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Introduction
The studying of a random processes in infinite dimension Banach spaces and its description by a partial differential equation for a functions on the Banach space are the important topics of contemporary mathematics (see [4, 8, 9]). To the investigation of the above topics and to construct the quantum theory of infinite dimension Hamiltonian systems the analogs of the Lebesgue measure on the infinite dimension linear space are introduced in the works [1, 10, 13, 17].
To study the random walks in the real Hilbert space E we introduce a class of measures on the Hilbert space which are invariant with respect to a shift on an arbitrary vector of the space E (see [10, 14]). For any choice of such measure we construct the Hilbert space ℋ of complex-valued functions any of each is square integrable with respect to this measure. We study operators of argument shifts on the spaces ℋ.
We study the random shift operator on the vector whose distribution on Hilbert space E is given by a semi-group γt, t ≥ 0, of Gaussian measures with respect to the convolution. We prove that the mean values of the random shift operator form the one-parametric semigroup U(t), t ≥ 0, of self-adjoint contractions in the space ℋ. The criteria of strong continuity of this semigroup U is obtained.
We prove that if the semigroup U is strongly continuous in the space ℋ then for any t > 0 the image U(t) f of any vector f ∈ ℋ has the derivatives of any order in the direction of any eigenvector of covariation operator D of Gaussian measure γ1. Therefore the space of smooth functions is defined as the image of the space ℋ under the actions of the operators U(t), t > 0, of semigroup U.
For any non-negative non-degenerated trace-class operator D in the space E the Sobolev space
W_{2,{\mathbf{D}}}^m(E) is defined as the space of functions u ∈ ℋ such that (∂k)lu ∈ ℋ for any l ∈ {1,...,m} and any k ∈ ℕ; and the following series converges
\sum\limits_{k = 1}^\infty {d_m}\parallel {({\partial _k})^m}u\parallel _\mathcal{H}^2 < + \infty.
Here {dk} is the sequence of eigenvalues of the operator D and {ek} is the sequence of corresponding eigenvectors. The function u ∈ ℋ has the derivative ∂hu ∈ ℋ in the direction of the unite vector h ∈ E if the following equality
\mathop {\lim }\limits_{t \to 0} \parallel \frac{1}{t}({{\mathbf{S}}_{th}} - {\mathbf{I}})u - {\partial _h}u{\parallel _\mathcal{H}} = 0 holds.
We study the connection of the random walks in the space E with the self-adjoint analogue of Laplace operator whose domain is the Sobolev space. We prove that the analogue of Laplace operator is the generator of the semigroup of self-adjoint operators arising as the mean value of random shift operators. The properties of smooth function space embedding into the Sobolev spaces are studied. The analogue of Gauss theorem is obtained for the simplest domains such as the rectangle in the space E. The correctness of the problem for Poisson equation in the rectangle with homogeneous Dirichlet condition is obtained and the variational approach of the solving of this problem is constructed.
A class of shift invariant measures on a Hilbert space
According to A. Weil theorem there is no Lebesgue measure on the infinite dimensional separable normed real linear space E, i.e. there is no Borel σ-additive σ-finite locally finite measure on the space E which is translation-invariant. Therefore an analogue of the Lebesgue measure is defined as an additive function on some ring of subsets of the space E which is translation-invariant. In this paper we present the analogue of the Lebesgue measure which is σ-finite and locally finite but not Borel and not σ-additive measure (see [10, 11, 12]). In the papers [1, 16, 17] the analogue of the Lebesgue measure is considered as the measure which is Borel and σ-additive but not σ-finite and not locally finite.
We study invariant measures on a real separable Hilbert space E, which are invariant with respect to any shift. In this article finite-additive analogues of the Lebesgue measure are constructed. The non-negative finite-additive translation-invariant measure λ is defined on the special ring ℛ of subsets from a space E in the work [10]. The ring ℛ contains all infinite-dimensional rectangles whose products of side lengths are absolutely convergent.
Now we describe some class of translation-invariant measures on separable Hilbert space E any of each is the restriction of measure λ from the work [10] on a ring ℛℰ depending on the chois of an orthonormal basis ℰ = {ej} in te space E. Let 𝒮 be a set of orthonormal bases in the space E. Firstly we describe a class of measures on the space E which are invariant with respect to the shift on any vector of this space.
Let us introduce the following family of the elementary sets. Rectangle in the real separable Hilbert space E is the set Π ⊂ E such that there is an orthonormal basis {ej} ≡ ℰ in E and there is an elements a,b ∈ l∞ such that
\Pi = \{ x \in E:(x,{e_j}) \in [{a_j},{b_j})\;\forall \;j \in {\mathbf{N}}\}.
The rectangle (1) is noted by the symbol Πℰ,a,b.
The rectangle (1) is called measurable if it either empty set, or the following condition holds
\sum\limits_{j = 1}^\infty {\text{max}}\{ 0,{\text{ln}}({b_j} - {a_j})\} < + \infty.
Let λ(Π) = 0 if Π = ∅, and let
\lambda ({\Pi _{\mathcal{E},a,b}}) = {\text{exp}}\left( {\sum\limits_{j = 1}^\infty {\text{ln}}({b_j} - {a_j})} \right)
for any nonempty measurable rectangle Πℰ,a,b.
For any orthonormal basis ℱ = {fk} of the space E the symbol K(ℱ) note the set of measurable rectangles with the edges collinear to the vectors of ONB ℱ. Let the symbol rℱ notes the minimal ring of subsets containing the set of measurable rectangles K(ℱ).
Theorem 1
[11] For any orthonormal basis ℱ = {fk} of the space E there exists the unique measure λℱ : rℱ → [0, +∞) such that the equality (3) holds for any rectangle Πℱ,a,b ∈ 𝒦ℱ. The measure λℱ has the unique completion onto the ring ℛℱ which is completion of the ring rℱ by measure λℱ.
Note 2. Here the ring ℛℱ consists on the sets A ⊂ E such that
\overline {{\lambda _\mathcal{F}}} (A) = \underline {{\lambda _\mathcal{F}}} (A) \in \mathbb{R} where
\overline {{\lambda _\mathcal{F}}} (A) = \mathop {{\text{inf}}}\limits_{B \in {r_\mathcal{F}},{\kern 1pt} B \supset A} {\lambda _\mathcal{F}}(A),
\underline {{\lambda _\mathcal{F}}} (A) = \mathop {{\text{sup}}}\limits_{B \in {r_\mathcal{F}},{\kern 1pt} B \subset A} {\lambda _\mathcal{F}}(A) are external and inner measure of a set A with respect to the measure λℱ.
Note 3. Note that there are translation-invariant measures on the space E of another type which is countable additive but not σ-finite (see [17]). There are measures on infinite dimensional topological vector spaces which are translation-invariant with respect to only some subspace of acceptable vectors (see [14]).
Quadratically integrable functions
Now we define space of quadratically integrable functions with respect to λℰ. Since we will use it very often, we define it concisely ℋℰ = L2(E, ℛℰ, λℰ, ℂ).
Let 𝒮 (E, ℛℰ, ℂ) be the linear space hull over field ℂ of indicator functions of the sets from the ring ℛℰ. Let βℰ be the sesquilinear form on the space 𝒮 (E, ℛℰ, ℂ) which is defined by the following conditions:
{\beta _\mathcal{E}}({\chi _A},{\chi _B}) = {\lambda _\mathcal{E}}(A\bigcap\nolimits_ B) for any sets A, B ∈ 𝒦ℰ; for any functions u, v ∈ 𝒮 (E, ℛℰ, ℂ) such that
u = \sum\limits_{j = 1}^n {c_j}{\chi _{{A_j}}},
v = \sum\limits_{k = 1}^m {b_k}{\chi _{{B_k}}} the value βℰ (u, v) is given by the equality
{\beta _\mathcal{E}}(u,v) \equiv {(u,v)_{\mathcal{H}(E)}} = \left( {\sum\limits_{j = 1}^n {c_j}{\chi _{{A_j}}},\sum\limits_{k = 1}^m {b_k}{\chi _{{B_k}}}} \right) = \sum\limits_{k = 1}^m \sum\limits_{j = 1}^n {\bar b_k}{c_j}({\chi _{{A_j}}},{\chi _{{B_k}}}).
