Dirichlet Problem for Poisson Equation on the Rectangle in Infinite Dimensional Hilbert Space

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 γ_t, 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 $W_{2, D}^{m} (E)$ 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 $\sum_{k = 1}^{\infty} d_{m} ‖ {(\partial_{k})}^{m} u ‖_{ℋ}^{2} < + \infty .$ \sum\limits_{k = 1}^\infty {d_m}\parallel {({\partial _k})^m}u\parallel _\mathcal{H}^2 < + \infty.

Here {d_k} is the sequence of eigenvalues of the operator D and {e_k} 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 $lim_{t \to 0} ‖ \frac{1}{t} (S_{th} - I) u - \partial_{h} u ‖_{ℋ} = 0$ \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 ℛ_ℰ depending on the chois of an orthonormal basis ℰ = {e_j} 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 {e_j} ≡ ℰ in E and there is an elements a,b ∈ l_∞ such that (1) $Π = {x \in E : (x, e_{j}) \in [a_{j}, b_{j}) \forall j \in N} .$ \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 Π_ℰ,a,b.

The rectangle (1) is called measurable if it either empty set, or the following condition holds (2) $\sum_{j = 1}^{\infty} max {0, ln (b_{j} - a_{j})} < + \infty .$ \sum\limits_{j = 1}^\infty {\text{max}}\{ 0,{\text{ln}}({b_j} - {a_j})\} < + \infty.

Let λ(Π) = 0 if Π = ∅, and let (3) $λ (Π_{ℰ, a, b}) = exp (\sum_{j = 1}^{\infty} ln (b_{j} - a_{j}))$ \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 Π_ℰ,a,b.

For any orthonormal basis ℱ = {f_k} 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 r_ℱ notes the minimal ring of subsets containing the set of measurable rectangles K(ℱ).

Theorem 1

[11] For any orthonormal basis ℱ = {f_k} of the space E there exists the unique measure λ_ℱ : r_ℱ → [0, +∞) such that the equality (3) holds for any rectangle Π_ℱ,a,b ∈ 𝒦_ℱ. The measure λ_ℱ has the unique completion onto the ring ℛ_ℱ which is completion of the ring r_ℱ by measure λ_ℱ.

Note 2. Here the ring ℛ_ℱ consists on the sets A ⊂ E such that $\bar{λ_{ℱ}} (A) = \underline{λ_{ℱ}} (A) \in ℝ$ \overline {{\lambda _\mathcal{F}}} (A) = \underline {{\lambda _\mathcal{F}}} (A) \in \mathbb{R} where $\bar{λ_{ℱ}} (A) = inf_{B \in r_{ℱ}, B \supset A} λ_{ℱ} (A)$ \overline {{\lambda _\mathcal{F}}} (A) = \mathop {{\text{inf}}}\limits_{B \in {r_\mathcal{F}},{\kern 1pt} B \supset A} {\lambda _\mathcal{F}}(A), $\underline{λ_{ℱ}} (A) = sup_{B \in r_{ℱ}, B \subset A} λ_{ℱ} (A)$ \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 λ_ℱ.

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]).

2.1

Quadratically integrable functions

Now we define space of quadratically integrable functions with respect to λ_ℰ. Since we will use it very often, we define it concisely ℋ_ℰ = L₂(E, ℛ_ℰ, λ_ℰ, ℂ).

Let 𝒮 (E, ℛ_ℰ, ℂ) be the linear space hull over field ℂ of indicator functions of the sets from the ring ℛ_ℰ. Let β_ℰ be the sesquilinear form on the space 𝒮 (E, ℛ_ℰ, ℂ) which is defined by the following conditions: $β_{ℰ} (χ_{A}, χ_{B}) = λ_{ℰ} (A \cap B)$ {\beta _\mathcal{E}}({\chi _A},{\chi _B}) = {\lambda _\mathcal{E}}(A\bigcap\nolimits_ B) for any sets A, B ∈ 𝒦_ℰ; for any functions u, v ∈ 𝒮 (E, ℛ_ℰ, ℂ) such that $u = \sum_{j = 1}^{n} c_{j} χ_{A_{j}}$ u = \sum\limits_{j = 1}^n {c_j}{\chi _{{A_j}}}, $v = \sum_{k = 1}^{m} b_{k} χ_{B_{k}}$ v = \sum\limits_{k = 1}^m {b_k}{\chi _{{B_k}}} the value β_ℰ (u, v) is given by the equality (1) $β_{ℰ} (u, v) \equiv {(u, v)}_{ℋ (E)} = (\sum_{j = 1}^{n} c_{j} χ_{A_{j}}, \sum_{k = 1}^{m} b_{k} χ_{B_{k}}) = \sum_{k = 1}^{m} \sum_{j = 1}^{n} {\bar{b}}_{k} c_{j} (χ_{A_{j}}, χ_{B_{k}}) .$ {\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, ℛ_ℰ, ℂ) is Hermitian and nonnegative.

The function u ∈ 𝒮 (E, ℛ_ℰ, ℂ) is called equivalent to the function u ∈ 𝒮 (E, ℛ_ℰ, ℂ) iff β_ℰ (u − v, u − v) = 0. The linear space 𝒮₂(E, ℛ_ℰ, λ_ℰ, ℂ) of the equivalence classes of functions of the space 𝒮 (E, ℛ_ℰ, ℂ) endowing with the sesquilnear form β_ℰ is the pre-Hilbert space. The Hilbert space L₂(E, ℛ_ℰ, λ_ℰ, ℂ) ≡ ℋ_ℰ is defined as the completion of the space 𝒮₂(E, ℛ_ℰ, λ_ℰ, ℂ).

Thus for any ONB ℰ in the space E there are the ring of subsets ℛ_ℰ of λ_ℰ -measurable sets, the measure λ_ℰ : ℛ_ℰ → [0, +∞) and the Hilbert space ℋ_ℰ of complex valued λ_ℰ -measurable square integrable functions on the space E. Since the pre-Hilbert space 𝒮₂(E, ℛ_ℰ, λ_ℰ, ℂ) of a simple functions is dense linear manifold in the space ℋ_ℰ then the Hibert space ℋ_ℰ is the space of continuous linear functionals on the pre-Hilbert space 𝒮₂(E, ℛ_ℰ, λ_ℰ, ℂ).

Lemma 4

[10] The space ℋ_ℰ is not separable.

2.2

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 λ_ℰ be a complete translation invariant measure on the space E such that the measure λ_ℰ is defined on the ring ℛ_ℰ by the theorem 1. Let ℛ_ℰ₁ = 𝒥 (ℛ_ℰ) and λ_ℰ₁ is the measure on the space (E₁, ℛ_ℰ₁) such that λ_ℰ₁(A) = λ_ℰ (𝒥⁻¹(A)) ∀ A ∈ ℛ_ℰ₁.

Let l_ℝ 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 K_ℰ and K_ℰ₁ 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; r_ℰ and r_ℰ₁ are the minimal rings containing the collections of sets K_ℰ and K_ℰ₁ 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 ${\underline{λ}}_{ℰ} (X) = sup_{\cup_{k = 1}^{n} Q_{k} \subseteq X, Q_{k} \in K_{ℰ}} λ_{ℰ} (\cup_{k = 1}^{n} Q_{k})$ {\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 r_ℰ.

Lemma 6

([2], lemma 3.4). Let A = g × Π, where Π ∈ 𝒦_ℰ₁and λ_ℰ₁(Π) ≠ 0. Then A ∈ ℛ_ℰ iff g ∈ r(ℝ). In this case the following equality λ_ℰ (A) = l_ℝ(g)λ_ℰ₁(Π) holds.

The collection 𝒦_ℰ of measurable rectangles is the part of the following collection of sets {A₀ × A₁, A₀ ∈ r( ℝ), A₁ ∈ ℛ_ℰ₁}; the last collection of sets is the part of the ring ℛ_ℰ. Since the ring r_ℰ is the minimal ring containing the collection of sets 𝒦_ℰ and the ring ℛ_ℰ is the completion of the ring r_ℰ by the measure λ_ℰ, then the ring ℛ_ℰ is the comletion by measure λ_ℰ of the minimal ring, containing the collection of the sets {A₀ × A₁, A₀ ∈ r( ℝ), A₁ ∈ ℛ_ℰ₁}. Hence the following statement holds.

