Z. ClESIELSKI (Sopot)On the spectrum oî the Laplace operator1. Introduction. Let us consider in the Euclidean space

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I V (1970) ROCZNIKÏ POLSKIEGO TO W AR ZYSTW A MATEM AT Y CZNE G O

Séria I : P RACE MATEMATYCZNE X I V (1970)

Z. ClESIELSKI (Sopot)

O n the spectrum oî the Laplace operator

1. Introduction. Let us consider in the Euclidean space

1, a bounded domain TJ and let its boundary be denoted by dU. In various physical considerations one is led to the eigenvalue problem

\Au = fiu, “ u =

on dZT” .

For U with smooth boundary the condition “ u = 0 on d Z7” simply means that u ( x ) -*■ 0 as x -> zedTJ, xe TJ. It is well known that under certain smoothness hypothesis on dU the domain В {A) of the Laplace operator corresponding to this boundary condition can be chosen in such a way that \A as an operator in L 2 ( TJ) is symmetric, has discrete spectrum and the orthonormal set { y j of eigenfunctions corresponding to the eigenvalues { ^ } is complete.

H. Weyl established in 1915 (cf. [13], pp. 41-45) the asymptotic formula

(

1

1

lim

n—>co

— f*n

n2llc = 2n Г ( Р + 1 )

\ U \

2 Ik

?

where | U\ is the Lebesgne measure of U.

About twenty years later T. Carleman succeeded in proving (cf. [3])

(

1

2

lim — V * w2 Ax) = - i - for x e U.

n->oo n

LJ \U\

7 = 1

There are known several proofs of these results but all of them require unpleasent restrictions on the boundary of TJ.

The aim of this paper is to choose suitable domain for A and then to prove formulas (1.1) and (1.2) for arbitrary bounded domain TJ.

Our argument is very much related to the proof given by Kac in [

], pp. 205-209 (see also [9], pp. 301-303).

A short discussion of the physical questions related to the results

of Weyl and Carleman can be found in the beautiful paper of Kac [11].

2. Potential theory. Most of the results and definitions mentioned here can be found in [

].

The fundamental solution for \A in Ek is known to be

(

2

1

h{x)

— log \x\

■к for Jc =

,

1

2nm Щ к - i ) N 2~fc for Tc >

; i.e. in the distribution sense

(2.2 ) I M V = - Ô V, y e R k,

where hy{x) = Ti(x—y) and Ôy is the é-Dirac distribution concentrated at y.

The letter U is reserved for fixed but arbitrary bounded domain in Rk.

The harmonic measure for U corresponding to x e U is denoted by C(dU) denotes the set of all continuous functions on dJJ. The Wiener generalized solution of the Dirichlet problem with the boundary function f eC(dU) is then equal to

Hf (x) = f f(z)jux(dz), XeU.

dU

A point zedU is said to be regular if and only if for all feC(dU) (2.3) Hf(x) - *f( z) as x z, x e U .

The set of all regular points is denoted by dr U. Now we are ready to write the formula for the Green function of U

(2.4) G(x, y) = h { x - y ) - H hy(x), x , y e U . It can be shown that

Ehu(x) = Hhx (y) for x, y e U, whence the symmetry of G(x, y) follows.

The set of all real valued bounded and continuous functions on U is denoted by C( U) and the set of all bounded Borel functions on U by B(U). For f e B ( U ) we define ||/|| = sup{|/(#)|: xeU) . Clearly [C(U), || ||]

and [B( U), | | ||] are Banach spaces.

Let us write

Д Г И = / M® — y)f{y)dy,

и

Spectrum o f the Laplace operator 43

It can be seen from (2.1) that

(2.5) H: B( U) - >C( Bk),

where C {Rk) is the set of all continnons functions on Rk.

The Green operator G is defined by formula (

.

) Gf{x) = f G( x, y) f ( y) dy, x e U.

и

This is a good place to introduce the set

C0(U) = {feC{TJ): for each zedr U, f ( x )

as x -> z, xe Ü}', it is immediate that CQ{U) is a closed subspace of the Banach space C(U).

Definitions (2.1) and (2.4) can be used to show that G is a continuous operator on B( U) and, moreover, that G: B( U) -> G(U).

Now, let f e B( U) . According to (

.

), (2.4) and (2.5) we have Gf(x) = д{ х) —Щ(х) with g = Щ

whence by (2.3) GfeC0(U). Thus,

(2.7) G: B ( U ) - + C 0(U).