This sesquilinear form on the space 𝒮 (E, ℛℰ, ℂ) is Hermitian and nonnegative.
The function u ∈ 𝒮 (E, ℛℰ, ℂ) is called equivalent to the function u ∈ 𝒮 (E, ℛℰ, ℂ) iff βℰ (u − v, u − v) = 0. The linear space 𝒮2(E, ℛℰ, λℰ, ℂ) of the equivalence classes of functions of the space 𝒮 (E, ℛℰ, ℂ) endowing with the sesquilnear form βℰ is the pre-Hilbert space. The Hilbert space L2(E, ℛℰ, λℰ, ℂ) ≡ ℋℰ is defined as the completion of the space 𝒮2(E, ℛℰ, λℰ, ℂ).
Thus for any ONB ℰ in the space E there are the ring of subsets ℛℰ of λℰ -measurable sets, the measure λℰ : ℛℰ → [0, +∞) and the Hilbert space ℋℰ of complex valued λℰ -measurable square integrable functions on the space E. Since the pre-Hilbert space 𝒮2(E, ℛℰ, λℰ, ℂ) of a simple functions is dense linear manifold in the space ℋℰ then the Hibert space ℋℰ is the space of continuous linear functionals on the pre-Hilbert space 𝒮2(E, ℛℰ, λℰ, ℂ).
The products of the spaces with finite additive measures
Let ℰ = {e1, e2, ...} be an ONB in the space E. Let E1 be the Hilbert space with the ONB ℰ1 = {e2, e3, ...}. Then E = ℝ ⊕ E1, E ∋ x = (x1, ξ) ∈ ℝ ⊕ E1, where x1 ∈ ℝ; ξ ∈ E1.
Let 𝒥 be the isomorphism of the Hilbert space E onto the Hilbert space E1 such that ℰ1 = 𝒥 (ℰ). Let λℰ be a complete translation invariant measure on the space E such that the measure λℰ is defined on the ring ℛℰ by the theorem 1. Let ℛℰ1 = 𝒥 (ℛℰ) and λℰ1 is the measure on the space (E1, ℛℰ1) such that λℰ1(A) = λℰ (𝒥−1(A)) ∀ A ∈ ℛℰ1.
Let lℝ be the Jordan measure on the real line ℝ. Let r( ℝ) be a ring of measurable by Jordan subsets of real line ℝ. Remind that Kℰ and Kℰ1 are the collections of measurable rectangles in the spaces E and E1 whose edges are collinear to the vectors of ONB ℰ and ℰ1 respectively; rℰ and rℰ1 are the minimal rings containing the collections of sets Kℰ and Kℰ1 respectively.
We will use the following notations Π = Π′ × Π″ ⊂ ℝ × E1 where Π′ the finite segment of real line and Π, Π″ are the measurable rectangles in the spaces E, E1 respectively.
Lemma 5
([2], lemma 3.3) The inner measure of the set X ⊂ E is defined by the equality{\underline \lambda _\mathcal{E}}(X) = \mathop {\sup }\limits_{\bigcup\limits_{k = 1}^n {Q_k} \subseteq X,{Q_k} \in {K_\mathcal{E}}} {\lambda _\mathcal{E}}\left( {\bigcup\limits_{k = 1}^n {Q_k}} \right)where supremum is defind over the set of finite union of measurable rectangles but not on the hole ring rℰ.
Lemma 6
([2], lemma 3.4). Let A = g × Π, where Π ∈ 𝒦ℰ1and λℰ1(Π) ≠ 0. Then A ∈ ℛℰ iff g ∈ r(ℝ). In this case the following equality λℰ (A) = lℝ(g)λℰ1(Π) holds.
The collection 𝒦ℰ of measurable rectangles is the part of the following collection of sets {A0 × A1, A0 ∈ r( ℝ), A1 ∈ ℛℰ1}; the last collection of sets is the part of the ring ℛℰ. Since the ring rℰ is the minimal ring containing the collection of sets 𝒦ℰ and the ring ℛℰ is the completion of the ring rℰ by the measure λℰ, then the ring ℛℰ is the comletion by measure λℰ of the minimal ring, containing the collection of the sets {A0 × A1, A0 ∈ r( ℝ), A1 ∈ ℛℰ1}. Hence the following statement holds.
Proof
The space with finite additive measure (E, rℰ, λℰ) is the prodict of the spaces with finite additive measures ( ℝ, r( ℝ), lℝ) and (E1, rℰ1, λℰ1)
Proof
According to the lemma 1 [5] (page 222) the space with finite additive measure (E, rℰ, λℰ) is the product of the spaces with the finite additive measures ( ℝ, r( ℝ), lℝ) and (E1, rℰ1, λℰ1). In fact, since the rings ℛℰ and ℛℰ1 is obtained by using of completions by measures λℰ and λℰ1 procedure from the rings rℰ and rℰ1 respectively then the measure λℰ be a unique finite additive measure which is defined on the ring ℛℰ and satisfies the conditions λℰ (A0 × A1) = lℝ(A0)λℰ1(A1) ∀ A0 ∈ r( ℝ), A1 ∈ ℛℰ1.
Definition 1
A tensor product of the finite additive measures μ = μ1 ⊗ μ2 on the space X = X1 × X2 is the measure μ on the space X which the completion of the measure μ1 × μ2. Here μ1 × μ2 is the measure which satisfies following two conditions:
it is defined on the minimal ring containing the collection of sets {A1 × A2, A1 ∈ R1, A2 ∈ R2},
it satisfies the equality μ1 × μ2(A1 × A2) = μ1(A1)μ2(A2) ∀ A1 ∈ R1, A2 ∈ R2.
Lemma 8
The following equality λℰ = lℝ ⊗ λℰ1holds in the sense of definition 1.
Proof
In fact, the procedures of definition of the measure λℰ and the measure lℝ ⊗ λℰ1 have the following common constructions.
The definition the measure λℰ consists of three parts:
The function of a set is defined firstly on the collection of measurable rectangles 𝒦ℰ;
the function 𝒦ℰ → ℝ is extended onto the measure λ on the minimal ring rℰ containing the collection of sets 𝒦ℰ;
the measure λ : rℰ → ℝ is extended onto the ring ℛℰ which is completion of the ring rℰ by measure λ.
In the case of the measure lℝ ⊗ λℰ1 the collection of a sets {A1 × A2 | A1 ∈ r( ℝ), A2 ∈ 𝒦ℰ1} is used instead of the collection 𝒦ℰ on the first step. According to definition the equality lℝ ⊗ λℰ1(I1 × Π1) = λℰ (I1 × Π1) holds for any measurable rectangle I1 × Π1 where I1 ∈ r( ℝ), Π1 ∈ 𝒦ℰ1. Therefore the inequalities
{\underline \lambda _\mathcal{E}}(A) \leqslant \overline {{l_\mathbb{R}} \otimes {\lambda _{{\mathcal{E}_1}}}} (A) \leqslant \underline {{l_\mathbb{R}} \otimes {\lambda _{{\mathcal{E}_1}}}} (A) \leqslant {\overline \lambda _\mathcal{E}}(A) hold for any set A ⊂ E. Hence if a set A ⊂ E is measurable with respect to the measure λℰ then it is measurable with respect measure lℝ ⊗ λℰ1 and the the extensions of the measures λℰ and lℝ ⊗ λℰ1 coincides on the ring ℛℰ.