Proof

The space with finite additive measure (E, r_ℰ, λ_ℰ) is the prodict of the spaces with finite additive measures ( ℝ, r( ℝ), l_ℝ) and (E₁, r_ℰ₁, λ_ℰ₁)

Proof

According to the lemma 1 [5] (page 222) the space with finite additive measure (E, r_ℰ, λ_ℰ) is the product of the spaces with the finite additive measures ( ℝ, r( ℝ), l_ℝ) and (E₁, r_ℰ₁, λ_ℰ₁). In fact, since the rings ℛ_ℰ and ℛ_ℰ₁ is obtained by using of completions by measures λ_ℰ and λ_ℰ₁ procedure from the rings r_ℰ and r_ℰ₁ respectively then the measure λ_ℰ be a unique finite additive measure which is defined on the ring ℛ_ℰ and satisfies the conditions λ_ℰ (A₀ × A₁) = l_ℝ(A₀)λ_ℰ₁(A₁) ∀ A₀ ∈ r( ℝ), A₁ ∈ ℛ_ℰ₁.

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: 1)

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 λ_ℰ = l_ℝ ⊗ λ_ℰ₁holds in the sense of definition 1.

Proof

In fact, the procedures of definition of the measure λ_ℰ and the measure l_ℝ ⊗ λ_ℰ₁ have the following common constructions.

The definition the measure λ_ℰ consists of three parts:

The function of a set is defined firstly on the collection of measurable rectangles 𝒦_ℰ;

the function 𝒦_ℰ → ℝ is extended onto the measure λ on the minimal ring r_ℰ containing the collection of sets 𝒦_ℰ;

the measure λ : r_ℰ → ℝ is extended onto the ring ℛ_ℰ which is completion of the ring r_ℰ by measure λ.

In the case of the measure l_ℝ ⊗ λ_ℰ₁ the collection of a sets {A₁ × A₂ | A₁ ∈ r( ℝ), A₂ ∈ 𝒦_ℰ₁} is used instead of the collection 𝒦_ℰ on the first step. According to definition the equality l_ℝ ⊗ λ_ℰ₁(I₁ × Π₁) = λ_ℰ (I₁ × Π₁) holds for any measurable rectangle I₁ × Π₁ where I₁ ∈ r( ℝ), Π₁ ∈ 𝒦_ℰ₁. Therefore the inequalities ${\underline{λ}}_{ℰ} (A) \leq \bar{l_{ℝ} \otimes λ_{ℰ_{1}}} (A) \leq \underline{l_{ℝ} \otimes λ_{ℰ_{1}}} (A) \leq {\bar{λ}}_{ℰ} (A)$ {\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 λ_ℰ then it is measurable with respect measure l_ℝ ⊗ λ_ℰ₁ and the the extensions of the measures λ_ℰ and l_ℝ ⊗ λ_ℰ₁ coincides on the ring ℛ_ℰ.

On the other hand any set of type {A₁ × A₂ | A₁ ∈ r( ℝ), A₂ ∈ 𝒦_ℰ₁} belongs to the ring ℛ_ℰ and in this case the equality λ_ℰ (A₁ × A₂) = (l_ℝ ⊗ λ_ℰ₁)(A₁ × A₂) holds. Since the measure λ_ℰ : ℛ_ℰ → ℝ is complete then the continuation of the function of a set (l_ℝ ⊗ λ_ℰ₁) : {A₁ × A₂ | A₁ ∈ r( ℝ), A₂ ∈ 𝒦_ℰ₁} → ℝ by means of continuation on the minimal ring (steps 2)) and completion (step 3)) coincides with the measure λ_ℰ.

Lemma 9

The linear space span({χ_Π, Π ∈ 𝒦_ℰ }) is dense in the Hilbert space L₂(E, ℛ_ℰ, λ_ℰ, ℂ).

Proof

In fact, according to definition the Hilbert space L₂(E, ℛ_ℰ, λ_ℰ, ℂ) is the closure of the linear space $\bar{span ({χ_{A}, A \in ℛ_{ℰ}})}$ \overline {{\text{span}}(\{ {\chi _A},A \in {\mathcal{R}_\mathcal{E}}\} )} endowed with the norm of the space L₂(E, ℛ_ℰ, λ_ℰ, ℂ). Since the ring ℛ_ℰ is the completion of the ring r_ℰ by the measure λ_ℰ then the following equality $L_{2} (E, ℛ_{ℰ}, λ_{ℰ}, ℂ) = \bar{span ({χ_{A}, A \in r_{ℰ}})}$ {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 ∈ r_ℰ is the finite union of disjoint sets $A = \cup_{k = 1}^{N} B_{k}$ A = \bigcup\limits_{k = 1}^N {B_k} where B_k is the complement of a measurable rectangle to finite union of measurable rectangles: $\forall k \in \bar{1, N} B_{k} = Π_{k, 0} \ \cup_{j = 1}^{m_{k}} Π_{k, j}$ \forall \,k \in \overline {1,N} \;{B_k} = {\Pi _{k,0}}\backslash \bigcup\limits_{j = 1}^{{m_k}} {\Pi _{k,j}}, $Π k, i \in 𝒦_{ℰ} \forall i \in \bar{0, m_{k}}$ \Pi k,i \in {\mathcal{K}_\mathcal{E}}\;\forall \;i \in \overline {0,{m_k}} . So, χ_A ∈ span({χ_Π, Π ∈ 𝒦_ℰ }) for any A ∈ r_ℰ. Consequently, $L_{2} (E, ℛ_{ℰ}, λ_{ℰ}, ℂ) = \bar{span ({χ_{Π}, Π \in 𝒦_{ℰ}})}$ {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, ℛ_ℰ₁, λ_ℰ₁,ℂ), which is limit of fundamental sequence f_k ⊗ v_k into limit of sequence f_k ⊗ v_k in space L₂(E, ℛ_ℰ, λ_ℰ, ℂ) provides us with canonical isomorphism between these two space.

Proof

Space L₂(E, ℛ_ℰ, λ_ℰ, ℂ) according to lemma 9 is a completion of span({χ_A, A ∈ 𝒦_ℰ }) with respect to norm ‖ · ‖_{L₂(E,ℛ_ℰ, λ_ℰ}, _ℂ), defined by sesquilinear form β_ℰ, see (1). So, L₂(E, ℛ_ℰ, λ_ℰ, ℂ) is completion of the space span({χ_A₀×A₁, A₀ ∈ 𝒦_ℝ, A₁ ∈ 𝒦_ℰ₁}) with respect to norm ‖ · ‖_{L₂ (E,ℛ_ℰ, λ_ℰ}, _ℂ), defined by sesquilinear form β_ℰ. (Here we define by 𝒦_ℝ a set of all bounded intervals of ℝ).

Space L₂( ℝ) ⊗ L₂(E₁, ℛ_ℰ₁, λ_ℰ₁, ℂ) is completion of linear span ℒ of elements f ⊗ v, where f ∈ L₂( ℝ), v ∈ L₂(E₁, ℛ_ℰ₁, λ_ℰ₁, 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 𝒦_ℝ 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 ∈ 𝒦_ℝ, A₁ ∈ 𝒦ℰ₁} with respect to norm ‖ · ‖_⊗.