This property will still hold after we replace B( U) by the set of all bounded Lebesgue measurable functions on Ü.

In the sequel we are going to employ the following maximum prin­

ciple: I f h is harmonic on U and heC0(TJ), then h =

. The domain of the Laplace operator is defined as follows (2.8) D( A) = { f e C0(U) n C®(U): AfeC(U)},

where

(2) ( U) is the set of all functions / on U with continuous partial derivatives of order two.

It is very convenient to state in this place the following identity (2.9) / = G ( - * 4 f ) for f e B ( A ) .

For the proof let f e D ( A ) and g = G( — ^Af). It follows from (2.7) and (2.8) that / — </e(7

( U). According to (2.2) and (2.4) we have Ag — Af in the distribution sense and therefore g —/ i s harmonic. Thus, the above maximum principle gives the required equality g = /.

3. Heat equation. In the time-space (0

the fundamental solution for the heat equation

/о *

du

(3.1) 1 Л и = - ^ 7

• at

is known to be

(3.2) p(t, x, y) = (2TrO_ifce x p | - ~ \x-y\^.

For fixed y e B k the function is the nniqne positive solution corre­

sponding to the initial distribution ôy and to the boundary condition P( h x, у) ->

as x oo for each positive t.

Now let U be bounded domain in Bk. We are interested in the funda­

mental solution for equation (3.1) on (0, o o ) x U corresponding to the initial distribution 6y and to the boundary values zero, i.e. we would like to have for each y e U on (

, o o ) x U a positive function q { ’ , - , y ) satisfying (3.1) and such that q(t, •, y) -> ôy with t 0+ and for each t >

, q(t, x , y ) 0 with x zedr U , xeTJ. The existence and uniqueness of such q(t, x, y) can be established either probabilistically or with the aid of the axiomatic potential theory of H. Bauer (cf. [

], [9], [5] and [1]).

Either approach can be used to derive the following properties of q(t, x , y ) :

1

°

< q(t, x, y) < p { t , x, y) on (

, o o ) x U xTJ.

2

° q{t, x, y ) is symmetric in x and y.

3° For t > 0, s > 0 and x,yeTJ we have

q ( t + s , x, y) = f q(t, x, z)q(s,

, y)dz.

и

4° For fixed t > 0 and xe TJ, q(t, x, •) is in C0(U).

5° Let

Qtfix) = f f {y) q. { t , x, y) dy, и

and let f eC(U) . Then for each x e U we have (3.3) Qtf ( x) - +f ( x) with t 0+ .

6

° {Qt, t > 0} is a semigroup of operators in B( U) with ||Qf||< 1 and, moreover,

(3.4) Qt: B ( U ) - + 0 ( U ) .

7° For x, y e U we have

00

(3.5) 0 ( x , y ) = f q( t , x, y) dt .

0

8

° For each x e U there exists on (

,

) x U a Borel probability measure vx such that

(3.6) q(t, x, y) = p(t, x, y ) - J p ( t —s, z , y)vx(ds, dz), Et

where Et — (0, t) x U .

Spectrum of the Laplace operator 45

In the Banach space B ( U) we can introdnce in a natural way the notion of weak convergence (cf. [

], pp. 77-79) which is characterized as follows: / = wlimfn for f n, f e B (U ), i.e. / is a weak limit of the sequence { f n} if and only if the sequence {||/J} is bounded and f n(x) ->f(x) for x e ü .

For the semigroup Qt: B( U) ->• B( U) the set B 0(U) = { f e B { U ) : f = wlimQtf }

t->o+

is called the invariant subspace.

Notice that from (3.3) and from the inequality \\Qt\ \ < 1 it follows

(3.7) C( U) <=B0(U).

The weak infinitesimal operator of the semigroup {Qt} is defined by the formula

О f — f

(3.8) A f — wlim —---.

*->o+ t

The domain of A is denoted by В (A) and it is defined as the set of a l l / e B

(Z7) for which the right-hand side of (3.8) exists and belongs to B0(U).

The resolvent operator Bx for the semigroup {Qt} is defined for f eB ( U) as follows

oo

(3.9) RJ(æ) = J e~uQJ{x)dt, A > 0 , x e U .

