SOLUTIONS OF LINEAR ELLIPTIC EQUATIONS IN GAUSS-SOBOLEV SPACES

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1 Communications on Stochastic Analysis Vol. 6, No. 3 (2012) Serials Publications SOLUTIONS OF LINEAR ELLIPTIC EQUATIONS IN GAUSS-SOBOLEV SPACES PAO-LIU COW Abstract. The paper is concerned with a class of linear elliptic equations in a Gauss-Sobolev space setting. They arise from the stationary solutions of the corresponding parabolic equations. For nonhomogeneous elliptic equations, under appropriate conditions, the existence and uniqueness theorem for strong solutions is given. Then it is shown that the associated resolvent operator is compact. Based on this result, we shall prove a Fredholm Alternative theorem for the elliptic equation and a Sturm-Liouville type of theorem for the eigenvalue problem of a symmetric elliptic operator. 1. Introduction The subject of parabolic equations in infinite dimensions has been studied by many authors, (see e.g., the papers [6, 7, 17, 3] and in the books [8, 9]). As in finite dimensions, an elliptic equation may be regarded as the equation for the stationary solution of some parabolic equation, if exists, when the time goes to infinity. For early works, in the abstract Wiener space, the infinite-dimensional Laplace equation was treated in the context of potential theory by Gross [13] and a nice exposition of the connection between the infinite-dimensional elliptic and parabolic equations was given by Daleskii [6]. More recently, in the book [9] by Da Prato and Zabczyk, the authors gave a detailed treatment of infinitedimensional elliptic equations in the spaces of continuous functions, where the solutions are considered as the stationary solutions of the corresponding parabolic equations. Similarly, in [4], we considered a class of semilinear parabolic equations in an L 2 -Gauss-Sobolev space and showed that, under suitable conditions, their stationary solutions are the mild solutions of the related elliptic equations. So far, in studying the elliptic problem, most results rely on its connection to the parabolic equation which is the Kolmogorov equation of some diffusion process in a ilbert space. owever, for partial differential equations in finite dimensions, the theory of elliptic equations is considered in its own rights, independent of related parabolic equations [12]. Therefore it is worthwhile to generalize this approach to elliptic equations in infinite dimensions. In the present paper, similar to the finite-dimensional case, we shall begin with a class of linear elliptic equations in a L 2 -Sobolev space setting with respect to a suitable Gaussian measure. It will be Received ; Communicated by the editors Mathematics Subject Classification. Primary 60; Secondary 60G, 35K55, 35K99, 93E. Key words and phrases. Elliptic equation in infinite dimensions, Gauss-Sobolev space, strong solutions, compact resolvent, eigenvalue problem. 471

2 472 PAO-LIU COW shown that several basic results for linear elliptic equations in finite dimensions can be extended to the infinite-dimensional counter parts. In passing it is worth noting that the infinite-dimensional Laplacians on a Lévy - Gel fand triple was treated by Barhoumi, Kuo and Ouerdian [1]. To be specific, the paper is organized as follows. In Section 2, we recall some basic results in Gauss-Sobolev spaces to be needed in the subsequent sections. Section 3 pertains to the strong solutions of some linear elliptic equations in a Gauss-Sobolev space, where the existence and uniqueness Theorem 3.2 is proved. Section 4 contains a key result (Theorem 4.1) showing that the resolvent of the elliptic operator is compact. Based on this result, the Fredholm Alternative Theorem 4.4 is proved. In Section 5, we first characterize the spectral properties of the elliptic operator in Theorem 5.1. Then the eigenvalue problem for the symmetric part of the elliptic operator is studied and the results are summarized in Theorem 5.2 and Theorem 5.3. They show that the eigenvalues are positive, nondecreasing with finite multiplicity, and the set of normalized eigenfunctions forms a complete orthonormal basis in the ilbert space consisting of all L 2 (µ) functions, where µ is an invariant measure defined in Theorem 2.1. Moreover the principal eigenvalue is shown to be simple and can be characterized by a variational principle. 2. Preliminaries Let be a real separable ilbert space with inner product (, ) and norm. Let V be a ilbertsubspacewith norm. Denote the dual spaceofv byv and their duality pairing by,. Assume that the inclusions V = V are dense and continuous [15]. Suppose that A : V V is a continuous closed linear operator with domain D(A) dense in, and W t is a -valued Wiener process with the covariance operator R. Consider the linear stochastic equation in a distributional sense: du t = Au t dt+dw t, t > 0, u 0 = h. (2.1) Assume that the following conditions (A) hold: (A.1) Let A : V V be a self-adjoint, coercive operator such that Av,v β v 2, for some β > 0, and ( A) has positive eigenvalues 0 < α 1 α 2 α n, counting the finite multiplicity, with α n as n. The corresponding orthonormal set of eigenfunctions {e n } is complete. (A.2) The resolvent operator R λ (A) and covariance operator R commute, so that R λ (A)R = RR λ (A), where R λ (A) = (λi A) 1, λ 0, with I being the identity operator in. (A.3) The covariance operator R : is a self-adjoint operator with a finite trace such that TrR <.