On the other hand any set of type {A1 × A2 | A1 ∈ r( ℝ), A2 ∈ 𝒦ℰ1} belongs to the ring ℛℰ and in this case the equality λℰ (A1 × A2) = (lℝ ⊗ λℰ1)(A1 × A2) holds. Since the measure λℰ : ℛℰ → ℝ is complete then the continuation of the function of a set (lℝ ⊗ λℰ1) : {A1 × A2 | A1 ∈ r( ℝ), A2 ∈ 𝒦ℰ1} → ℝ by means of continuation on the minimal ring (steps 2)) and completion (step 3)) coincides with the measure λℰ.
Lemma 9
The linear space span({χΠ, Π ∈ 𝒦ℰ }) is dense in the Hilbert space L2(E, ℛℰ, λℰ, ℂ).
Proof
In fact, according to definition the Hilbert space L2(E, ℛℰ, λℰ, ℂ) is the closure of the linear space
\overline {{\text{span}}(\{ {\chi _A},A \in {\mathcal{R}_\mathcal{E}}\} )} endowed with the norm of the space L2(E, ℛℰ, λℰ, ℂ). Since the ring ℛℰ is the completion of the ring rℰ by the measure λℰ then the following equality
{L_2}(E,{\mathcal{R}_\mathcal{E}},{\lambda _\mathcal{E}},\mathbb{C}) = \overline {{\text{span}}(\{ {\chi _A},{\kern 1pt} A \in {r_\mathcal{E}}\} )} holds. Note that any set A ∈ rℰ is the finite union of disjoint sets
A = \bigcup\limits_{k = 1}^N {B_k} where Bk is the complement of a measurable rectangle to finite union of measurable rectangles:
\forall \,k \in \overline {1,N} \;{B_k} = {\Pi _{k,0}}\backslash \bigcup\limits_{j = 1}^{{m_k}} {\Pi _{k,j}},
\Pi k,i \in {\mathcal{K}_\mathcal{E}}\;\forall \;i \in \overline {0,{m_k}} . So, χA ∈ span({χΠ, Π ∈ 𝒦ℰ }) for any A ∈ rℰ. Consequently,
{L_2}(E,{\mathcal{R}_\mathcal{E}},{\lambda _\mathcal{E}},\mathbb{C}) = \overline {{\text{span}}(\{ {\chi _\Pi },{\kern 1pt} \Pi \in {\mathcal{K}_\mathcal{E}}\} )} .
Theorem 10
Morphism ℐ mapping element of L2( ℝ) ⊗ L2(E1, ℛℰ1, λℰ1,ℂ), which is limit of fundamental sequence fk ⊗ vk into limit of sequence fk ⊗ vk in space L2(E, ℛℰ, λℰ, ℂ) provides us with canonical isomorphism between these two space.
Proof
Space L2(E, ℛℰ, λℰ, ℂ) according to lemma 9 is a completion of span({χA, A ∈ 𝒦ℰ }) with respect to norm ‖ · ‖L2(E,ℛℰ, λℰ, ℂ), defined by sesquilinear form βℰ, see (1). So, L2(E, ℛℰ, λℰ, ℂ) is completion of the space span({χA0×A1, A0 ∈ 𝒦ℝ, A1 ∈ 𝒦ℰ1}) with respect to norm ‖ · ‖L2 (E,ℛℰ, λℰ, ℂ), defined by sesquilinear form βℰ. (Here we define by 𝒦ℝ a set of all bounded intervals of ℝ).
Space L2( ℝ) ⊗ L2(E1, ℛℰ1, λℰ1, ℂ) is completion of linear span ℒ of elements f ⊗ v, where f ∈ L2( ℝ), v ∈ L2(E1, ℛℰ1, λℰ1, ℂ) with respect to norm ‖ · ‖⊗, defined by sesquilinear form β⊗ on ℒ with restriction β⊗ (f1 ⊗ v1, f2 ⊗ v2) = (f1, f2)L2( ℝ)(v1, v2)L2(E1,ℛℰ1,λℰ1,ℂ). Note that in space L2( ℝ) linear span ℒ0 of set of characteristic functions from 𝒦ℝ is dense linear submanifold, and in space L2(E1, ℛℰ1, λℰ1, ℂ) according to lemma 9 linear span ℒ1 of set of characteristic functions of set from 𝒦ℰ1 is also a dense linear submanifold. That’s why space L2( ℝ) ⊗ L2(E1, ℛℰ1, λℰ1, ℂ) is exactly a comletion of linear span span{χA0 ⊗ χA1, A0 ∈ 𝒦ℝ, A1 ∈ 𝒦ℰ1} with respect to norm ‖ · ‖⊗.
Since for any interval Δ ∈ 𝒦ℝ and any measurable rectangle Π′∈ 𝒦ℰ1 holds ‖χΔ ⊗ χΠ′ ‖⊗ = ‖χΠ‖L2(E,ℛℰ, λℰ,ℂ), where Π = Δ × Π′, then for any sets A0 ∈ 𝒦ℝ, A1 ∈ 𝒦ℰ1 holds
\parallel {\chi _{{A_0}}} \otimes {\chi _{{A_1}}}{\parallel _ \otimes } = \parallel {\chi _{{A_0} \times {A_1}}}{\parallel _{{L_2}(E,{\mathcal{R}_\mathcal{E}},{\lambda _\mathcal{E}},\mathbb{C})}}.
Since linear span span({χA0×A1, A0 ∈ 𝒦ℝ, A1 ∈ 𝒦ℰ1}) is dense in space L2(E, ℛℰ, λℰ, ℂ), and linear span span({χA0 ⊗ χA1, A0 ∈ 𝒦ℝ, A1 ∈ 𝒦ℰ1}) is dense in space L2( ℝ) ⊗ L2(E1, ℛℰ1, λℰ1, ℂ), then (dee equation 2) spaces L2(E, ℛℰ, λℰ, ℂ) and L2( ℝ) ⊗ L2(E1, ℛℰ1λℰ1, ℂ) are isometrically isomorphic and L2(E, ℛℰ, λℰ, ℂ) = ℐ (L2( ℝ) ⊗ L2(E1, ℛℰ1λℰ1, ℂ)).
Partial Fourier transforms
Fourier transform of the the space L2( ℝ) is unitary mapping of the space L2( ℝ) into itself. Therefore the partial Fourier transform ℱ1 with respect to the first coordinate is defined on the space L2(E, ℛℰ, λℰ, ℂ) = L2( ℝ) ⊗ L2(E1, ℛℰ1, λℰ1,ℂ). The partial Fourier transform ℱ1 is defined on the linear hull of the elements of type u1(x1)u2(x), u1 ∈ L2( ℝ), u2 ∈ L2(E1, ℛℰ1, λℰ1, ℂ) by the equality
{\mathcal{F}_1}\left( {\sum\limits_{k = 1}^n {u_{1,k}}{u_{2,k}}} \right) = \sum\limits_{k = 1}^n {\hat u_{1,k}}{u_2},
where û1,k ∈ L2( ℝ) is Fourier transform of the function u1,k ∈ L2( ℝ). Since the linear hull of the elements of type u1(x1)u2(ξ), u1 ∈ L2( ℝ), u2 ∈ L2(E1, ℛℰ1, λℰ1, ℂ) is dense in the space L2(E, ℛℰ, λℰ, ℂ) then the partial Fourier transform ℱ1 has the unique continuation up to the unitary transform of the space L2(E, ℛℰ, λℰ, ℂ) into itself.