о

The case of A = 0 needs to be discussed separately. Notice that for / >

according to (3.9) and (3.5) we have B J / Of as A \

and therefore for arbitrary feB{TJ), BJ( x ) -+Gf(x) as A

+ and xeU. Moreover,

\\RJ\\<№M II <1164/1 K i m ll/ll- Thus,

Gf = w ]im BJ, f e B ( U) , Л->-

+

but this means that G is the potential operator of the semigroup {Qt}.

Since G is bounded we can apply a known result on weak infinitesimal operators for semigroups of contractions (cf. [

], p. 65) to obtain

(3.10) G: B 0(U) - + D( A) ,

- A : D( A) ^ B 0(U), and that G is the inverse to — A, i.e.

(3.11) ( - A ) G f = f ÎOT f e B 0{U),

G { —A ) f = f for f e ! ) { A ) .

Combining (3.7) and (3.10) we find that G[C(U)] <= B( A) whence by (2.9) we get

(3.12) B{ A) c= JD(A).

Now we would like to check

(3.13) A f = \Af for f e D( A) .

This can be seen as follows. Let f eD( A) . According to (2.9) f = G( — ^Af), where — %AfeC{TJ) c= B 0(U) and therefore (3.11) gives A f — AG( — %Af) = %Af. Thus we have shown that A is an extension of A.

4. The eigenvalue problem. Consider the Hilbert space L2(U) with the scalar product

(f,9) = J f(x)g{x)doc, ||/||2 = V ( f , f ) . и

The Green operator G is a self-adjoint weakly singular integral operator in L 2(JJ) with the kernel G( x, y) . The weak singularity follows from (

.

) and (2.4), i.e. we have

0 < G ( ^ , y )< | — — p, C X , y e U

with some constants

and

< ô < к. Defining

G(0){x, у) = G( x , y ) ,

г(т+

)(гс, y) = j G ( x , z)G{m)(z, y)dz и

we find that (cf. [12], p. 75)

(4.1)

г(т)(ж, у) < Cm, Cm = const, X,yeTJ.

This inequality allows to establish the compactness of G in L 2(U) and in particular that its spectrum is discrete with

as the only limit point.

Using Property 3° of Section 3 and (3.5) one shows (cf. [4]) that G is positive, i.e. (G f , f ) >

for / Ф

, f e L z(U). This implies that

is not an eigenvalue and that the spectrum of G is non-negative. Thus, there exists in L 2(U) an orthonormal complete set of eigenvectors {(pj}

such that the corresponding eigenvalues Xj satisfy the inequalities

^ -V It is clear that Gm(p,; — X™(pj and therefore (4.1) gives

(4.2) Aj

Spectrum o f the Laplace operator 47

Consequently the eigenfunctions are bounded and Lebesgue measurable and therefore by the modified property (2.7) we get

Applying to both sides of (4.3) the Laplace operator in the distri­

bution sens we obtain

Since the operator \A + Xj1 is elliptic we can apply the H. Weyl Lemma ([7], p. 140, Corollary 4.1.2) to find that the functions щ are in C°°(U). Moreover, (4.2) gives ||fz%|| = Xf1 Н^-Ц < oo whence ^А^е С{ U) and therefore cpj eD(A). Consequently, if we write /q for — Xf1, then in the classical sense

(4.4) \Acpj =

, cpjdD{A).

Since {(pj) is a basis it follows that D(A) is dense in L 2(U). Using (2.9) we get (Af, g) = ( /, Ag) for / , geD(A). This and (4.4) show that the operator \A with the domain D{ A) is symmetric, it has a discrete spectrum and an orthonormal complete set of eigenvectors {(pj} to which there correspond the eigenvalues { /q} ,/q = —Xj1.

5. The resolvent equation. It is known that for Я >

the operator {XI—J

maps D( A) onto B 0{U) in one-to-one way and that it is equal to the resolvent operator Вл (cf. [

], p. 65) defined in (3.9). Thus for ge B0(U) and / = Rxg we have

Applying to both sides the Green operator G we obtain by (3.11) (4.3) Gcpj = Xj<pf and

Ü) for all j .

4

X f - A f = g.

X G f+ f = Gg.

Introducing gÀ = XRÀg we get

XGgx+ gx — XGg.

This and inequalities (4.1) and (4.2) give for some m

m

9я{оо) = £ ( - } . ) ie tg(x) + ( - Z r +1 ^

and the series converges uniformly and absolutely on U.

