []

The author expresses his gratitude to A. L. Skubachevskii for his attention to this paper and for his valuable remarks.

References

1. M. N. Vishik, "Strongly elliptic systems of differential equations," Mat. Sb. [Math. USSR-Sb.], 29, No. 3, 615-676 (1951). 2. L. G~xding, "Dirichlet's problem for linear elliptic partial differential equations," Math. Scand., 1, No. 1, 55-72 (1953). 3. S. Agmon, "The coerciveness problem for integro-differential forms," J. Analyse Math., 6, No. 1,183-223 (1958). 4. D. G. Figueiredo, "The coerciveness problem for forms over vector-valued functions," Comm. Pure Appl. Math., 16,

No. 1, 63-94 (1963). 5. J. Necas, "Sur les normes dquivalentes dans W(k)(•) et sur la eoercivitd des formes formellement positives," in: S~min.

de Math. Super., Montr6al (1965). 6. A. Skubachevskii, "The first boundary value problem for strongly elliptic differential-difference equations," J. Differential

Equations, 63, No. 3, 332-361 (1986). 7. W. Rudin, Functional Analysis, McGraw-Hill, New York-Toronto (1973).

Moscow AVIATION INSTITUTE

Translated by M. A. Shishkova

82