3 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES 473 It follows from (A.2) and (A.3) that {e n } is also the set of eigenfunctions of R with eigenvalues {ρ n } such that where ρ n > 0 and Re n = ρ n e n, n = 1,2,,n,, (2.2) ρ n <. n=1 By applying Theorem 4.1 in [5] for invariant measures and a direct calculation, we have the following theorem. Theorem 2.1. Under conditions (A), the stochastic equation (2.1) has a unique invariant measure µ on, which is a centered Gaussian measure with covariance operator Γ = 1 2 A 1 R. Remark 2.2. We make the following two remarks: (1) It is easy to check that e n s are also eigenfunctions of Γ so that Γe n = γ n e n, n = 1,2,,n,, (2.3) where γ n = ρ n. 2α n (2) Let e ta, t 0, denote the semigroup of operators on generated by A. Without condition (A.2), the covariance operator of the invariant measure µ is given by Γ = 0 e ta Re ta dt, which cannot be evaluated in a closed form. Though an L 2 (µ) theory can be developed in the subsequent analysis, one needs to impose some conditions which are not easily verifiable. Let = L 2 (,µ) be a ilbert space consisting of real-valued functions Φ on with norm defined by Φ = { Φ(v) 2 µ(dv)} 1/2, and the inner product [, ] given by [Θ,Φ] = Θ(v)Φ(v)µ(dv), for Θ, Φ. Let n = (n 1,n 2,,n k, ), where n k Z +, be a sequence of nonnegative integers, and let Z = {n : n = n = n k < }, so that n k = 0 except for a finite number of n k s. Let h m(r) be the normalized one-dimensional ermite polynomial of degree m. For v, define a ermite (polynomial) functional of degree n by n (v) = h nk [l k (v)], where we set l k (v) = (v,γ 1/2 e k ) and Γ 1/2 denotes a pseudo-inverse of Γ 1/2. For a smooth functional Φ on, let DΦ and D 2 Φ denote the Fréchet derivatives of the first and second orders, respectively. The differential operator AΦ(v) = 1 2 Tr[RD2 Φ(v)]+ Av,DΦ(v) (2.4)

4 474 PAO-LIU COW is well defined for a polynomial functional Φ with DΦ(v) lies in the domain D(A) of A. owever this condition is rather restrictive on Φ. For ease of calculations, in place ofermite functionals, introduce an exponential family E A () of functionals as follows [8]: E A (). = Span{ReΦ h, ImΦ h : h D(A)}, (2.5) where Φ h (v). = exp{i(h,v)}. It is known that E A () D(A) is dense in. For v E A (), the equation (2.4) is well defined. Returning to the ermite functionals, it is known that the following holds [2]: Proposition 2.3. The set of all ermite functionals { n : n Z} forms a complete orthonormal system in. Moreover we have where λ n = n k α nk. A n (v) = λ n n (v), n Z, We now introduce the L 2 Gauss-Sobolev spaces. For Φ, by Proposition 2.2, it can be expressed as Φ = n ZΦ n n, where Φ n = [Φ, n ] and Φ 2 = n Φ n 2 <. Let m denote the Gauss-Sobolev space of order m defined by for any integer m, where the norm m = {Φ : Φ m < } Φ m = (I A) m/2 Φ = { n (1+λ n ) m Φ n 2 } 1/2, (2.6) with I being the identity operator in = 0. For m 1, the dual space m of m is denoted by m, and the duality pairing between them will be denoted by, m with, 1 =,. Clearly, the sequence of norms { Φ m } is increasing, that is, Φ m < Φ m+1, for any integer m, and, by identify with its dual, we have m m m+1 m, form 1, and the inclusions are dense and continuous. Of course the spaces m can be defined for any real number m, but they are not needed in this paper. Owingtotheuseoftheinvariantmeasureµ, itispossibletodevelopal 2 -theory of infinite-dimensional parabolic and elliptic equations connected to stochastic PDEs. To do so, similar to the finite-dimensional case, the integration by parts is an indispensable technique. In the abstract Wiener Space, the integration by parts with respect the Wiener measure was obtained by Kuo [14]. As a generalization to the Gaussian invariant measure µ, the following integration by parts formula

