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

V.M. Busovikov 1 , 2 , 3 , 4  and V.Zh. Sakbaev 1 , 2 , 3 , 4
  • 1 Moscow Institute of Physics and Technology, , Instiutskii per. 9, Dolgoprudny, Moscow region
  • 2 Institute of Mathematics with Computer Center of the Ufa Science Center of the Russian Academy of Sciences, , Chernyshevskii st. 112, Ufa
  • 3 Institute of Information Technology, Mathematics and Mechanics of the Nizhniy Novgorod State University, Gagarin ave 23, Nizhniy Novgorod
  • 4 Steklov Mathematical Institute of Russian Academy of Sciences, , Gubkin srt.8, Moscow
V.M. Busovikov
  • Moscow Institute of Physics and Technology, Instiutskii per. 9, Dolgoprudny, Moscow region, 141701, Russia
  • Institute of Mathematics with Computer Center of the Ufa Science Center of the Russian Academy of Sciences, Chernyshevskii st. 112, Ufa, 450008, Russia
  • Institute of Information Technology, Mathematics and Mechanics of the Nizhniy Novgorod State University, Gagarin ave 23, Nizhniy Novgorod, 603950, Russia
  • Steklov Mathematical Institute of Russian Academy of Sciences, Gubkin srt.8, Moscow, 119991, Russia
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and V.Zh. Sakbaev
  • Corresponding author
  • Moscow Institute of Physics and Technology, Instiutskii per. 9, Dolgoprudny, Moscow region, 141701, Russia
  • Institute of Mathematics with Computer Center of the Ufa Science Center of the Russian Academy of Sciences, Chernyshevskii st. 112, Ufa, 450008, Russia
  • Institute of Information Technology, Mathematics and Mechanics of the Nizhniy Novgorod State University, Gagarin ave 23, Nizhniy Novgorod, 603950, Russia
  • Steklov Mathematical Institute of Russian Academy of Sciences, Gubkin srt.8, Moscow, 119991, Russia
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Abstract

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 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.

1 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 of complex-valued functions any of each is square integrable with respect to this measure. We study operators of argument shifts on the spaces .

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 . The criteria of strong continuity of this semigroup U is obtained.

We prove that if the semigroup U is strongly continuous in the space then for any t > 0 the image U(t) f of any vector f 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 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 W2,Dm(E) is defined as the space of functions u such that (∂k)lu for any l ∈ {1,...,m} and any k ∈ ℕ; and the following series converges
k=1dm(k)mu2<+.

Here {dk} is the sequence of eigenvalues of the operator D and {ek} is the sequence of corresponding eigenvectors. The function u has the derivative ∂hu in the direction of the unite vector hE if the following equality limt01t(SthI)uhu=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.

2 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 of subsets from a space E in the work [10]. The ring 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 depending on the chois of an orthonormal basis = {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} ≡ in E and there is an elements a,bl such that
Π={xE:(x,ej)[aj,bj)jN}.

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
j=1max{0,ln(bjaj)}<+.
Let λ(Π) = 0 if Π = ∅, and let
λ(Π,a,b)=exp(j=1ln(bjaj))
for any nonempty measurable rectangle Πℰ,a,b.

For any orthonormal basis = {fk} of the space E the symbol K() note the set of measurable rectangles with the edges collinear to the vectors of ONB . Let the symbol r notes the minimal ring of subsets containing the set of measurable rectangles K().

Theorem 1

[11] For any orthonormal basis ℱ = {fk} of the space E there exists the unique measure λ : r → [0, +∞) such that the equality (3) holds for any rectangle Πℱ,a,b𝒦. The measure λ has the unique completion onto the ring ℛ which is completion of the ring r by measure λ.

Note 2. Here the ring consists on the sets AE such that λ¯(A)=λ_(A) where λ¯(A)=infBr,BAλ(A), λ_(A)=supBr,BAλ(A) are external and inner measure of a set A with respect to the measure λ.

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. Since we will use it very often, we define it concisely = L2(E, , λ, ℂ).