Analogously, partial Fourier transform ℱn with respect to first n coordinates is defined on the space L2(E, ℛℰ, λℰ, ℂ). According to the properties of Fourier transform of the space L2( ℝn) with some n ∈ ℕ; the following statement holds: partial Fourier transform of the space L2(E, ℛℰ, λℰ, ℂ) with respect to first n coordinates is unitary mapping of the space L2(E, ℛℰ, λℰ, ℂ) into itself for any n ∈ ℕ;.
Partial Fourier transform will useful in the studying of the operators of multiplication on coordinate and momentum operator with respect to direction of a vector ej of the basis ℰ. It also be used further in the studying of generators of diffusion semigroups and its fraction powers.
Sobolev spaces and spaces of smooth functions
Averaging of random shifts and space of smooth functions
Let D ∈ B(E) be a nonnegative trace class operator with the orthonormal basis ℰ of eigenvectors. Any operator D of the above class defines the centered countable additive Gaussian measure νD on the space E such that the measure νD has the covariance operator D and zero mean value.
Shift operator on the vector h ∈ E is defined on the space ℋℰ = L2(E, ℛℰ.λℰ, ℂ) by the equality
{{\mathbf{S}}_h}u(x) = u(x - h).
It is obvious that for any h ∈ H operator Sh belongs to the Banach space B(ℋℰ) of bounded linear operators in the space ℋℰ endowed with the operator norm; moreover Sh is the unitary operator in the space ℋℰ. Let h be a random vector of the space E whose distribution is given by the measure ν. Then the mean value U ∈ B(ℋℰ) of random shift operator Sh is given by the Pettis integral
\int\limits_E {{\mathbf{S}}_h}d\nu (h) = {\mathbf{U}}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,({\mathbf{U}}f,g) = \int\limits_E ({{\mathbf{S}}_h}f,g)d\nu (h)\,\,\forall f,g \in {\mathcal{H}_\mathcal{E}}.
According to the paper [12] (see also [15]) the following statement holds.
Theorem 11
LetD ∈ B(E) be a nonnegative trace class operator with the orthonormal basis ℰ of eigenvectors. Then one-parametric family of operators{{\mathbf{U}}_t} = \int\limits_E {{\mathbf{S}}_h}d{\nu _{t{\mathbf{D}}}}(h),\,\,t \geqslant 0,is a one-parametric semigroup of self-adjoint contractions in the space ℋℰ. The semigroupUt, t ≥ 0, is strong continuous in the space ℋℰ if and only if{{\mathbf{D}}^{\frac{1}{2}}}is trace class operator.
Definition 2
A function u′j ∈ ℋℰ is called the derivative of the function u ∈ ℋℰ in the direction of a unite vector ej if the following equality holds
\mathop {{\text{lim}}}\limits_{s \to 0} \parallel \frac{1}{s}({{\mathbf{S}}_{s{e_j}}}u - u) - {u'_j}{\parallel _{{\mathcal{H}_\mathcal{E}}}} = 0.
Lemma 12
[see [11], lemmas 7.1, 7.2]. Let {ej} ≡ ℰ be the orthonormal basis of eigenvectors of positive trace class operatorD. If νDbe a probability Gaussian measure on the space E with covariance operatorDand{\mathcal{U}_{\mathbf{D}}}(t) = \int\limits_E {{\mathbf{S}}_{\sqrt t h}}d{\nu _{\mathbf{D}}}(h), t ≥ 0, u ∈ ℋℰ then for any l ∈ ℕ; there is the number cl > 0 such that for any u ∈ ℋℰ, j ∈ ℕ;, t > 0 there is the derivative of the power l\partial _j^l{\mathcal{U}_{\mathbf{D}}}(t)u \in {\mathcal{H}_\mathcal{E}}and the following estimates take place\parallel \partial _j^l{\mathcal{U}_{\mathbf{D}}}(t)u{\parallel _{{\mathcal{H}_\mathcal{E}}}} \leqslant \frac{{{c_l}}}{{{{(\sqrt {t{d_j}} )}^l}}}\parallel u{\parallel _{{\mathcal{H}_\mathcal{E}}}}.
Let D be a positive trace class operator in the space E. Let
C_{\mathbf{D}}^\infty (E) be a linear hull of the following system of elements {𝒰D(t)u, t > 0, u ∈ ℋℰ}. The linear manifold
C_{\mathbf{D}}^\infty (E) is called the space of smooth functions according to lemma 12.
Sobolev spaces and embedding theorem
For any a > 0 the symbol
W_{2,{{\mathbf{D}}^a}}^1(E) notes the linear space of elements u of the space ℋℰ such that the following two condition hold
1) for any j ∈ ℕ; there is the derivative
{u_j} \equiv \frac{\partial }{{\partial {x_j}}}u \in {\mathcal{H}_\mathcal{E}} with respect to the direction of eigenvector ej of operator D;
2)\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{j = 1}^n d_j^a\parallel {u_j}\parallel _{{\mathcal{H}_\mathcal{E}}}^2 < + \infty.
The space
W_{2,{{\mathbf{D}}^a}}^1(E) endowed with the norm
\parallel u{\parallel _{W_{2,{{\mathbf{D}}^a}}^1(E)}} = {(\parallel u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \sum\limits_{j = 1}^n d_j^a\parallel {u_j}\parallel _{{\mathcal{H}_\mathcal{E}}}^2)^{\frac{1}{2}}} is the Hilbert space which is continuously embedded into the space ℋℰ (see [3, 11]).
For any numbers a > 0 and any l ∈ ℕ; the symbol
W_{2,{{\mathbf{D}}^a}}^1(E) notes the linear space of elements u of the space ℋℰ such that the following two conditions hold:
1) for any j ∈ ℕ; there is the l-order derivative
\frac{{{\partial ^2}}}{{\partial e_j^l}}u \in {\mathcal{H}_\mathcal{E}};
2)\,\,\,\,\,\,\,\,\,\sum\limits_{j = 1}^n d_j^a\left\| {\frac{{{\partial ^l}}}{{\partial e_j^l}}u} \right\|_{{\mathcal{H}_\mathcal{E}}}^2 < + \infty.
Analogously, the space
W_{2,{{\mathbf{D}}^a}}^1(E) endowed with the norm
\parallel u{\parallel _{W_{2,{{\mathbf{D}}^a}}^l(E)}} = {(\parallel u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \sum\limits_{j = 1}^n d_j^a\parallel \frac{{{\partial ^l}}}{{\partial e_j^l}}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2)^{\frac{1}{2}}}is the Hilbert space which is continuously embedded into the space ℋℰ (see [3, 11]).
Theorem 13
[see [11], lemmas 7.1, 7.2]. Let u ∈ ℋℰ. LetDbe a positive trace class operator in the space E such that{{\mathbf{D}}^{\frac{1}{2}}}is the trace class operator. Then for any t > 0 the inclusion{\mathcal{U}_{{{\mathbf{D}}^{\frac{1}{2}}}}}(t)u \in W_{2,{\mathbf{D}}}^1(E)holds.
Theorem 14
LetDbe a positive trace class operator in Hilbert space E such thatDγ is trace class operator with some γ > 0. Let l ∈ ℕ;. If b ≥ lα + γ with some α ∈ [γ, +∞) thenC_{{{\mathbf{D}}^\alpha }}^\infty (E) \subset W_{2,{{\mathbf{D}}^b}}^l(E).