5 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES 475 holds (see Lemma [9]). In this case, instead of the usual derivative DΦ, it is more natural to use the R derivative D R Φ = R 1/2 DΦ, which can be regarded as a Gross derivative or the derivative of Φ in the direction of R. = R 1/2. Proposition 2.4. Let g R and Φ,Ψ 1. Then we have (D R Φ,g)Ψdµ = Φ(D R Ψ,g)dµ+ (v,γ 1/2 g)φψdµ. (2.7) The following properties of A are crucial in the subsequent analysis. For now let the differential operator A given by (2.4) be defined in the set of ermite polynomial functionals. In fact it can be extended to a self-adjoint linear operator in. To this end, let P N be a projection operator in onto its subspace spanned by the ermite polynomial functionals of degree N and define A N = P N A. Then the following theorem holds (Theorem 3.1, [2]). Theorem 2.5. The sequence {A N } converges strongly to a linear symmetric operator A : 2, so that, for Φ,Ψ 2, the second integration by parts formula holds: (AΦ,Ψ)dµ = = (AΨ)Φdµ 1 (D R Φ,D R Ψ)dµ, (2.8) 2 Moreover A has a self-adjoint extension, still denoted by A with domain dense in. In particular, for m = 2, it follows from (2.6) and (2.8) that Corollary 2.6. The 1 -norm can be defined as for all Φ 1, where D R Φ. = R 1/2 DΦ. Φ 1 = { Φ D R Φ 2 } 1/2, (2.9) Remark 2.7. In (2.9)the factor 1 2 was notdeleted for convenience. Also it becomes clear that the space 1 consists of all L 2 (µ)-functions whose R-derivatives are µ square-integrable. Let the functions F : and G : R be bounded and continuous. For Q L 2 ((0,T);) and Θ, consider the initial-value problem for the parabolic equation: t Ψ t(v) = AΨ t (v) (F(v),D R Ψ t (v)) G(v)Ψ t (v)+q t (v), Ψ 0 (v) = Θ(v), (2.10) for 0 < t < T, v, where A is given by (2.4). Suppose that the conditions for Theorem 4.2 in [3] are met. Then the following proposition holds.

6 476 PAO-LIU COW Proposition 2.8. The initial-value problem for the parabolic equation (2.10) has a unique solution Ψ C([0,T];) L 2 ((0,T); 1 ) such that sup 0 t T Ψ t 2 + T 0 Ψ s 2 1 ds K(T){1+ Θ 2 + where K(T) is a positive constant depending on T. T 0 Q s 2 ds}, (2.11) Moreover, when Q t = Q independent of t, it was shown that, as t, the solution Φ t of (2.10) approaches the mild solution Φ of the linear elliptic equation AΦ(v)+(F(v),D R Φ(v))+G(v)Φ(v) = Q(v), (2.12) or, for α > 0 and A α = A+α, Φ satisfies the equation Φ(v) = A 1 α {(F(v),D RΦ(v))+G(v)Φ(v) Q(v)}, v. In what follows, we shall study the strong solutions (to be defined) of equation (2.12) in an L 2 -Gauss-Sobolev space setting and the related eigenvalue problems. 3. Solutions of Linear Elliptic Equations Let L denote an linear elliptic operator defined by where A is given by (2.4) and LΦ = AΦ+F Φ+GΦ, Φ E A (), (3.1) F Φ = (F ( ), D R Φ( )). (3.2) Then, for Φ 1 and Q, the elliptic equation (2.12) can be written as LΦ = Q, (3.3) in a generalized sense. Multiplying the equation (3.1) by Ψ 1 and integrating the resulting equation with respect to µ, we obtain (LΦ)Ψdµ = { 1 2 (D RΦ,D R Ψ) + (FΦ)Ψ + (GΦ)Ψ}dµ, (3.4) where the second integration by parts formula (2.8) was used. Associated with L, we define a bilinear form B(, ) : 1 1 R as follows B(Φ,Ψ) = { 1 2 (D RΦ,D R Ψ) + (FΦ)Ψ + (GΦ)Ψ}dµ (3.5) = 1 2 [D RΦ,D R Ψ] + [FΦ,Ψ]+[GΦ,Ψ], for Φ,Ψ 1. Now consider a generalized solution of the elliptic equation (3.3). There are several versions of generalized solutions, such as mild solution, strict solution and so on (see [9]). ere, for Q 1, a generalized solution Φ is said to be a strong (or variational) solution of problem (3.3) if Φ 1 and it satisfies the following equation B(Φ,Ψ) = Q,Ψ, for all Ψ 1. (3.6)