Since for any interval Δ ∈ 𝒦_ℝ and any measurable rectangle Π′∈ 𝒦_ℰ₁ holds ‖χ_Δ ⊗ χ_Π′ ‖_⊗ = ‖χ_Π‖_L₂₍_E,ℛℰ, _λℰ,_ℂ), where Π = Δ × Π′, then for any sets A₀ ∈ 𝒦_ℝ, A₁ ∈ 𝒦_ℰ₁ holds (2) $‖ χ_{A_{0}} \otimes χ_{A_{1}} ‖_{\otimes} = ‖ χ_{A_{0} \times A_{1}} ‖_{L_{2} (E, ℛ_{ℰ}, λ_{ℰ}, ℂ)} .$ \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({χ_A₀×A₁, A₀ ∈ 𝒦_ℝ, A₁ ∈ 𝒦_ℰ₁}) is dense in space L₂(E, ℛ_ℰ, λ_ℰ, ℂ), and linear span span({χ_A₀ ⊗ χ_A₁, A₀ ∈ 𝒦_ℝ, A₁ ∈ 𝒦_ℰ₁}) is dense in space L₂( ℝ) ⊗ L₂(E₁, ℛ_ℰ₁, λ_ℰ₁, ℂ), then (dee equation 2) spaces L₂(E, ℛ_ℰ, λ_ℰ, ℂ) and L₂( ℝ) ⊗ L₂(E₁, ℛ_ℰ₁λ_ℰ₁, ℂ) are isometrically isomorphic and L₂(E, ℛ_ℰ, λ_ℰ, ℂ) = ℐ (L₂( ℝ) ⊗ L₂(E₁, ℛ_ℰ₁λ_ℰ₁, ℂ)).

2.3

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, ℛ_ℰ, λ_ℰ, ℂ) = L₂( ℝ) ⊗ L₂(E₁, ℛ_ℰ₁, λ_ℰ₁,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, ℛ_ℰ₁, λ_ℰ₁, ℂ) by the equality $ℱ_{1} (\sum_{k = 1}^{n} u_{1, k} u_{2, k}) = \sum_{k = 1}^{n} {\hat{u}}_{1, k} u_{2},$ {\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, ℛ_ℰ₁, λ_ℰ₁, ℂ) is dense in the space L₂(E, ℛ_ℰ, λ_ℰ, ℂ) then the partial Fourier transform ℱ₁ has the unique continuation up to the unitary transform of the space L₂(E, ℛ_ℰ, λ_ℰ, ℂ) into itself.

Analogously, partial Fourier transform ℱ_n with respect to first n coordinates is defined on the space L₂(E, ℛ_ℰ, λ_ℰ, 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, ℛ_ℰ, λ_ℰ, ℂ) with respect to first n coordinates is unitary mapping of the space L₂(E, ℛ_ℰ, λ_ℰ, 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 e_j 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

3.1

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 ν_D on the space E such that the measure ν_D has the covariance operator D and zero mean value.

Shift operator on the vector h ∈ E is defined on the space ℋ_ℰ = L₂(E, ℛ_ℰ.λ_ℰ, ℂ) by the equality $S_{h} u (x) = u (x - h) .$ {{\mathbf{S}}_h}u(x) = u(x - h).

It is obvious that for any h ∈ H operator S_h belongs to the Banach space B(ℋ_ℰ) of bounded linear operators in the space ℋ_ℰ endowed with the operator norm; moreover S_h is the unitary operator in the space ℋ_ℰ. Let h be a random vector of the space E whose distribution is given by the measure ν. Then the mean value U ∈ B(ℋ_ℰ) of random shift operator S_h is given by the Pettis integral $\int_{E} S_{h} d ν (h) = U \Leftrightarrow (U f, g) = \int_{E} (S_{h} f, g) d ν (h) \forall f, g \in ℋ_{ℰ} .$ \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 $U_{t} = \int_{E} S_{h} d ν_{t D} (h), t \geq 0,$ {{\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 ℋ_ℰ. The semigroupU_t, t ≥ 0, is strong continuous in the space ℋ_ℰ if and only if $D^{\frac{1}{2}}$ {{\mathbf{D}}^{\frac{1}{2}}}is trace class operator.

Definition 2

A function u′_j ∈ ℋ_ℰ is called the derivative of the function u ∈ ℋ_ℰ in the direction of a unite vector e_j if the following equality holds $lim_{s \to 0} ‖ \frac{1}{s} (S_{s e_{j}} u - u) - {u^{'}}_{j} ‖_{ℋ_{ℰ}} = 0$ \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 {e_j} ≡ ℰ be the orthonormal basis of eigenvectors of positive trace class operatorD. If ν_Dbe a probability Gaussian measure on the space E with covariance operatorDand $𝒰_{D} (t) = \int_{E} S_{\sqrt{t} h} d ν_{D} (h)$ {\mathcal{U}_{\mathbf{D}}}(t) = \int\limits_E {{\mathbf{S}}_{\sqrt t h}}d{\nu _{\mathbf{D}}}(h), t ≥ 0, u ∈ ℋ_ℰ then for any l ∈ ℕ; there is the number c_l > 0 such that for any u ∈ ℋ_ℰ, j ∈ ℕ;, t > 0 there is the derivative of the power l $\partial_{j}^{l} 𝒰_{D} (t) u \in ℋ_{ℰ}$ \partial _j^l{\mathcal{U}_{\mathbf{D}}}(t)u \in {\mathcal{H}_\mathcal{E}}and the following estimates take place(3) $‖ \partial_{j}^{l} 𝒰_{D} (t) u ‖_{ℋ_{ℰ}} \leq \frac{c_{l}}{{(\sqrt{t d_{j}})}^{l}} ‖ u ‖_{ℋ_{ℰ}} .$ \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 $C_{D}^{\infty} (E)$ C_{\mathbf{D}}^\infty (E) be a linear hull of the following system of elements {𝒰_D(t)u, t > 0, u ∈ ℋ_ℰ}. The linear manifold $C_{D}^{\infty} (E)$ C_{\mathbf{D}}^\infty (E) is called the space of smooth functions according to lemma 12.

3.2

Sobolev spaces and embedding theorem

For any a > 0 the symbol $W_{2, D^{a}}^{1} (E)$ W_{2,{{\mathbf{D}}^a}}^1(E) notes the linear space of elements u of the space ℋ_ℰ such that the following two condition hold

1) for any j ∈ ℕ; there is the derivative $u_{j} \equiv \frac{\partial}{\partial x_{j}} u \in ℋ_{ℰ}$ {u_j} \equiv \frac{\partial }{{\partial {x_j}}}u \in {\mathcal{H}_\mathcal{E}} with respect to the direction of eigenvector e_j of operator D; (4) $2) \sum_{j = 1}^{n} d_{j}^{a} ‖ u_{j} ‖_{ℋ_{ℰ}}^{2} < + \infty .$ 2)\,\,\,\,\,\,\,\,\,\,\,\,\,\sum\limits_{j = 1}^n d_j^a\parallel {u_j}\parallel _{{\mathcal{H}_\mathcal{E}}}^2 < + \infty.

The space $W_{2, D^{a}}^{1} (E)$ W_{2,{{\mathbf{D}}^a}}^1(E) endowed with the norm $‖ u ‖_{W_{2, D^{a}}^{1} (E)} = {(‖ u ‖_{ℋ_{ℰ}}^{2} + \sum_{j = 1}^{n} d_{j}^{a} ‖ u_{j} ‖_{ℋ_{ℰ}}^{2})}^{\frac{1}{2}}$ \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 ℋ_ℰ (see [3, 11]).

For any numbers a > 0 and any l ∈ ℕ; the symbol $W_{2, D^{a}}^{1} (E)$ W_{2,{{\mathbf{D}}^a}}^1(E) notes the linear space of elements u of the space ℋ_ℰ such that the following two conditions hold:

1) for any j ∈ ℕ; there is the l-order derivative $\frac{\partial^{2}}{\partial e_{j}^{l}} u \in ℋ_{ℰ}$ \frac{{{\partial ^2}}}{{\partial e_j^l}}u \in {\mathcal{H}_\mathcal{E}}; (5) $2) \sum_{j = 1}^{n} d_{j}^{a} {‖ \frac{\partial^{l}}{\partial e_{j}^{l}} u ‖}_{ℋ_{ℰ}}^{2} < + \infty .$ 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 $W_{2, D^{a}}^{1} (E)$ W_{2,{{\mathbf{D}}^a}}^1(E) endowed with the norm $‖ u ‖_{W_{2, D^{a}}^{l} (E)} = {(‖ u ‖_{ℋ_{ℰ}}^{2} + \sum_{j = 1}^{n} d_{j}^{a} ‖ \frac{\partial^{l}}{\partial e_{j}^{l}} u ‖_{ℋ_{ℰ}}^{2})}^{\frac{1}{2}}$ \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 ℋ_ℰ (see [3, 11]).