It is seen from (3.9) that the last equality can be written by means of the Laplace transform. The nniqueness theorem on Laplace transform and a simple additional argument lead to the formula

00

(5.1) Qtg{x) = ' £ e ~ t^{g,(pj)(pj {x), x e U, t > 0, ge B0(U).

3 = 1

How, G(U) cz B 0( U) and therefore (5.1) gives

oo

(5.2) q(t,oo,y) = ^ e ~ tlÀJ'(P].(x)(pj {y),

7 = 1

where t >

and x, ye U.

For U with smooth boundary formula (5.2) is well known.

6. Theorems of H. Weyl and T. Carleman. To formulate the final results it is more natural to consider the semigroup {Qt} and its potential G as operators acting in L2(TJ).

Denoting by \\Qt\\2 the norm of Qt in L2(U) we check easily with the help of (5.2) that

< 1.

Let Q denote the L 2{TJ) infinitesimal operator of the semigroup and let D(Q) be its domain. We are going to show that

(

.

) J){Q) = {g = G f : f e L2( U) } .

Let D denote the right-hand side of (6.1). The positiveness of G implies that the mapping G: L2{U) -> D is one-to-one. It should be clear that formula (5.1) gives

O O

(

.

) Q,g = у е - ч^( д, щ)ç>, for g tL \ V )

3 = 1

with the right-hand side convergent absolutely and uniformly on U.

Suppose that gel), i.e. g = Gf,. f e L 2{U). The completeness of {

,}

and (

.

) give Qtg - g

t t as t ->

+ •

Thus, geB(Q) and, moreover,

(6.3) QGf = —/ for f e L 2(U).

Conversely, let geD(Q). Then

G ^{Q t g - g} _{+ g} as t ->

+ >

2

t

Spectrum of the Laplace operator 49

and therefore

(6.4) GQg = —g for geB(Q),

whence geB.

We conclude from (6.3) and (6.4) that Q is the inverse to ~G. Since G is self-adjoint it follows that Q is self-adjoint too.

The Lebesgue dominated convergence theorem implies that В (A) c B{Q) hence by (3.12) В (A) c B(Q). Now, (2.9) and (6.3) give Qf = \Af for f e B( A) .

Becapitulating this discussion we can make the following statement:

The symmetric operator \A given on В (A) has Q = —G~l as its self- adjoint extension.

It remains to prove asymptotic formulas (1.1) and (1.2).

For the proof of (1.1) let Uô — { xeU: \x—z\ > ô for z i U} . It is clear that \U0\ \ TJ\~1 -> 1 with ô -> 0+ . Property

° of Section 3 gives for x e ü ô and t < ô2/k

(6.5) 0 < (2nt)~ik— q(t, x, x) < (2nt)~ik exp .

Since 0 < q(t, x, y) < p(t, x, y) we have for x e U \ U 0 and t ^ ô 2/k (

.

)

< (

тvt)-ik — q(t, x, x) <

nt)~ik.

Integrating (6.5) over Ud and (

.

) over U\Ud we obtain

0

< I TJ\ (2nt)~ik— j q{t, x, x)dx и

< I Ut\ (2izt)~ikexp ( - A j + I XJ\ VsI (2irf)-‘ *, hence for t < ô2jk

0

<

(2nt)

~W\

Ik

- x , x)dx < exp и

Now let t = <53; then, for small t, t

1

ЕШ

\ U \ r

ô3 < ô2/k and therefore

J i(t , x, x)dx

и

1 ^ 1

(27

rf

as f

4_.

This and (5.2) give

2 ⁽

^{тг t)ik for t}

^{+ .}

4 — Prace matematyczne X IV

The Tauberian theorem ([14], p. 192) gives for Я

I TJ\Xik (

Tc)*fcr ( i f c + l ) * Substituting X = — jun we obtain (1.1).

The proof of (1.2) is now easy. Let weU be fixed and let <3 be such that we Us, then by (6.5) we obtain

Substituting X = — fin and then applying (

.

) we obtain (1.2).

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Soc. 81 (1956), pp. 294-319.

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[11] — Can one hear the shape of a drum11!, Amer. Math. Monthly 73 (1966), pp. 1-23.

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[14] D . V . W i d d e r , The Laplace transform, Princeton 1946.

q{t, w, w) ~ (2-Kt) ik, t

+ , whence again by (5.2)

OO

- (

ттt)~*k.

The same Tauberian theorem gives for X

References