7 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES 477 Lemma 3.1. (Energy inequalities) Suppose that F : and G : R are bounded and continuous. Then the following inequalities hold. There exists a constant b > 0 such that B(Φ,Ψ) b Φ 1 Ψ 1, for Φ,Ψ 1, (3.7) and, for any ε (0,1/2), B satisfies the coercivity condition: B(Φ,Φ) ( 1 2 ε) D R Φ 2 + (δ β2 4ε ) Φ 2, for Φ 1, (3.8) where β = sup F(v) and δ = inf G(v). v v Proof. From the equations (3.2) and (3.5), we have B(Φ,Ψ) = { 1 2 (D RΦ,D R Ψ) + (F, D R Φ)Ψ + (GΦ)Ψ}dµ 1 2 D R Φ D R Ψ + (F,D R Φ) Ψ + GΦ Ψ (3.9) where β = sup v 1 2 D R Φ D R Ψ + β D R Φ Ψ +γ Φ Ψ, F(v) and γ = sup G(v). It follows from (3.9) that v B(Φ,Ψ) b Φ 1 Ψ 1, for some suitable constant b > 0. By setting Ψ = Φ in (3.2) and (3.5), we obtain B(Φ, Φ) = { 1 2 (D RΦ,D R Φ) + (F, D R Φ)Φ + (GΦ)Φ}dµ = 1 2 D R Φ 2 + [(F,D R Φ),Φ] + [GΦ,Φ] (3.10) where δ = inf v G(v). For any ε > 0, we have 1 2 D R Φ 2 β D R Φ Φ + δ Φ 2, β D R Φ Φ ε D R Φ 2 + β2 4ε Φ 2. (3.11) By making use of (3.11) in (3.10), we can get the desired inequality (3.8): B(Φ,Φ) ( 1 2 ε) D R Φ 2 + (δ β2 4ε ) Φ 2, which completes the proof. With the aid of the energy estimates, under suitable conditions on F and G, the following existence theorem can be established.

8 478 PAO-LIU COW Theorem 3.2. (Existence of strong solutions) Suppose the functions F : and G : R are bounded and continuous. Then there is a constant α 0 0 such that for each α > α 0 and for any Q 1, the following elliptic problem has a unique strong solution Φ 1. L α Φ. = LΦ + αφ = Q (3.12) Proof. By definition of a strong solution, we have to show the that there exists a unique solution Φ 1 satisfying the variational equation B α (Φ,Ψ). = B(Φ,Ψ) + α[φ,ψ] = Q,Ψ, (3.13) for all Ψ 1. To this end we will apply the Lax-Milgram Theorem [18] in the real separable ilbert space 1. By Lemma 3.1, the inequality (3.7) holds similarly for B α with a different constant b 1 > 0, B α (Φ,Ψ) b 1 Φ 1 Ψ 1, for Φ,Ψ 1. (3.14) In particular we take ε = 1 4 and α 0 = δ β 2 in the inequality (3.11), which is then used in (3.13) to give B α (Φ,Φ) 1 4 D R Φ 2 + η Φ 2, (3.15) where η = α α 0 > 0 by assumption. It follows that B α (Φ,Φ) κ Φ 2 1, (3.16) for κ = min{ 1 4,η}. In view of (3.14) and (3.15), the bilinear form B α(, ) satisfies the hypotheses for the Lax-Milgram Theorem. For Q 1, Q, defines a bounded linear functional on 1. ence there exists a function Φ 1 which is the unique solution of the equation B α (Φ,Ψ) = Q,Ψ for all Ψ 1. Remark 3.3. By writing B α (Φ,Ψ) = L α Φ, Ψ, it follows from Theorem 3.2 that the mapping L α : 1 1 is an isomorphism. Corollary 3.4. Suppose that F C b (;) and G C b () such that inf G(v) > sup F(v) 2. (3.17) v v Then there exists a unique strong solution Φ 1 of the equation LΦ = Q. Proof. This follows from the fact that, under condition(3.17), the inequality(3.16) holds with α = 0.