Theorem 13

[see [11], lemmas 7.1, 7.2]. Let u ∈ ℋ_ℰ. LetDbe a positive trace class operator in the space E such that $D^{\frac{1}{2}}$ {{\mathbf{D}}^{\frac{1}{2}}}is the trace class operator. Then for any t > 0 the inclusion $𝒰_{D^{\frac{1}{2}}} (t) u \in W_{2, D}^{1} (E)$ {\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 thatD^γ is trace class operator with some γ > 0. Let l ∈ ℕ;. If b ≥ lα + γ with some α ∈ [γ, +∞) then $C_{D^{α}}^{\infty} (E) \subset W_{2, D^{b}}^{l} (E)$ C_{{{\mathbf{D}}^\alpha }}^\infty (E) \subset W_{2,{{\mathbf{D}}^b}}^l(E).

If, moreover, α ≥ 2γ then the linear manifold $C_{D^{α}}^{\infty} (E)$ C_{{{\mathbf{D}}^\alpha }}^\infty (E)is dense in the space $W_{2, D^{b}}^{l} (E)$ W_{2,{{\mathbf{D}}^b}}^l(E).

The proof of this theorem is published in the work [3].

3.3

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 ℋ_ℰ₁ = L₂(E₁, ℛ₁, λ_ℰ₁, ℂ). Let ℛ( ℝ) be a ring of Lebesgue integrable subsets of the space ℝ with finite Lebesgue measure. Let λ_L be the Lebesgue measure on the space ℝ.

Then accordind to the theorem 10 ℋ_ℰ = L₂( ℝ, ℛ( ℝ), λ_L, ℋ_ℰ₁)) where L₂( ℝ, ℛ( ℝ), λ_L, ℋ_ℰ₁)) is the space of integrable in Bokhner sense with respect to Lebesgue measure λ_L mappings ℝ → ℋ_ℰ₁.

We define a linear space $W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}}) = {u \in ℋ_{ℰ} : \frac{\partial}{\partial e_{1}} u \in ℋ_{ℰ}}$ 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 (6) $| u ‖_{W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}}))} = {(‖ u ‖_{ℋ_{ℰ}}^{2} + ‖ \frac{\partial}{\partial e_{1}} u ‖_{ℋ_{ℰ}}^{2})}^{\frac{1}{2}} .$ |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 $‖ u ‖_{W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})}^{2} = \int_{ℝ} (1 + ξ^{2}) ‖ {\hat{u}}_{1} (ξ) ‖_{ℋ_{ℰ_{1}}}^{2} d ξ$ \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 $u \in W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ 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 $W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}}))$ 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

If $u \in W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ u \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, ℋ_ℰ₁). Moreover, there is the constant C > 0 such that(7) $‖ \tilde{u} ‖_{C (ℝ, ℋ_{ℰ_{1}})} \leq C ‖ u ‖_{W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})} .$ \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 ℋ_ℰ₁ the proof of this theorem is given in the monograph [6]. In the case under consideration the space ℋ_ℰ₁ is not separable. But according to the condition $u \in W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ 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 ℋ_ℰ₁ containing the values of the mapping u : ℝ → ℋ_ℰ₁. Therefore the proof of the theorem 3.1 by [6] can be applied to the obtaining of the statement of theorem 15.

If $u \in W_{2, D}^{1} (E)$ u \in W_{2,{\mathbf{D}}}^1(E) then the function u can be considered as the function of the space $W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}}). Therefore the function $u \in W_{2, D}^{1} (E)$ u \in W_{2,{\mathbf{D}}}^1(E) can be considered as the continuous maping ũ : ℝ → ℋ_ℰ₁ according to theorem 15. Hence we can use the following definition of the trace of function.

Definition 3

The trace of the function $u \in W_{2, D}^{1} (E)$ u \in W_{2,{\mathbf{D}}}^1(E)\ at the hyperplane x₁ = t₀, t₀ ∈ ℝ, is the value of function ũ ∈ C( ℝ, ℋ_ℰ₁) at the point t₀.

Corollary 16

If $u \in W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ u \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})then for any t₀ ∈ ℝ the estimate $∥ \tilde{u} (t_{0}) ∥_{ℋ_{ℰ_{1}}} \leq C ∥ u ∥_{W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})}$ \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

If $u \in W_{2}^{1} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ u \in W_2^1(\mathbb{R},\mathcal{R}(\mathbb{R}),{\lambda _L},{\mathcal{H}_{{\mathcal{E}_1}}})and $\frac{d}{d s} S_{s e_{1}} u = v \in L_{2} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{1}})$ \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 holds $u (t_{2}) - u (t_{1}) = \int_{t_{1}}^{t_{2}} v (s) ds .$ u({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 Π_a,b ∈ 𝒦 (ℰ) be a measurable rectangle. For any j ∈ ℕ; the equality E = ℝ ⊕ E_j where ℝ = span(e_j) and E_j = (span(e_j))^⊥. Let ℰ_j = {e₁,...,e_j₋₁, e_j₊₁,...}. Therefore the following equality $Π_{a, b} = [a_{j}, b_{j}) \times {\hat{Π}}_{{\hat{a}}_{j}, {\hat{b}}_{j}}$ {\Pi _{a,b}} = [{a_j},{b_j}) \times {\hat \Pi _{{{\hat a}_j},{{\hat b}_j}}} holds where ${\hat{Π}}_{{\hat{a}}_{j}, {\hat{b}}_{j}} \in 𝒦 (ℰ_{j})$ {\hat \Pi _{{{\hat a}_j},{{\hat b}_j}}} \in \mathcal{K}({E_j}) is the measurable rectangle in the space E_j.

Let $u \in W_{2, D}^{2} (E)$ u \in W_{2,{\mathbf{D}}}^2(E). Then for any j ∈ ℕ; and any a ∈ R there is the trace u|_x_{_j=}_a ∈ ℋ_ℰj of the function u according to the theorem 3.4. Since for any j ∈ ℕ; the function ∂_ju has the derivative $\partial_{j}^{2} u \in ℋ_{ℰ}$ \partial _j^2u \in {\mathcal{H}_\mathcal{E}} in the direction e_j, then according to the theorem 3.4 there is the trace (∂_ju)|_x_j₌_a ∈ ℋ_ℰ_j.