If, moreover, α ≥ 2γ then the linear manifoldC_{{{\mathbf{D}}^\alpha }}^\infty (E)is dense in the spaceW_{2,{{\mathbf{D}}^b}}^l(E).
The proof of this theorem is published in the work [3].
The traces of a functions on the codimension 1 hyperplane
Let ℰ be an orthonormal basis in the space E. Let e1 ∈ ℰ and ℰ1 = ℰ \{e1}. Let E1 = (span(e1))⊥ and ℋℰ1 = L2(E1, ℛ1, λℰ1, ℂ). Let ℛ( ℝ) be a ring of Lebesgue integrable subsets of the space ℝ with finite Lebesgue measure. Let λL be the Lebesgue measure on the space ℝ.
Then accordind to the theorem 10 ℋℰ = L2( ℝ, ℛ( ℝ), λL, ℋℰ1)) where L2( ℝ, ℛ( ℝ), λL, ℋℰ1)) is the space of integrable in Bokhner sense with respect to Lebesgue measure λL mappings ℝ → ℋℰ1.
We define a linear space
W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}) = \{ u \in {\mathcal{H}_\mathcal{E}}:\frac{\partial }{{\partial {e_1}}}u \in {\mathcal{H}_\mathcal{E}}\} endowed with the Sobolev norm
|u{\parallel _{W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}))}} = {(\parallel u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel \frac{\partial }{{\partial {e_1}}}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2)^{\frac{1}{2}}}.
According to the definition of partial Fourier transform
\parallel u\parallel _{W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})}^2 = \int\limits_\mathbb{R} (1 + {\xi ^2})\parallel {\hat u_1}(\xi )\parallel _{{\mathcal{H}_{{\mathcal{E}_1}}}}^2d\xi
for any
u \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}) where û = ℱ1(u). Hence the space
W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})) endoved with the norm (6) is the Hilbert space.
Theorem 15
Ifu \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})then the equivalence class u contains the continuous function ũ ∈ C( ℝ, ℋℰ1). Moreover, there is the constant C > 0 such that\parallel \tilde u{\parallel _{C(\mathbb{R},{\mathcal{H}_{{\mathcal{E}_1}}})}} \leqslant C\parallel u{\parallel _{W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})}}.
Proof
In the case of separable space ℋℰ1 the proof of this theorem is given in the monograph [6]. In the case under consideration the space ℋℰ1 is not separable. But according to the condition
u \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}) there is the separable subspace of the space ℋℰ1 containing the values of the mapping u : ℝ → ℋℰ1. Therefore the proof of the theorem 3.1 by [6] can be applied to the obtaining of the statement of theorem 15.
If
u \in W_{2,{\mathbf{D}}}^1(E) then the function u can be considered as the function of the space
W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}). Therefore the function
u \in W_{2,{\mathbf{D}}}^1(E) can be considered as the continuous maping ũ : ℝ → ℋℰ1 according to theorem 15. Hence we can use the following definition of the trace of function.
Definition 3
The trace of the function
u \in W_{2,{\mathbf{D}}}^1(E)\ at the hyperplane x1 = t0, t0 ∈ ℝ, is the value of function ũ ∈ C( ℝ, ℋℰ1) at the point t0.
Corollary 16
Ifu \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})then for any t0 ∈ ℝ the estimate\parallel \tilde u({t_0}){\parallel _{{\mathcal{H}_{{\mathcal{E}_1}}}}} \leqslant C\parallel u{\parallel _{W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})}}holds.
Corollary 17
Ifu \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})and\frac{d}{{ds}}{{\mathbf{S}}_{s{e_1}}}u = v \in {L_2}(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}), then for any t1, t2 ∈ ℝ the equality holdsu({t_2}) - u({t_1}) = \int\limits_{{t_1}}^{{t_2}} v(s)ds.
The analog of Gauss theorem for rectangle
Let ℰ be the ONB of positive trace-class covariation operator D of Gaussian measure γ. Let Πa,b ∈ 𝒦 (ℰ) be a measurable rectangle. For any j ∈ ℕ; the equality E = ℝ ⊕ Ej where ℝ = span(ej) and Ej = (span(ej))⊥. Let ℰj = {e1,...,ej−1, ej+1,...}. Therefore the following equality
{\Pi _{a,b}} = [{a_j},{b_j}) \times {\hat \Pi _{{{\hat a}_j},{{\hat b}_j}}} holds where
{\hat \Pi _{{{\hat a}_j},{{\hat b}_j}}} \in \mathcal{K}({E_j}) is the measurable rectangle in the space Ej.
Let
u \in W_{2,{\mathbf{D}}}^2(E). Then for any j ∈ ℕ; and any a ∈ R there is the trace u|xj=a ∈ ℋℰj of the function u according to the theorem 3.4. Since for any j ∈ ℕ; the function ∂ju has the derivative
\partial _j^2u \in {\mathcal{H}_\mathcal{E}} in the direction ej, then according to the theorem 3.4 there is the trace (∂ju)|xj=a ∈ ℋℰj.
Theorem 18
LetDbe a positive trace class operator in Hilbert space E. Letu \in W_{2,{\mathbf{D}}}^2(E), let Πa,b ∈ 𝒦 (ℰ) be a measurable rectangle. Then the equality\int\limits_{{\Pi _{a,b}}} {\Delta _{\mathbf{D}}}u\phi d{\lambda _\mathcal{E}} = - \int\limits_{{\Pi _{a,b}}} {(\nabla u,{\mathbf{D}}\nabla \phi )_E}d{\lambda _\mathcal{E}} + \int\limits_{\partial {\Pi _{a,b}}} ({\mathbf{n}},{\mathbf{D}}\nabla u)\phi ds,holds for any function\phi \in W_{2,{\mathbf{D}}}^1(E). Here\int\limits_{\partial {\Pi _{a,b}}} ({\mathbf{n}},{\mathbf{D}}\nabla u)\phi ds = \sum\limits_{j = 1}^\infty {d_j}\int\limits_{{{\hat \Pi }_{{{\hat a}_j},{{\hat b}_j}}}} [(\phi {\kern 1pt} {\partial _j}u){|_{{x_j} = {b_j}}} - (\phi {\kern 1pt} {\partial _j}u){|_{x = {a_j}}}]d{\lambda _{{\mathcal{E}_j}}}.