9 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES Compact Resolvent and Fredholm Alternative For U, consider the elliptic problem: L λ Φ. = LΦ + λφ = U, (4.1) where λ > α 0 is a real parameter. By Theorem 3.2, the problem (4.1) has a unique strong solution Φ 1 satisfying B λ (Φ,Ψ) = L λ Φ, Ψ = U,Ψ, (4.2) for all Ψ 1. For each U, let us express the solution of (4.1) as Denote the resolvent operator K λ of L on by Φ = L 1 λ U. (4.3) K λ U. = L 1 λ U (4.4) for all U. In the following theorem, we will show that the resolvent operator K λ : is compact. Theorem 4.1. (Compact Resolvent) Under the conditions of Theorem 3.2 with λ = α, the resolvent operator K α : is bounded, linear and compact. Proof. By the estimate (3.16) and equation (3.13), we have κ Φ 2 1 B α (Φ,Φ) = [U, Φ] U Φ 1, which, in view of (4.3) and (4.4), implies Φ 1 = K α U 1 C U, (4.5) for C = 1 κ. ence the linear operator K α : is bounded. To show compactness, let {U n } be a bounded sequence in with U n C 0 for some C 0 > 0 and for each n 1. Define Φ n = K α U n. Then, by (4.5), we obtain Φ n 1 1 κ U n C 1, (4.6) where C 1 = C 0 κ. It follows that {Φ n} is a bounded sequence in the separable ilbert space 1 and, hence, there exists a subsequence, to be denoted by {Φ k } for simplicity, which converges weakly to Φ or Φ k Φ in 1. To show that the subsequence will converge strongly in, by Proposition 2.2, we can express Φ = n Z φ n n and Φ k = n Zφ n,k n, (4.7) where φ n = [Φ, n ] and φ n,k = [Φ k, n ]. For any integer N > 0, let Z N = {n Z : 1 n N} and Z + N = {n Z : n > N}. By the orthogonality of ermite

10 480 PAO-LIU COW functionals n s, we can get Φ Φ k 2 = (φ n φ n,k ) 2 + (φ n φ n,k ) 2 n Z N n Z + N (φ n φ n,k ) (1+λ n )(φ n φ n,k ) 2 N n Z N n Z + N [Φ n Φ n,k, n ] 2 + α 1 N Φ Φ k 2 1. n Z N (4.8) Since Φ k Φ, by a theorem on the weak convergence in 1 (p.120, [18]), the subsequence {Φ k } is bounded such that sup Φ k C, and Φ C, some constantc > 0. k 1 Forthe lasttermin the inequality(4.8), givenanyε > 0, there is anintegern 0 > 0 such that α 1 N Φ Φ k 2 1 2α 1C 2 < ε N 2, for N > N 0. (4.9) Again, by the weak convergence of {Φ k }, for the given N 0, we have lim k n Z N0 (φ n φ n,k ) 2 = lim k Therefore there is an integer m > 0 n Z N [Φ n Φ n,k, n ] 2 < ε 2 for k > m. Now the estimates (4.8) (4.10) implies lim Φ Φ k = 0, k which proves the compactness of the resolvent Kγ. n Z N0 [Φ n Φ n,k, n ] 2 = 0. (4.10) Due to the coercivity condition(3.16), the compactness of the resolvent operator implies the following fact. Theorem 4.2. The embedding of 1 into is compact. In the Wiener space, a direct proof of this important result was given in [10] and [16]. To define the adjoint operator of L, we first introduce the divergence operator D. Let F C 1 b (;) be expanded in terms of the eigenfunctions {e k} of A: F = f k e k, where f k = (F,e k ). Then the divergence D F of F is defined as D F =. Tr(DF) = (Df k,e k ).