Theorem 18

LetDbe a positive trace class operator in Hilbert space E. Let $u \in W_{2, D}^{2} (E)$ u \in W_{2,{\mathbf{D}}}^2(E), let Π_a,b ∈ 𝒦 (ℰ) be a measurable rectangle. Then the equality(8) $\int_{Π_{a, b}} Δ_{D} u φ d λ_{ℰ} = - \int_{Π_{a, b}} {(\nabla u, D \nabla φ)}_{E} d λ_{ℰ} + \int_{\partial Π_{a, b}} (n, D \nabla u) φ ds,$ \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 $φ \in W_{2, D}^{1} (E)$ \phi \in W_{2,{\mathbf{D}}}^1(E). Here(9) $\int_{\partial Π_{a, b}} (n, D \nabla u) φ ds = \sum_{j = 1}^{\infty} d_{j} \int_{{\hat{Π}}_{{\hat{a}}_{j}, {\hat{b}}_{j}}} [(φ \partial_{j} u) |_{x_{j} = b_{j}} - (φ \partial_{j} u) |_{x = a_{j}}] d λ_{ℰ_{j}} .$ \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 $φ \in W_{2, D}^{1} (E)$ \phi \in W_{2,{\mathbf{D}}}^1(E) then the condition ϕ|_{x_j}=a ∈ ℋ_ℰ_{_|} holds for any j ∈ ℕ; and any a ∈ R according to the theorem 15, moreover $‖ φ_{x_{j} = a} ‖_{ℋ_{ℰ_{j}}} \leq C {(‖ φ ‖_{L_{2} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{j}})}^{2} + ‖ \partial_{j} φ ‖_{L_{2} (ℝ, ℛ (ℝ), λ_{L}, ℋ_{ℰ_{j}})}^{2})}^{\frac{1}{2}}$ \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 $\begin{matrix} | \int_{{\hat{Π}}_{{\hat{a}}_{j}, {\hat{b}}_{j}}} [(φ \partial_{j} u) |_{x_{j} = b} - (φ \partial_{j} u) |_{x_{j} = a}] d λ_{j} | = | {(φ |_{x = b}, \partial_{j} u |_{x = b})}_{ℋ_{ℰ_{j}}} - {(φ |_{x = a}, \partial_{j} u |_{x = a})}_{ℋ_{ℰ_{j}}} | \leq \\ \leq 2 ‖ \partial_{j} u ‖_{C (ℝ, ℋ_{ℰ_{j}})} ‖ φ ‖_{C (ℝ, ℋ_{ℰ_{j}})} \end{matrix}$ \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 $\begin{matrix} | \int_{{\hat{Π}}_{{\hat{a}}_{j}, {\hat{b}}_{j}}} [(φ \partial_{j} u) |_{x_{j} = b} - (φ \partial_{j} u) |_{x_{j} = a}] d λ_{j} | \leq \\ \leq 2 C^{2} {(‖ φ ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j} φ ‖_{ℋ_{ℰ}}^{2})}^{\frac{1}{2}} {(‖ \partial_{j} u ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j}^{2} u ‖_{ℋ_{ℰ}}^{2})}^{\frac{1}{2}} \leq \\ \leq C^{2} (‖ φ ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j} φ ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j} u ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j}^{2} u ‖_{ℋ_{ℰ}}^{2}) \end{matrix}$ \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 $\sum_{j = 1}^{\infty} d_{j} (‖ φ ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j} φ ‖_{ℋ_{ℰ}}^{2}) = (Tr D - 1) ‖ φ ‖_{ℋ_{ℰ}}^{2} + ‖ φ ‖_{W_{2, D}^{1} (E)}^{2} .$ \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 x_j is unitary operator in the space ℋ_ℰ and according to the inequality k² ≤ 1 + k⁴, k ∈ ℝ, we obtain $\sum_{j = 1}^{\infty} d_{j} ‖ \partial_{j} u ‖_{ℋ_{ℰ}}^{2} \leq \sum_{j = 1}^{\infty} d_{j} (‖ u ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j}^{2} u ‖_{ℋ_{ℰ}}^{2})$ \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)

Hence $\sum_{j = 1}^{\infty} d_{j} (‖ \partial_{j} u ‖_{ℋ_{ℰ}}^{2} + ‖ \partial_{j}^{2} u ‖_{ℋ_{ℰ}}^{2}) \leq 2 ‖ u ‖_{W_{2, D}^{2} (E)}^{2} + (Tr D - 2) ‖ u ‖_{ℋ_{ℰ}}^{2} .$ \sum\limits_{j = 1}^\infty {d_j}(\parallel {\partial _j}u\parallel _{{\mathcal{H}_\mathcal{E}}}^2 + \parallel \partial _j^2u\parallel _{{\mathcal{H}_\mathcal{E}}}^2) \leqslant 2\parallel u\parallel _{W_{2,{\mathbf{D}}}^2(E)}^2 + ({\text{Tr}}{\mathbf{D}} - 2)\parallel u\parallel _{{\mathcal{H}_\mathcal{E}}}^2.

Therefore the series in the right hand side (9) converges.

Since $\partial_{j} (φ d_{j} \partial_{j} u) = d_{j} \partial_{j} φ \partial_{j} u + d_{j} \partial_{j}^{2} u φ,$ {\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 (10) $\int_{\partial Π_{a, b}} (n, e_{j} d_{j} \partial_{j} u) φ ds = \int_{Π_{a, b}} d_{j} \partial_{j}^{2} u φ d λ + \int_{Π_{a, b}} d_{j} \partial_{j} u \partial_{j} φ d λ .$ \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 $W_{2, D}^{2} (E)$ 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 Π_0,1 ∈ 𝒦_ℰ. Let symbol L₂(Π_0,1) notes the subspace of the space ℋ_ℰ with the support in the set Π_0,1. Let us note that f = χ_Π0,1f for any f ∈ L₂(Π_0,1).

Let symbol ${\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) notes the space of function $W_{2, D}^{1} (E) \cap L_{2} (Π_{0, 1})$ W_{2,{\mathbf{D}}}^1(E)\bigcap\nolimits_ {L_2}({\Pi _{0,1}}) with the support in the set Π_0,1. It should be note that for any $u \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) the following statement holds: $u |_{\partial Π_{0, 1}} = 0; \partial_{x_{k}} u \in L_{2} (Π_{0, 1}) \forall k \in ℕ .$ 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 ${\dot{W}}_{2, D}^{2} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) of functions $u \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) such that $\begin{matrix} \exists g_{j k} \in L_{2} (Π_{0, 1}) : (D \partial_{k} u, \partial_{j} φ) = - (g_{j k}, φ) \forall φ \in {\dot{W}}_{2, D}^{1} (Π_{0, 1}), \forall j, k \in ℕ; \\ \sum_{j = 1}^{\infty} ‖ g_{j j} ‖_{ℋ_{ℰ}}^{2} < + \infty . \end{matrix}$ \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(Π_0,1) and a given number a ≥ 0 we should find a function $u \in {\dot{W}}_{2, D}^{2} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) such that (11) $Δ_{D} u = au + f,$ {\Delta _{\mathbf{D}}}u = au + f,(12) $u |_{\partial Π_{0, 1}} = 0 .$ u{|_{\partial {\Pi _{0,1}}}} = 0.

To investigate the above problem we apply the variation approach (see [7]). At first we study the space ${\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).

Let us introduce the trapezoid-like function ψ_δ : ℝ → ℝ which is given by the equalites $ψ_{δ} (x) = 0, x \in (- \infty, 0] \cup [1, + \infty)$ {\psi _\delta }(x) = 0,x \in ( - \infty ,0]\bigcup\nolimits [1, + \infty ); ψ_δ (x) = 1, x ∈ [δ, 1 − δ]; $ψ_{δ} (x) = \frac{1}{δ} x, x \in (0, δ)$ {\psi _\delta }(x) = \frac{1}{\delta }x,x \in \left( {0,\delta } \right); $ψ_{δ} (x) = - \frac{1}{δ} (x - 1), x \in (1 - δ, 1)$ {\psi _\delta }(x) = - \frac{1}{\delta }(x - 1),x \in (1 - \delta ,1) for any $δ \in (0, \frac{1}{2})$ \delta \in (0,\frac{1}{2}). Then $ψ_{δ} \in {\dot{W}}_{2}^{1} ([0, 1])$ {\psi _\delta } \in \dot W_2^1([0,1]), ‖ψ_δ‖_L_2([0,1]) ∈ [1 − 2δ, 1], $‖ \partial_{x} ψ ‖_{L_{2} ([0, 1])} = \sqrt{2 / δ}$ {\partial _x}\psi {\parallel _{{L_2}([0,1])}} = \sqrt {2/\delta } .

Let D a the nonnegative trace-class operator such that $D^{\frac{1}{2}}$ {{\mathbf{D}}^{\frac{1}{2}}} is trace-class operator. Let {e_j} be the ONB of eigenvectors of the operator D and {d_j} be the corresponding sequence of eigenvalues. For any sequence ${δ_{k}} : ℕ \to (0, \frac{1}{2})$ \{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2}) the function $Ψ_{{δ}} (x) = Π_{j = 1}^{\infty} ψ_{δ_{j}} (x_{j})$ {\Psi _{\{ \delta \} }}(x) = \Pi _{j = 1}^\infty {\psi _{{\delta _j}}}({x_j}) is defined.