Proof
Since
\phi \in W_{2,{\mathbf{D}}}^1(E) then the condition ϕ|xj=a ∈ ℋℰ| holds for any j ∈ ℕ; and any a ∈ R according to the theorem 15, moreover
\parallel {\phi _{{x_j} = a}}{\parallel _{{\mathcal{H}_{{\mathcal{E}_j}}}}} \leqslant C{(\parallel \phi \parallel _{{L_2}(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_j}}})}^2 + \parallel {\partial _j}\phi \parallel _{{L_2}(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_j}}})}^2)^{\frac{1}{2}}}
according to theorem 15 and corollary (16). Therefore
\begin{array}{*{20}{c}} {|\int\limits_{{{\hat \Pi }_{{{\hat a}_j},{{\hat b}_j}}}} [(\phi {\kern 1pt} {\partial _j}u){|_{{x_j} = b}} - (\phi {\partial _j}u){|_{{x_j} = a}}]d{\lambda _j}| = |{{(\phi {|_{x = b}},{\partial _j}u{|_{x = b}})}_{{\mathcal{H}_{{\mathcal{E}_j}}}}} - {{(\phi {|_{x = a}},{\partial _j}u{|_{x = a}})}_{{\mathcal{H}_{{\mathcal{E}_j}}}}}| \leqslant } \\ { \leqslant 2\parallel {\partial _j}u{\parallel _{C(\mathbb{R},{\mathcal{H}_{{\mathcal{E}_j}}})}}\parallel \phi {\parallel _{C(\mathbb{R},{\mathcal{H}_{{\mathcal{E}_j}}})}}} \end{array}
for any j ∈ ℕ;. Hence the estimates
\begin{array}{*{20}{c}} {|\int\limits_{{{\hat \Pi }_{{{\hat a}_j},{{\hat b}_j}}}} [(\phi {\partial _j}u){|_{{x_j} = b}} - (\phi {\kern 1pt} {\partial _j}u){|_{{x_j} = a}}]d{\lambda _j}| \leqslant } \\ { \leqslant 2{C^2}{{(\parallel \phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel {\partial _j}\phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2)}^{\frac{1}{2}}}{{(\parallel {\partial _j}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel \partial _j^2u\parallel _{{\mathcal{H}_\mathcal{E}}}^2)}^{\frac{1}{2}}} \leqslant } \\ { \leqslant {C^2}(\parallel \phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel {\partial _j}\phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel {\partial _j}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel \partial _j^2u\parallel _{{\mathcal{H}_\mathcal{E}}}^2)} \end{array}
hold for any j ∈ ℕ;. Let us note that
\sum\limits_{j = 1}^\infty {d_j}(\parallel \phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel {\partial _j}\phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2) = ({\text{Tr}}{\mathbf{D}} - 1)\parallel \phi \parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel \phi \parallel _{W_{2,{\mathbf{D}}}^1(E)}^2.
Since partial Fourier transform with respect to coordinate xj is unitary operator in the space ℋℰ and according to the inequality k2 ≤ 1 + k4, k ∈ ℝ, we obtain
\sum\limits_{j = 1}^\infty {d_j}\parallel {\partial _j}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 \leqslant \sum\limits_{j = 1}^\infty {d_j}(\parallel u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel \partial _j^2u\parallel _{{\mathcal{H}_\mathcal{E}}}^2)
Therefore the series in the right hand side (9) converges.
Since
{\partial _j}(\phi {d_j}{\partial _j}u) = {d_j}{\partial _j}\phi {\partial _j}u + {d_j}\partial _j^2u\phi ,
then according to the corollary 17 the equality
\int\limits_{\partial {\Pi _{a,b}}} ({\mathbf{n}},{{\mathbf{e}}_j}{d_j}{\partial _j}u)\phi ds = \int\limits_{{\Pi _{a,b}}} {d_j}\partial _j^2u\phi d\lambda + \int\limits_{{\Pi _{a,b}}} {d_j}{\partial _j}u{\partial _j}\phi d\lambda.
holds for any j ∈ ℕ;. As it shown above, under the summation of the equalities (10) by j ∈ ℕ; the series of left hand side absolutely converges. The series of first components in right hand side absolutely converges by definition of the space
W_{2,{\mathbf{D}}}^2(E). Hence the series in right hand side of (10) absolutely converges and the equality (8) takes place.
Dirichlet problem for Poisson equation
Let us consider the unique rectangle Π0,1 ∈ 𝒦ℰ. Let symbol L2(Π0,1) notes the subspace of the space ℋℰ with the support in the set Π0,1. Let us note that f = χΠ0,1f for any f ∈ L2(Π0,1).
Let symbol
\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) notes the space of function
W_{2,{\mathbf{D}}}^1(E)\bigcap\nolimits_ {L_2}({\Pi _{0,1}}) with the support in the set Π0,1. It should be note that for any
u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}})
the following statement holds:
u{|_{\partial {\Pi _{0,1}}}} = 0;\,\,\,{\partial _{{x_k}}}u \in {L_2}({\Pi _{0,1}})\,\,\forall k \in \mathbb{N}.
We also introduce the space
\dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) of functions
u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) such that
\begin{array}{*{20}{c}} {\exists {g_{jk}} \in {L_2}({\Pi _{0,1}}):({\mathbf{D}}{\partial _k}u,{\partial _j}\phi ) = - ({g_{jk}},\phi )\;\forall \;\phi \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}),\;\forall j,k \in \mathbb{N};} \\ {\sum\limits_{j = 1}^\infty \parallel {g_{jj}}\parallel _{{\mathcal{H}_\mathcal{E}}}^2 < + \infty.} \end{array}
We pose the following problem. For a given function f ∈ L2(Π0,1) and a given number a ≥ 0 we should find a function
u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) such that
{\Delta _{\mathbf{D}}}u = au + f,u{|_{\partial {\Pi _{0,1}}}} = 0.
To investigate the above problem we apply the variation approach (see [7]). At first we study the space
\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).
Let us introduce the trapezoid-like function ψδ : ℝ → ℝ which is given by the equalites
{\psi _\delta }(x) = 0,x \in ( - \infty ,0]\bigcup\nolimits [1, + \infty ); ψδ (x) = 1, x ∈ [δ, 1 − δ];
{\psi _\delta }(x) = \frac{1}{\delta }x,x \in \left( {0,\delta } \right);
{\psi _\delta }(x) = - \frac{1}{\delta }(x - 1),x \in (1 - \delta ,1) for any
\delta \in (0,\frac{1}{2}). Then
{\psi _\delta } \in \dot W_2^1([0,1]), ‖ψδ‖L2([0,1]) ∈ [1 − 2δ, 1],
{\partial _x}\psi {\parallel _{{L_2}([0,1])}} = \sqrt {2/\delta } .
Let D a the nonnegative trace-class operator such that
{{\mathbf{D}}^{\frac{1}{2}}} is trace-class operator. Let {ej} be the ONB of eigenvectors of the operator D and {dj} be the corresponding sequence of eigenvalues. For any sequence
\{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2}) the function
{\Psi _{\{ \delta \} }}(x) = \Pi _{j = 1}^\infty {\psi _{{\delta _j}}}({x_j}) is defined.
The inclusion Ψ{δ} ∈ ℋℰ for the nonnegative function Ψ{δ} is equivalent to the condition
\forall \varepsilon > 0\,\exists g,G \in {\mathcal{H}_\mathcal{E}}:g \leqslant {\Psi _{\{ \delta \} }} \leqslant G\,\,\,\user1{and}\quad \parallel G - g{\parallel _{\mathcal{H}\mathcal{E}}} < \varepsilon.
Let us define
{G_n}(x) = \Pi _{k = 1}^n{\psi _{{\delta _k}}}({x_k})\Pi _{k = n + 1}^\infty {\chi _{[0,1]}}({x_k})
and
{g_n}(x) = \Pi _{k = 1}^n{\psi _{{\delta _k}}}({x_k})\Pi _{k = n + 1}^\infty {\chi _{[{\delta _k},1 - {\delta _k}]}}({x_k}) for some n ∈ ℕ;. Then gn ≤ Ψ{δ} ≤ Gn and
\parallel {G_n} - {g_n}{\parallel _\mathcal{H}} < 1 - {\text{exp}}(\sum\limits_{k = n + 1}^\infty {\text{ln}}(1 - 2{\delta _k})) for any n ∈ ℕ;. Since
\mathop {{\text{lim}}}\limits_{n \to \infty } \sum\limits_{k = n + 1}^\infty {\delta _k} = 0 then
\mathop {{\text{lim}}}\limits_{n \to \infty } \parallel {G_n} - {g_n}{\parallel _\mathcal{H}} = 0. Therefore Ψ{δ} ∈ ℋ.
Let us note that 0 ≤ χΠδ,1−δ ≤ Ψδ ≤ χΠ0,1. Hence
{\lambda _\mathcal{E}}({\Pi _{\delta ,1 - \delta }}) \leqslant \parallel {\Psi _\delta }\parallel _\mathcal{H}^2 \leqslant {\lambda _\mathcal{E}}({\Pi _{0,1}}). Therefore the statements of the 19 are proved.