11 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES 481 Recall the covariant operator Γ = ( 1)A 1 R for µ. We shall need the following integration by parts formula. Lemma 4.3. Suppose that the function F : Γ() is bounded continuous and differentiable such that sup TrDF(v) <. (4.11) v and sup (Γ 1 F(v),v) <. (4.12) v Then, for Φ,Ψ 1, the following equation holds (DΦ,F)Ψdµ = ΦD (ΨF)dµ + Γ 1 F(v),v ΦΨdµ. (4.13) Proof. The proof is similar to that of Lemma in [8], it will only be sketched. Let. n F n = f k e k, (4.14) with f k = (F,e k ). Then the sequence {F n } converges strongly to F in. In view of (4.14), we have n (DΦ,F n )Ψdµ = (DΦ,e k )f k Ψdµ. (4.15) By invoking the first integral-by-parts formula (2.7), (DΦ,e k )f k Ψ = Φ(D(Ψf k ),e k )dµ+ so that (4.15) yields n (DΦ,F n )Ψdµ = + n ΦD (Ψf k e k )dµ (v,γ 1 e k )f k ΦΨdµ (v,γ 1 e k ))f k ΦΨdµ, (4.16) = ΦD (ΨF n )dµ + (v,γ 1 F n (v))φψdµ(v). Now the formula (4.13) follows from (4.15) by taking the limit termwise as n. Let Φ,Ψ E A () and let F,G be given as in by Lemma 4.3. we can write [LΦ,Ψ] = [Φ,L Ψ], where L is the formal adjoint of L defined by L Ψ. = AΨ+D (ΨF) Γ 1 F(v),v Ψ GΨ. (4.17) The associated bilinear form B : 1 1 R is given by B (Φ,Ψ) = B(Ψ,Φ),

12 482 PAO-LIU COW for all Φ,Ψ 1. For Q, consider the adjoint problem L Ψ = Q. (4.18) A function Ψ 1 is said to be a strong solution of (4.16) provided that B (Ψ,Φ) = [Q,Φ] for all Φ 1. Now, for U, consider the nonhomogeneous problem and the related homogeneous problems LΦ = U, (4.19) LΦ = 0, (4.20) and L Ψ = 0. (4.21) Let N and N denote, respectively, the subspaces of solutions of (4.20) and (4.21) in 1. Then, by applying the Fredholm theory of compact operators [18], we can prove the following theorem. Theorem 4.4. (Fredholm Alternative) Let L and L be defined by (3.1) and (4.17) respectively, in which F satisfies the conditions (4.11) and (4.12) in Lemma 4.3. (1) Exactly one of the following statements is true: (a) For each Q, the nonhomogeneous problem (4.19) has a unique strong solution. (b) The homogeneous problem (4.20) has a nontrivial solution. (2) If case (b) holds, the dimension of null space N is finite and equals to the dimension of N. (3) The nonhomogeneous problem (4.19) has a solution if and only if [U,Ψ] = 0, for all Ψ N. Proof. To prove the theorem, we shall convert the differential equations into equivalent Fredholm type of equations involving a compact operator. To proceed let α be given as in Theorem 4.1 and rewrite equation (4.19) as L α Φ. = LΦ+αΦ = αφ+u. By theorem 4.1, the equation (4.19) is equivalent to the equation Φ = K α (αφ+u), which can be rewritten as the Fredholm equation where I is the identity operator on, (I T )Φ = Q, (4.22) T = αk α and Q = K α U. Since K α : is compact, T is also compact and Q belongs to. By applying the Fredholm Alternative Theorem [11] to equation (4.22), the equivalent