Lemma 19

If ${δ_{k}} : ℕ \to (0, \frac{1}{2})$ \{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2})and $\sum_{j = 1}^{\infty} δ_{j} < + \infty$ \sum\limits_{j = 1}^\infty {\delta _j} < + \infty then Ψ_{_δ_} ∈ ℋ_ℰ and the estimates $Π_{k = 1}^{\infty} (1 - 2 δ_{k}) \leq ‖ Ψ_{δ} ‖_{L_{2}} \leq 1$ \Pi _{k = 1}^\infty (1 - 2{\delta _k}) \leqslant \parallel {\Psi _\delta }{\parallel _{{L_2}}} \leqslant 1hold.

Proof

The inclusion Ψ_{_δ_} ∈ ℋ_ℰ for the nonnegative function Ψ_{_δ_} is equivalent to the condition $\forall ε > 0 \exists g, G \in ℋ_{ℰ} : g \leq Ψ_{{δ}} \leq G and ‖ G - g ‖_{ℋ ℰ} < ε .$ \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 $G_{n} (x) = Π_{k = 1}^{n} ψ_{δ_{k}} (x_{k}) Π_{k = n + 1}^{\infty} χ_{[0, 1]} (x_{k})$ {G_n}(x) = \Pi _{k = 1}^n{\psi _{{\delta _k}}}({x_k})\Pi _{k = n + 1}^\infty {\chi _{[0,1]}}({x_k}) and $g_{n} (x) = Π_{k = 1}^{n} ψ_{δ_{k}} (x_{k}) Π_{k = n + 1}^{\infty} χ_{[δ_{k}, 1 - δ_{k}]} (x_{k})$ {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 g_n ≤ Ψ_{_δ_} ≤ G_n and $‖ G_{n} - g_{n} ‖_{ℋ} < 1 - exp (\sum_{k = n + 1}^{\infty} ln (1 - 2 δ_{k}))$ \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 $lim_{n \to \infty} \sum_{k = n + 1}^{\infty} δ_{k} = 0$ \mathop {{\text{lim}}}\limits_{n \to \infty } \sum\limits_{k = n + 1}^\infty {\delta _k} = 0 then $lim_{n \to \infty} ‖ G_{n} - g_{n} ‖_{ℋ} = 0$ \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 ≤ χ_Π_δ,₁₋_δ ≤ Ψδ ≤ χ_Π0,1. Hence $λ_{ℰ} (Π_{δ, 1 - δ}) \leq ‖ Ψ_{δ} ‖_{ℋ}^{2} \leq λ_{ℰ} (Π_{0, 1})$ {\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 $λ (Π_{δ, 1 - δ}) = exp (\sum_{k = 1}^{\infty} ln (1 - 2 δ_{k}))$ \lambda ({\Pi _{\delta ,1 - \delta }}) = {\text{exp}}(\sum\limits_{k = 1}^\infty {\text{ln}}(1 - 2{\delta _k})). Since $δ_{k} \in (0, \frac{1}{2})$ {\delta _k} \in (0,\frac{1}{2}) for any k ∈ ℕ; then the series $\sum_{k = 1}^{\infty} ln (1 - 2 δ_{k})$ \sum\limits_{k = 1}^\infty {\text{ln}}(1 - 2{\delta _k}) converges if and only if the series $\sum_{k = 1}^{\infty} δ_{k}$ \sum\limits_{k = 1}^\infty {\delta _k} converges. In this case λ (Π_δ,₁₋_δ) > 0. In the other case λ (Π_δ,₁₋_δ) = 0.

Lemma 20

The condition $\sum_{k = 1}^{\infty} \frac{d_{k}}{δ_{k}} < + \infty$ \sum\limits_{k = 1}^\infty {d_k}{\delta _k} < + \infty is necessary and sufficient to the inclusion $Ψ_{{δ}} \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ {\Psi _{\{ \delta \} }} \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).

Proof

We should proof that ∂_jΨ_{_δ_} ∈ ℋ_ℰ for any j ∈ ℕ; and the series $\sum_{j = 1}^{\infty} d_{j} ‖ \partial_{j} Ψ_{{δ}} ‖_{ℋ_{ℰ}}^{2}$ \sum\limits_{j = 1}^\infty {d_j}\parallel {\partial _j}{\Psi _{\{ \delta \} }}\parallel _{{\mathcal{H}_\mathcal{E}}}^2 converges. Note that $Ψ_{{δ}} (x) = ψ_{δ_{j}} (x_{j}) Ψ_{{\hat{δ}}} (\hat{x}), x \in E$ {\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 $\hat{x} = {x_{1}, ..., x_{j - 1}, x_{j + 1}, ...}$ \hat x = \{ {x_1},...,{x_{j - 1}},{x_{j + 1}},...\} and $\hat{δ} = {δ_{1}, ..., δ_{j - 1}, δ_{j + 1}, ...}$ \hat \delta = \{ {\delta _1},...,{\delta _{j - 1}},{\delta _{j + 1}},...\} . Therefore for any j ∈ ℕ; the following equality holds $\partial_{j} Ψ_{{δ}} (x) = \partial_{j} ψ_{δ_{j}} (x_{j}) Ψ_{{\hat{δ}}} (\hat{x}), x \in E .$ {\partial _j}{\Psi _{\{ \delta \} }}(x) = {\partial _j}{\psi _{{\delta _j}}}({x_j}){\Psi _{\{ \hat \delta \} }}(\hat x),{\kern 1pt} x \in E.

Hence ∂_jΨ_{_δ_} ∈ ℋ_ℰ for any j ∈ ℕ; and $‖ \partial_{j} Ψ_{{δ}} ‖_{ℋ_{ℰ}} = ‖ \partial_{j} ψ_{δ_{j}} ‖_{L_{2} (R)} ‖ Ψ_{{\hat{δ}}} ‖_{ℋ_{E_{j}}} \leq {(2 / δ_{j})}^{\frac{1}{2}} .$ \parallel {\partial _j}{\Psi _{\{ \delta \} }}{\parallel _{{\mathcal{H}_\mathcal{E}}}} = \parallel {\partial _j}{\psi _{{\delta _j}}}{\parallel _{{L_2}(R)}}\parallel {\Psi _{\{ \hat \delta \} }}{\parallel _{{\mathcal{H}_{{E_j}}}}} \leqslant {(2/{\delta _j})^{\frac{1}{2}}}.

Therefore the series $\sum_{j = 1}^{\infty} d_{j} ‖ \partial_{j} Ψ_{{δ}} ‖_{ℋ}^{2}$ \sum\limits_{j = 1}^\infty {d_j}\parallel {\partial _j}{\Psi _{\{ \delta \} }}\parallel _\mathcal{H}^2 converges if and only if the series $\sum_{j = 1}^{\infty} \frac{d_{j}}{δ_{j}}$ \sum\limits_{j = 1}^\infty \frac{{{d_j}}}{{{\delta _j}}} converges.

Note 21.LetDbe a positive operator in the space E such that $\sqrt{D}$ \sqrt {\mathbf{D}} is trace class operator. Then there is the sequence ${δ_{k}} : ℕ \to (0, \frac{1}{2})$ \{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2})such that $\sum_{j = 1}^{\infty} δ_{j} < + \infty$ \sum\limits_{j = 1}^\infty {\delta _j} < + \infty and the condition $\sum_{k = 1}^{\infty} \frac{d_{k}}{δ_{k}} < + \infty$ \sum\limits_{k = 1}^\infty \frac{{{d_k}}}{{{\delta _k}}} < + \infty satisfies. For example, $δ_{k} = \sqrt{d_{k}}$ {\delta _k} = \sqrt {{d_k}} , k∈ ℕ.