Note that
\lambda ({\Pi _{\delta ,1 - \delta }}) = {\text{exp}}(\sum\limits_{k = 1}^\infty {\text{ln}}(1 - 2{\delta _k})). Since
{\delta _k} \in (0,\frac{1}{2}) for any k ∈ ℕ; then the series
\sum\limits_{k = 1}^\infty {\text{ln}}(1 - 2{\delta _k}) converges if and only if the series
\sum\limits_{k = 1}^\infty {\delta _k} converges. In this case λ (Πδ,1−δ) > 0. In the other case λ (Πδ,1−δ) = 0.
Lemma 20
The condition\sum\limits_{k = 1}^\infty {d_k}{\delta _k} < + \infty is necessary and sufficient to the inclusion{\Psi _{\{ \delta \} }} \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).
Proof
We should proof that ∂jΨ{δ} ∈ ℋℰ for any j ∈ ℕ; and the series
\sum\limits_{j = 1}^\infty {d_j}\parallel {\partial _j}{\Psi _{\{ \delta \} }}\parallel _{{\mathcal{H}_\mathcal{E}}}^2
converges. Note that
{\Psi _{\{ \delta \} }}(x) = {\psi _{{\delta _j}}}({x_j}){\Psi _{\{ \hat \delta \} }}(\hat x),{\kern 1pt} x \in E, x ∈ E, for any j ∈ ℕ; where
\hat x = \{ {x_1},...,{x_{j - 1}},{x_{j + 1}},...\} and
\hat \delta = \{ {\delta _1},...,{\delta _{j - 1}},{\delta _{j + 1}},...\} . Therefore for any j ∈ ℕ; the following equality holds
{\partial _j}{\Psi _{\{ \delta \} }}(x) = {\partial _j}{\psi _{{\delta _j}}}({x_j}){\Psi _{\{ \hat \delta \} }}(\hat x),{\kern 1pt} x \in E.
Therefore the series
\sum\limits_{j = 1}^\infty {d_j}\parallel {\partial _j}{\Psi _{\{ \delta \} }}\parallel _\mathcal{H}^2 converges if and only if the series
\sum\limits_{j = 1}^\infty \frac{{{d_j}}}{{{\delta _j}}} converges.
Note 21.LetDbe a positive operator in the space E such that\sqrt {\mathbf{D}} is trace class operator. Then there is the sequence\{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2})such that\sum\limits_{j = 1}^\infty {\delta _j} < + \infty and the condition\sum\limits_{k = 1}^\infty \frac{{{d_k}}}{{{\delta _k}}} < + \infty satisfies. For example,{\delta _k} = \sqrt {{d_k}} , k∈ ℕ.
Lemma 22
Letf \in W_{2,{\mathbf{D}}}^1(E). Let the sequence\{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2})satisfies the condition\sum\limits_{k = 1}^\infty \frac{{{d_k}}}{{{\delta _k}}} < + \infty . Then{\Psi _{\{ \delta \} }}f \in \dot W_2^1({\Pi _{0,1}}).
Proof
In fact, ‖Ψ{δ}f‖ℋℰ ≤ ‖ f ‖ℋℰ according to Cauchy inequality and lemma 1. Since ∂j(Ψ{δ}f) = ∂jΨ{δ}) f + Ψ{δ}∂jf then
\begin{array}{*{20}{c}} {\parallel {\partial _j}({\Psi _{\{ \delta \} }}f)\parallel _{{\mathcal{H}_\mathcal{E}}}^2 \leqslant 2\mathop {\sup }\limits_{x \in E} |{\Psi _{\{ \delta \} }}(x)|\parallel f\parallel _{W_{2,{\mathbf{D}}}^1}^2 + \frac{1}{{{\delta _j}}}(\int\limits_0^{{\delta _j}} + 2\int\limits_{1 - {\delta _j}}^1 )\int\limits_{{{\hat E}_j}} |f(x){|^2}d{\lambda _{{\mathcal{E}_j}}}({{\hat x}_j})d{x_j} \leqslant } \\ { \leqslant 2\parallel f\parallel _{W_{2,{\mathbf{D}}}^1}^2 + \frac{2}{{{\delta _j}}}\parallel f\parallel _{{\mathcal{H}_\mathcal{E}}}^2.} \end{array}\
Since {δ} ∈ l1 then
\mathop {\lim }\limits_{\varepsilon \to 0} {\lambda _\mathcal{E}}({\Pi _{0,1}}\backslash {\Pi _{\varepsilon \{ \delta \} ,1 - \varepsilon \{ \delta \} }}) = \mathop {{\text{lim}}}\limits_{\varepsilon \to 0} {\lambda _\mathcal{E}}({\Pi _{0,1}}\backslash {\Pi _{0,1 - 2\varepsilon \{ \delta \} }}) = 0 according to the theorem ? in [10]. Since f ∈ ℋ then for any σ > 0 there is the simple function g ∈ S2(E, ℛℰ, λℰ, C) such that ‖ f − g‖ℋ < σ.
Since the function g has the finite number of values then
M = \mathop {{\text{sup}}}\limits_{x \in E} |g(x)| \in [0, + \infty ). Therefore
\begin{array}{*{20}{c}} {\int\limits_{{\Pi _{0,1}}\backslash {\Pi _{\varepsilon \{ \delta \} ,1 - \varepsilon \{ \delta \} }}} |f(x){|^2}d{\lambda _\mathcal{E}} \leqslant } \\ {\parallel f - g\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \int\limits_{{\Pi _{0,1}}\backslash {\Pi _{\varepsilon \{ \delta \} ,1 - \varepsilon \{ \delta \} }}} |g(x){|^2}d{\lambda _\mathcal{E}} \leqslant \parallel f - g\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + {M^2}{\lambda _\mathcal{E}}({\Pi _{0,1}}\backslash {\Pi _{\varepsilon \{ \delta \} ,1 - \varepsilon \{ \delta \} }}).} \end{array}
Hence for any σ > 0 there is a number ε0 > 0 such that
\int\limits_{{\Pi _{0,1}}\backslash {\Pi _{\varepsilon \{ \delta \} ,1 - \varepsilon \{ \delta \} }}} |f(x{)|^2}d{\lambda _\mathcal{E}} \leqslant 2\sigma for any ε ∈ (0, ε0). Therefore
\mathop {{\text{lim}}}\limits_{\varepsilon \to + 0} \parallel f - {\Psi _{\varepsilon \{ \delta \} }}f{\parallel _{{L_2}({\Pi _{0,1}})}} = 0.
The consequence of the lemma 4 is the following statement.
Theorem 24
LetDbe a nonnegative operator in the space E such that\sqrt {\mathbf{D}} is trace class operator. Then the set of functions{S_1} = \{ {\Psi _{\varepsilon \{ \delta \} }}f,\;f \in f \in W_{2,{\mathbf{D}}}^1(E),{\kern 1pt} \delta \in (0,\frac{1}{2}),\{ \delta \} :\;\sum\limits_{k = 1}^\infty \frac{{{d_k}}}{{{\delta _k}}} < + \infty \} is dense in the space L2(Π0,1).
Let a ≥ 0 and f ∈ L2(Π0,1). Let the functional
{J_f}:\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) \to \mathbb{R} be defined by the equality
{J_{a,f}}(u) = \frac{1}{2}\int\limits_{{\Pi _{0,1}}} [{(\nabla \bar u,{\mathbf{D}}\nabla u)_E} + a|u{|^2} + \bar uf + u\bar f]d\lambda ,\;u \in W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).