13 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES 483 statements (1) (3) hold for the Fredholm operator (I T ). Due to the equivalence of the problems (4.19) and (4.22), the theorem is thus proved. 5. Spectrum and Eigenvalue Problem For λ R, consider strong solutions of the eigenvalue problem: LΦ = λφ. (5.1) ere, for simplicity, we only treat the case of real solutions. As usual a nontrivial solution Φ of (5.1) is called an eigenfunction and the corresponding real number λ, an eigenvalue of L. The (real) spectrum Σ of L consists of all of its eigenvalues. Theorem 5.1. (Spectral Property) The spectrum Σ of L is at most countable. If the set Σ is infinite, then Σ = {λ k R : k 1} with λ k λ k+1, each with a finite multiplicity, for k 1, and λ k as k. Proof. By taking a real number α, rewrite the equation (5.1) as L α. = LΦ+αΦ = (λ+α)φ. (5.2) By taking α > α 0 as in Theorem 4.2, the equation (5.2) can be converted into the eigenvalue problem for the resolvent operator K α Φ = ρφ, (5.3) where ρ = 1 λ+α. (5.4) By Theorem 4.1, the resolvent operator K α on is compact. Therefore its spectrum Σ α is discrete. If the spectrum is infinite, then Σ α = {ρ k R : k 1} with ρ k ρ k+1, each of a finite multiplicity, and lim ρ k = 0. Now it follows k from equation (5.4) that spectrum Σ of L has the asserted property with λ k = 1 α. ρ k Remark 5.2. As in finite dimensions, the eigenvalue problem (5.1) may be generalized to the case of complex-valued solutions in a complex ilbert space. In this case the eigenvalues λ k may be complex. As a special case, set F 0 in L and the reduced operator L 0 is given by L 0 Φ = ( A)Φ + GΦ. Clearly L 0 is a formal self-adjoint operator, or L 0 = L 0. The corresponding bilinear form B 0 (Φ,Ψ) = L 0 Φ, Ψ = 1 [RDΦ,DΨ] + [GΦ,Ψ], (5.5) 2 for Φ,Ψ 1 Consider the eigenvalue problem: L 0 Φ = λφ. (5.6) For the special case when G = 0, L 0 = A. The eigenvalues and the eigenfunctions of ( A) were given explicitly in Proposition 2.2. The results show that

14 484 PAO-LIU COW the eigenvalues can be ordered as a non-increasing sequence {λ k } and the corresponding eigenfunctions are orthonormal ermite polynomial functionals. With a smooth perturbation by G, similar results will hold for the eigenvalue problem (5.6) as stated in the following theorem. Theorem 5.3. (Symmetric Eigenvalue Problem) Suppose that G : R + be a bounded, continuous and positive function, and there is a constant δ > 0 such that G(v) δ, v. Then the following statements hold: (1) Each eigenvalue of L 0 is positive with finite multiplicity. The set Σ 0 of eigenvalues forms a nondecreasing sequence (counting multiplicity) 0 < λ 1 λ 2 λ k such that λ k as k. (2) There exists an orthonormal basis Φ k,k = 1,2, of, where Φ k is an eigenfunction as a strong solution of for k = 1,2,. L 0 Φ k = λ k Φ k, Proof. By the assumption on G, it is easy to check that the bilinear form B 0 :. 1 1 R satisfies the conditions of Theorem 4.1. ence the inverse K 0 = L 1 0 is a self-adjoint compact operator in. Similar to the proof of Theorem 5.1, by converting the problem (5.6) into an equivalent eigenvalue problem for K 0, the statements (1) and (2) are well-known spectral properties of a self-adjoint, compact operator in a separable ilbert space. As in finite dimensions the smallest or the principal eigenvalue λ 1 can be characterized by a variational principle. Theorem 5.4. (The Principal Eigenvalue) The principal eigenvalue can be obtained by the variational formula λ 1 = B 0 (Φ,Φ) inf Φ 1, Φ 0 Φ 2. (5.7) Proof. To verify the variation principle, a strong solution Φ of equation (5.6) must satisfy B 0 (Φ,Φ) = λ[φ,φ] or λ = J(Φ). = B 0(Φ,Φ) Φ 2, (5.8) provided that Φ 0. In view of the coercivity condition B 0 (Φ,Φ) κ Φ 2 1 (5.9) for some κ > 0, J is bounded from below so that the minimal value of J is given by λ = inf J(Ψ), (5.10) Ψ 1, Ψ 0