Lemma 22

Let $f \in W_{2, D}^{1} (E)$ f \in W_{2,{\mathbf{D}}}^1(E). Let the sequence ${δ_{k}} : ℕ \to (0, \frac{1}{2})$ \{ {\delta _k}\} :\mathbb{N} \to (0,\frac{1}{2})satisfies the condition $\sum_{k = 1}^{\infty} \frac{d_{k}}{δ_{k}} < + \infty$ \sum\limits_{k = 1}^\infty \frac{{{d_k}}}{{{\delta _k}}} < + \infty . Then $Ψ_{{δ}} f \in {\dot{W}}_{2}^{1} (Π_{0, 1})$ {\Psi _{\{ \delta \} }}f \in \dot W_2^1({\Pi _{0,1}}).

Proof

In fact, ‖Ψ_{_δ_}f‖_ℋ_{_ℰ} ≤ ‖ f ‖_ℋ_{_ℰ} according to Cauchy inequality and lemma 1. Since ∂_j(Ψ_{_δ_}f) = ∂_jΨ_{_δ_}) f + Ψ_{_δ_}∂_jf then $\begin{matrix} ‖ \partial_{j} (Ψ_{{δ}} f) ‖_{ℋ_{ℰ}}^{2} \leq 2 sup_{x \in E} | Ψ_{{δ}} (x) | ‖ f ‖_{W_{2, D}^{1}}^{2} + \frac{1}{δ_{j}} (\int_{0}^{δ_{j}} + 2 \int_{1 - δ_{j}}^{1}) \int_{{\hat{E}}_{j}} | f (x) |^{2} d λ_{ℰ_{j}} ({\hat{x}}_{j}) d x_{j} \leq \\ \leq 2 ‖ f ‖_{W_{2, D}^{1}}^{2} + \frac{2}{δ_{j}} ‖ f ‖_{ℋ_{ℰ}}^{2} . \end{matrix}$ \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}\

Therefore $\sum_{j = 1}^{\infty} d_{j} ‖ \partial_{j} (Ψ_{{δ}} f) ‖_{ℋ_{ℰ}}^{2} \leq 2 [Tr (D) ‖ f ‖_{W_{2, D}^{1}}^{2} + \sum_{j = 1}^{\infty} \frac{d_{j}}{δ_{j}} ‖ f ‖_{ℋ_{ℰ}}^{2}] .$ \sum\limits_{j = 1}^\infty {d_j}\parallel {\partial _j}({\Psi _{\{ \delta \} }}f)\parallel _{{\mathcal{H}_\mathcal{E}}}^2 \leqslant 2[{\text{Tr}}({\mathbf{D}})\parallel f\parallel _{W_{2,{\mathbf{D}}}^1}^2 + \sum\limits_{j = 1}^\infty \frac{{{d_j}}}{{{\delta _j}}}\parallel f\parallel _{{\mathcal{H}_\mathcal{E}}}^2].

Thus we obtain that $Ψ_{{δ}} f \in W_{2, D}^{1} (E)$ {\Psi _{\{ \delta \} }}f \in W_{2,{\mathbf{D}}}^1(E). Since (Ψ_{_δ_}f)|_∂_{Π_0,1} = Ψ_{_δ_}|_∂_Π0,1f|_∂_{Π_0,1} then (Ψ_{_δ_}f)|_∂_{Π_0,1} = 0 and $Ψ_{{δ}} f \in {\dot{W}}_{2, D}^{1} (E)$ {\Psi _{\{ \delta \} }}f \in \dot W_{2,{\mathbf{D}}}^1(E).

Lemma 23

Let $f \in W_{2, D}^{1} (E)$ f \in W_{2,{\mathbf{D}}}^1(E)and $\sum_{k = 1}^{\infty} \frac{d_{k}}{δ_{k}} < + \infty$ \sum\limits_{k = 1}^\infty \frac{{{d_k}}}{{{\delta _k}}} < + \infty . Then $lim_{ε \to + 0} ‖ f - Ψ_{ε {δ}} f ‖_{L_{2} (Π_{0, 1})} = 0$ \mathop {{\text{lim}}}\limits_{\varepsilon \to + 0} \parallel f - {\Psi _{\varepsilon \{ \delta \} }}f{\parallel _{{L_2}({\Pi _{0,1}})}} = 0.

Proof

In fact, $‖ f (1 - Ψ_{ε {δ}}) ‖_{L_{2} (Π_{0, 1})}^{2} = \int_{Π_{0, 1}} | f (x) |^{2} {(1 - Ψ_{ε {δ}})}^{2} d λ_{ℰ} \leq \int_{Π_{0, 1} \ Π_{ε {δ}, 1 - \int {δ}}} | f (x) |^{2} d λ_{ℰ} .$ \parallel f(1 - {\Psi _{\varepsilon \{ \delta \} }})\parallel _{{L_2}({\Pi _{0,1}})}^2 = \int\limits_{{\Pi _{0,1}}} |f(x){|^2}{(1 - {\Psi _{\varepsilon \{ \delta \} }})^2}d{\lambda _\mathcal{E}} \leqslant \int\limits_{{\Pi _{0,1}}\backslash {\Pi _{\varepsilon \{ \delta \} ,1 - \varepsilon \{ \delta \} }}} |f(x){|^2}d{\lambda _\mathcal{E}}.

Since {δ} ∈ l₁ then $lim_{ε \to 0} λ_{ℰ} (Π_{0, 1} \ Π_{ε {δ}, 1 - ε {δ}}) = lim_{ε \to 0} λ_{ℰ} (Π_{0, 1} \ Π_{0, 1 - 2 ε {δ}}) = 0$ \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, ℛ_ℰ, λ_ℰ, C) such that ‖ f − g‖_ℋ < σ.

Since the function g has the finite number of values then $M = sup_{x \in E} | g (x) | \in [0, + \infty)$ M = \mathop {{\text{sup}}}\limits_{x \in E} |g(x)| \in [0, + \infty ). Therefore $\begin{matrix} \int_{Π_{0, 1} \ Π_{ε {δ}, 1 - ε {δ}}} | f (x) |^{2} d λ_{ℰ} \leq \\ ‖ f - g ‖_{ℋ_{ℰ}}^{2} + \int_{Π_{0, 1} \ Π_{ε {δ}, 1 - ε {δ}}} | g (x) |^{2} d λ_{ℰ} \leq ‖ f - g ‖_{ℋ_{ℰ}}^{2} + M^{2} λ_{ℰ} (Π_{0, 1} \ Π_{ε {δ}, 1 - ε {δ}}) . \end{matrix}$ \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 $\int_{Π_{0, 1} \ Π_{ɛ {δ}, 1 - ɛ {δ}}} | f (x {) |}^{2} d λ_{ℰ} \leq 2 σ$ \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 $lim_{ε \to + 0} ‖ f - Ψ_{ε {δ}} f ‖_{L_{2} (Π_{0, 1})} = 0$ \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 $\sqrt{D}$ \sqrt {\mathbf{D}} is trace class operator. Then the set of functions $S_{1} = {Ψ_{ε {δ}} f, f \in f \in W_{2, D}^{1} (E), δ \in (0, \frac{1}{2}), {δ} : \sum_{k = 1}^{\infty} \frac{d_{k}}{δ_{k}} < + \infty}$ {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(Π_0,1).

Let a ≥ 0 and f ∈ L₂(Π_0,1). Let the functional $J_{f} : {\dot{W}}_{2, D}^{1} (Π_{0, 1}) \to ℝ$ {J_f}:\dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) \to \mathbb{R} be defined by the equality (13) $J_{a, f} (u) = \frac{1}{2} \int_{Π_{0, 1}} [{(\nabla \bar{u}, D \nabla u)}_{E} + a | u |^{2} + \bar{u} f + u \bar{f}] d λ, u \in W_{2, D}^{1} (Π_{0, 1}) .$ {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(Π_0,1). Let $u \in {\dot{W}}_{2, D}^{2} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}})be a stationary point of the functional J_a,f. Then u is the solution of Dirichlet problem (11), (12).