Theorem 25
Let a ≥ 0 and f ∈ L2(Π0,1). Letu \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}})be a stationary point of the functional Ja,f. Then u is the solution of Dirichlet problem (11), (12).
Proof
Let
u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) be a stationary point of the functional (13). Then the function Ja,f (u + tϕ), t ∈ ℝ satisfies the equality
\frac{d}{{dt}}{J_{a,f}}(u + t\phi ) = 0 for any ϕ ∈ S1. Therefore
\int\limits_{{\Pi _{0,1}}} [{(\nabla \bar \phi ,{\mathbf{D}}\nabla u)_E} + a\bar \phi u + \bar \phi f]d\lambda + \int\limits_{{\Pi _{0,1}}} [{(\nabla \bar u,{\mathbf{D}}\nabla \phi )_E} + a\bar u\phi + \bar f\phi ]d\lambda = 0
for any ϕ ∈ S1. Hence
\int\limits_{{\Pi _{0,1}}} [\bar \phi ({\Delta _{\mathbf{D}}}u - f - au)]d{\lambda _\mathcal{E}} = 0 for any ϕ ∈ S1 according to the theorem 24. Since the set S1 is dense in the space ℋ then the function u satisfies Poisson equation (11). Since
u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) then the equality (12) is satisfied.
Theorem 26
Letu \in \dot W_2^2({\Pi _{0,1}})be the solution of Dirichlet problem (11), (12). Then it is the critical point of the functional (13).
Proof
Let
\phi \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}). Then the function Ja,f (u + tϕ), t ∈ ℝ, has the derivative
{\left. {\frac{{d{J_{a,f}}(u + t\phi )}}{{dt}}} \right|_{t = 0}} = \int\limits_{{\Pi _{0,1}}} [{(\nabla \bar \phi ,{\mathbf{D}}\nabla u)_E} + a\bar \phi u + \bar \phi f]d\lambda + \int\limits_{{\Pi _{0,1}}} [{(\nabla \bar u,{\mathbf{D}}\nabla \phi )_E} + a\bar u\phi + \bar f\phi ]d\lambda.
Then according to the theorem 18
\frac{d}{{dt}}{J_{a,f}}(u + t\phi ){|_{t = 0}} = - \int\limits_{{\Pi _{0,1}}} [\bar \phi ({\Delta _{\mathbf{D}}}u - au - f)]d\lambda - \int\limits_{{\Pi _{0,1}}} [\phi ({\Delta _{\mathbf{D}}}\bar u - a\bar u - \bar f)]d\lambda.
Since
u \in \dot W_2^2({\Pi _{0,1}}) be the soluion of Dirichlet problem (11), (12) then the equality
\frac{d}{{dt}}{J_{a,f}}(u + t\phi ){|_{t = 0}} = 0 holds for any
\phi \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) and the function u is the stationary point of the functional Ja,f.
The inequality
\parallel u{\parallel _\mathcal{H}} \leqslant \parallel u{\parallel _{\dot W_2^1}} holds according to the definition of the space
\dot W_2^1. Let f ∈ H. Then for any
u \in \dot W_2^1 the inequality
|(f,u)| \leqslant c\parallel u{\parallel _{\dot W_2^1}} take place where c ≤ ‖ f ‖ℋ. Then according to R theorem there is the element
v \in \dot W_2^1 such that
{(f,u)_\mathcal{H}} = {(v,u)_{\dot W_2^1}}\;\forall \;u \in \dot W_2^1.
Let us endow the space
\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) with the equivalent norm
\parallel u{\parallel _{\dot W_{2,{\mathbf{D}},a}^1}} = {(a\parallel u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \sum\limits_{k = 1}^\infty {d_k}\parallel {\partial _k}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2)^{\frac{1}{2}}}
for arbitrary a > 0. The space
\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) endowed with the equivalent norm (15) is noted by
\dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}). The inequalities
\frac{1}{{1 + a}}\parallel u\parallel _{\dot W_{2,{\mathbf{D}},a}^1}^2 \leqslant \parallel u\parallel _{\dot W_{2,{\mathbf{D}}}^1}^2 \leqslant (1 + \frac{1}{a})\parallel u\parallel _{\dot W_{2,{\mathbf{D}},a}^1}^2
holds according to the definition of the space
\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}). Therefore the space
\dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) is the Hilbert space.
Theorem 27
Let a > 0 and f ∈ L2(Π0,1). Then the functional (13) has the unique point of the minimum in the space\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).
Proof
Let f ∈ ℋℰ. Then for any
u \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) the inequality
|(f,u)| \leqslant c\parallel u{\parallel _{\dot W_{2,{\mathbf{D}},a}^1}} take place where c ≤ ‖f‖ℋℰ. Then according to Riesz theorem for any a > 0 there is the element
{v_a} \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) such that
{(f,u)_\mathcal{H}} = {(v,u)_{\dot W_{2,{\mathbf{D}},a}^1}}\;\forall \;u \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}).
Therefore for any
u \in \dot W_{2,{\mathbf{D}}}^1(E) the following equality holds
{J_{a,f}}(u) = \frac{1}{2}{(u,u)_{\dot W_{2,{\mathbf{D}},a}^1}} - \frac{1}{2}{(v,u)_{\dot W_{2,{\mathbf{D}},a}^1}} - \frac{1}{2}{(u,v)_{\dot W_{2,{\mathbf{D}},a}^1}} = \frac{1}{2}{(u - v,u - v)_{\dot W_{2,{\mathbf{D}},a}^1}} - \frac{1}{2}{(v,v)_{\dot W_{2,{\mathbf{D}},a}^1}}.
Therefore the functional Ja,f has the unique point of the minimum in the space
\dot W_{2,{\mathbf{D}}}^1 which coincides with the element
v \in \dot W_{2,{\mathbf{D}},a}^1.
Definition 4
The function
v \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) is called the generalized solution of the equation (11) with the Dirichlet condition (12) if the equality
{(v,\phi )_{\dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}})}} + {(a + f,\phi )_{{L_2}({\Pi _{0,1}})}} = 0
satisfies for any
\phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}).
Theorem 28
Let a > 0 and f ∈ L2(Π0,1). Then the functionu \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}})is point of minimum of the functional (13) if and only if it is the generalized solution of Dirichlet problem (11), (12).
Proof
If the function
u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) is point of minimum of the functional (13) then
\frac{d}{{dt}}{J_{a,f}}(u + t\phi ){|_{t = 0}} = 0 for any
\phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}). Hence the equality (16) satisfies for any
\phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) according to the expression (14). Therefore u is is the generalized solution of Dirichlet problem (11), (12).
Let u is is the generalized solution of Dirichlet problem (11), (12). Then the right hand side of the expression (14) is equal to zero. Therefore for any
\phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) the folowing equality holds
{J_{a,f}}(u + \phi ) - J(u) = \frac{1}{2}\parallel \phi \parallel _{\dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}})}^2.
Hence the function
u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) is point of strong minimum of the functional (13).
Conclusions
In this paper we show that the theory of Sobolev spaces and its application to partial differential equation can be constructed for the function on domains in infinite dimension Hilbert space endowing with finite additive shift invariant measures. We study the class of finite additive shift invariant measures on the real separable Hilbert space E. For any choice of such a measure we consider the Hilbert space ℋ of complex-valued functions which are square-integrable with respect to this measure. Some analogs of Sobolev spaces of functions on the space E are introduced. The analogue of Gauss theorem is obtained for the simplest domains such as the rectangle in the space E. The correctness of the problem for Poisson equation in the rectangle with homogeneous Dirichlet condition is obtained and the variational approach of the solving of this problem is constructed.