15 ELLIPTIC EQUATIONS IN GAUSS-SOBELEV SPACES 485 which can also be written as λ = inf Q(Ψ), (5.11) Ψ 1, Ψ =1 where we set Q(Φ) = B 0 (Φ,Φ). To show that λ = λ 1 and the minimizer of Q gives rise to the principal eigenfunction Φ 1, choose a minimizing sequence {Ψ n } in 1 with Ψ n 1 = 1 such that Q(Ψ n ) λ 1 as n. By (5.9) and the boundedness of the sequence {Ψ n } in 1, the compact embedding Theorem 4-2 implies the existence of a subsequence, to be denoted by {Ψ k }, which converges to a function Ψ 1 with Ψ 1 = 1. Since Q is a quadratic functional, by the Parallelogram Law and equation (5.11), we have Q( Ψ j Ψ k 2 ) = 1 2 (Q(Ψ j)+q(ψ k )) Q( Ψ j +Ψ k ) (Q(Ψ j)+q(ψ k ) λ ( Ψ j +Ψ k ) 0 2 as j,k. Again, by (5.9), we deduce that {Ψ k } is a Cauchy sequence in 1 which converges to Ψ 1 and Q(Ψ) = λ. Now, for Θ 1 and t R, let f(t) = J(Ψ+tΘ). As well-known in the calculus of variations, for J to attain its minimum at Ψ, it is necessary that f (0) = 2{B 0 (Ψ,Θ) λ [Φ,Θ]} = 0, which shows Ψ is the strong solution of L 0 (Ψ) = λ Ψ. ence, in view of (5.8), we conclude that λ = λ 1 and Ψ is the eigenfunction associated with the principal eigenvalue. Acknowledgement. This work was done to be presented at the Special Session on Stochastic Analysis in honor of Professor ui-siung Kuo in the 2012 AMS Annual Meeting in Boston. References 1. Barhoumi, A., Kuo, -., Ouerdian,.: Infinite dimensional Laplacians on a Lévy - Gel fand triple; Comm. Stoch. Analy. 1 (2007) Chow, P. L.: Infinte-dimensional Kolmogorov equations in Gauss-Sobolev spaces; Stoch. Analy. and Applic. 14 (1996) Chow, P. L.: Infinte-dimensional parabolic equations in Gauss-Sobolev spaces; Comm. Stoch. Analy. 1 (2007) Chow, P. L.: Singular perturbation and stationary solutions of parabolic equations in Gauss- Sobolev spaces; Comm. Stoch. Analy. 2 (2008) Chow, P. L., Khasminski, R. Z.: Stationary solutions of nonlinear stochastic evolution equations; Stoch. Analy. and Applic. 15 (1997) Daleskii, Yu. L.: Infinite dimensional elliptic operators and parabolic equations connected with them; Russian Math. Rev. 22 (1967) Da Prato, G.: Some results on elliptic and parabolic equations in ilbert space; Rend. Mat. Accad Lencei. 9 (1996) Da Prato, G., Zabczyk,J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, UK, 1992

16 486 PAO-LIU COW 9. Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in ilbert Spaces. Cambridge University Press, Cambridge, Da Prato, G., Malliavin, P., Nualart, D.: Compact families of Wiener functionals; C.R. Acad. Sci. Paris. 315, , Dunford, N., Schwartz, J.: Linear Operators Part I. Interscience Pub., New York, Gilbarg, D., Truginger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Gross, L., Potential theory in ilbert spaces; J. Funct. Analy. 1 (1967) Kuo,..: Integration by parts for abstract Wiener measures; Duke Math. J. 41 (1974) Lions, J. L., Mangenes, E.: Nonhomogeneous Boundary-value Problems and Applications. Springer-Verlag, New York, Peszat, S.: On a Sobolev space of functions of infinite numbers of variables; Bull. Pol. Acad. Sci. 98 (1993) Piech, M. A.: The Ornstein-Uhleneck semigroup in an infinite dimensional L 2 setting; J. Funct. Analy. 18 (1975) Yosida, K.: Functional Analysis. Springer-Verlag, New York, P.L. Chow: Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA address: plchow@math.wayne.edu