Proof

Let $u \in {\dot{W}}_{2, D}^{2} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) be a stationary point of the functional (13). Then the function J_a,f (u + tϕ), t ∈ ℝ satisfies the equality $\frac{d}{d t} J_{a, f} (u + t φ) = 0$ \frac{d}{{dt}}{J_{a,f}}(u + t\phi ) = 0 for any ϕ ∈ S₁. Therefore $\int_{Π_{0, 1}} [{(\nabla \bar{φ}, D \nabla u)}_{E} + a \bar{φ} u + \bar{φ} f] d λ + \int_{Π_{0, 1}} [{(\nabla \bar{u}, D \nabla φ)}_{E} + a \bar{u} φ + \bar{f} φ] d λ = 0$ \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 = 0 for any ϕ ∈ S₁. Hence $\int_{Π_{0, 1}} [\bar{φ} (Δ_{D} u - f - au)] d λ_{ℰ} = 0$ \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 $u \in {\dot{W}}_{2, D}^{2} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^2({\Pi _{0,1}}) then the equality (12) is satisfied.

Theorem 26

Let $u \in {\dot{W}}_{2}^{2} (Π_{0, 1})$ u \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 $φ \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \phi \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}). Then the function J_a,f (u + t_ϕ), t ∈ ℝ, has the derivative (14) ${\frac{d J_{a, f} (u + t φ)}{d t} |}_{t = 0} = \int_{Π_{0, 1}} [{(\nabla \bar{φ}, D \nabla u)}_{E} + a \bar{φ} u + \bar{φ} f] d λ + \int_{Π_{0, 1}} [{(\nabla \bar{u}, D \nabla φ)}_{E} + a \bar{u} φ + \bar{f} φ] d λ .$ {\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 $\frac{d}{d t} J_{a, f} (u + t φ) |_{t = 0} = - \int_{Π_{0, 1}} [\bar{φ} (Δ_{D} u - au - f)] d λ - \int_{Π_{0, 1}} [φ (Δ_{D} \bar{u} - a \bar{u} - \bar{f})] d λ .$ \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 $u \in {\dot{W}}_{2}^{2} (Π_{0, 1})$ u \in \dot W_2^2({\Pi _{0,1}}) be the soluion of Dirichlet problem (11), (12) then the equality $\frac{d}{d t} J_{a, f} (u + t φ) |_{t = 0} = 0$ \frac{d}{{dt}}{J_{a,f}}(u + t\phi ){|_{t = 0}} = 0 holds for any $φ \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \phi \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) and the function u is the stationary point of the functional J_a,f.

The inequality $‖ u ‖_{ℋ} \leq ‖ u ‖_{{\dot{W}}_{2}^{1}}$ \parallel u{\parallel _\mathcal{H}} \leqslant \parallel u{\parallel _{\dot W_2^1}} holds according to the definition of the space ${\dot{W}}_{2}^{1}$ \dot W_2^1. Let f ∈ H. Then for any $u \in {\dot{W}}_{2}^{1}$ u \in \dot W_2^1 the inequality $| (f, u) | \leq c ‖ u ‖_{{\dot{W}}_{2}^{1}}$ |(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 $v \in {\dot{W}}_{2}^{1}$ v \in \dot W_2^1 such that ${(f, u)}_{ℋ} = {(v, u)}_{{\dot{W}}_{2}^{1}} \forall u \in {\dot{W}}_{2}^{1}$ {(f,u)_\mathcal{H}} = {(v,u)_{\dot W_2^1}}\;\forall \;u \in \dot W_2^1.

Let us endow the space ${\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) with the equivalent norm (15) $‖ u ‖_{{\dot{W}}_{2, D, a}^{1}} = {(a ‖ u ‖_{ℋ_{ℰ}}^{2} + \sum_{k = 1}^{\infty} d_{k} ‖ \partial_{k} u ‖_{ℋ_{ℰ}}^{2})}^{\frac{1}{2}}$ \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 ${\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) endowed with the equivalent norm (15) is noted by ${\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}). The inequalities $\frac{1}{1 + a} ‖ u ‖_{{\dot{W}}_{2, D, a}^{1}}^{2} \leq ‖ u ‖_{{\dot{W}}_{2, D}^{1}}^{2} \leq (1 + \frac{1}{a}) ‖ u ‖_{{\dot{W}}_{2, D, a}^{1}}^{2}$ \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 ${\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}). Therefore the space ${\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) is the Hilbert space.

Theorem 27

Let a > 0 and f ∈ L₂(Π_0,1). Then the functional (13) has the unique point of the minimum in the space ${\dot{W}}_{2, D}^{1} (Π_{0, 1})$ \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}).

Proof

Let f ∈ ℋ_ℰ. Then for any $u \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) the inequality $| (f, u) | \leq c ‖ u ‖_{{\dot{W}}_{2, D, a}^{1}}$ |(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 $v_{a} \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ {v_a} \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) such that ${(f, u)}_{ℋ} = {(v, u)}_{{\dot{W}}_{2, D, a}^{1}} \forall u \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ {(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 $u \in {\dot{W}}_{2, D}^{1} (E)$ u \in \dot W_{2,{\mathbf{D}}}^1(E) the following equality holds $J_{a, f} (u) = \frac{1}{2} {(u, u)}_{{\dot{W}}_{2, D, a}^{1}} - \frac{1}{2} {(v, u)}_{{\dot{W}}_{2, D, a}^{1}} - \frac{1}{2} {(u, v)}_{{\dot{W}}_{2, D, a}^{1}} = \frac{1}{2} {(u - v, u - v)}_{{\dot{W}}_{2, D, a}^{1}} - \frac{1}{2} {(v, v)}_{{\dot{W}}_{2, D, a}^{1}} .$ {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 J_a,f has the unique point of the minimum in the space ${\dot{W}}_{2, D}^{1}$ \dot W_{2,{\mathbf{D}}}^1 which coincides with the element $v \in {\dot{W}}_{2, D, a}^{1}$ v \in \dot W_{2,{\mathbf{D}},a}^1.

Definition 4

The function $v \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ 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 (16) ${(v, φ)}_{{\dot{W}}_{2, D, a}^{1} (Π_{0, 1})} + {(a + f, φ)}_{L_{2} (Π_{0, 1})} = 0$ {(v,\phi )_{\dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}})}} + {(a + f,\phi )_{{L_2}({\Pi _{0,1}})}} = 0 satisfies for any $φ \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ \phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}).

Theorem 28

Let a > 0 and f ∈ L₂(Π_0,1). Then the function $u \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ u \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 $u \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ u \in \dot W_{2,{\mathbf{D}}}^1({\Pi _{0,1}}) is point of minimum of the functional (13) then $\frac{d}{d t} J_{a, f} (u + t φ) |_{t = 0} = 0$ \frac{d}{{dt}}{J_{a,f}}(u + t\phi ){|_{t = 0}} = 0 for any $φ \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ \phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}). Hence the equality (16) satisfies for any $φ \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ \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 $φ \in {\dot{W}}_{2, D, a}^{1} (Π_{0, 1})$ \phi \in \dot W_{2,{\mathbf{D}},a}^1({\Pi _{0,1}}) the folowing equality holds $J_{a, f} (u + φ) - J (u) = \frac{1}{2} ‖ φ ‖_{{\dot{W}}_{2, D, a}^{1} (Π_{0, 1})}^{2} .$ {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 $u \in {\dot{W}}_{2, D}^{1} (Π_{0, 1})$ 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.

eISSN:: 2444-8656
Langue:: Anglais

Périodicité:: Volume Open
Sujets de la revue:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Dirichlet Problem for Poisson Equation on the Rectangle in Infinite Dimensional Hilbert Space

Publié en ligne: 30 juin 2020

Pages: 329 - 344

Reçu: 26 janv. 2020

Accepté: 10 mars 2020

DOI: https://doi.org/10.2478/amns.2020.2.00016

Mots clésShift invariant measure on Banach space, random walks, Laplas operator, Sobolev space, Dirichlet problem

© 2020 V.M. Busovikov et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
Shift invariant measure on Banach space, random walks, Laplas operator, Sobolev space, Dirichlet problem