Let 𝒮 (E, , ℂ) be the linear space hull over field ℂ of indicator functions of the sets from the ring . Let β be the sesquilinear form on the space 𝒮 (E, , ℂ) which is defined by the following conditions: β(χA,χB)=λ(AB) for any sets A, B𝒦; for any functions u, v𝒮 (E, , ℂ) such that u=j=1ncjχAj, v=k=1mbkχBk the value β (u, v) is given by the equality
β(u,v)(u,v)(E)=(j=1ncjχAj,k=1mbkχBk)=k=1mj=1nb¯kcj(χAj,χBk).

This sesquilinear form on the space 𝒮 (E, , ℂ) is Hermitian and nonnegative.

The function u𝒮 (E, , ℂ) is called equivalent to the function u𝒮 (E, , ℂ) iff β (uv, uv) = 0. The linear space 𝒮2(E, , λ, ℂ) of the equivalence classes of functions of the space 𝒮 (E, , ℂ) endowing with the sesquilnear form β is the pre-Hilbert space. The Hilbert space L2(E, , λ, ℂ) ≡ is defined as the completion of the space 𝒮2(E, , λ, ℂ).

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

Lemma 4

[10] The space ℋ is not separable.

2.2 The products of the spaces with finite additive measures

Let = {e1, e2, ...} be an ONB in the space E. Let E1 be the Hilbert space with the ONB 1 = {e2, e3, ...}. Then E = ℝ ⊕ E1, Ex = (x1, ξ) ∈ ℝ ⊕ E1, where x1 ∈ ℝ; ξE1.

Let 𝒥 be the isomorphism of the Hilbert space E onto the Hilbert space E1 such that 1 = 𝒥 (). Let λ be a complete translation invariant measure on the space E such that the measure λ is defined on the ring by the theorem 1. Let 1 = 𝒥 () and λ1 is the measure on the space (E1, 1) such that λ1(A) = λ (𝒥−1(A)) ∀ A1.

Let l 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 and K1 are the collections of measurable rectangles in the spaces E and E1 whose edges are collinear to the vectors of ONB and 1 respectively; r and r1 are the minimal rings containing the collections of sets K and K1 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 XE is defined by the equality
λ_(X)=supk=1nQkX,QkKλ(k=1nQk)
where supremum is defind over the set of finite union of measurable rectangles but not on the hole ring r.
Lemma 6

([2], lemma 3.4). Let A = g × Π, where Π ∈ 𝒦1and λ1(Π) ≠ 0. Then A iff gr(ℝ). In this case the following equality λ (A) = l(g)λ1(Π) holds.

The collection 𝒦 of measurable rectangles is the part of the following collection of sets {A0 × A1, A0r( ℝ), A11}; the last collection of sets is the part of the ring . Since the ring r is the minimal ring containing the collection of sets 𝒦 and the ring is the completion of the ring r by the measure λ, then the ring is the comletion by measure λ of the minimal ring, containing the collection of the sets {A0 × A1, A0r( ℝ), A11}. Hence the following statement holds.

Proof

The space with finite additive measure (E, r, λ) is the prodict of the spaces with finite additive measures ( ℝ, r( ℝ), l) and (E1, r1, λ1)

Proof

According to the lemma 1 [5] (page 222) the space with finite additive measure (E, r, λ) is the product of the spaces with the finite additive measures ( ℝ, r( ℝ), l) and (E1, r1, λ1). In fact, since the rings and 1 is obtained by using of completions by measures λ and λ1 procedure from the rings r and r1 respectively then the measure λ be a unique finite additive measure which is defined on the ring and satisfies the conditions λ (A0 × A1) = l(A0)λ1(A1) ∀ A0r( ℝ), A11.

Definition 1

A tensor product of the finite additive measures μ = μ1μ2 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. 1)it is defined on the minimal ring containing the collection of sets {A1 × A2, A1R1, A2R2},
  2. 2)it satisfies the equality μ1 × μ2(A1 × A2) = μ1(A1)μ2(A2) ∀ A1R1, A2R2.

Lemma 8

The following equality λ = lλ1holds in the sense of definition 1.

Proof

In fact, the procedures of definition of the measure λ and the measure lλ1 have the following common constructions.

The definition the measure λ consists of three parts:

  1. The function of a set is defined firstly on the collection of measurable rectangles 𝒦;
  2. the function 𝒦 → ℝ is extended onto the measure λ on the minimal ring r containing the collection of sets 𝒦;
  3. the measure λ : r → ℝ is extended onto the ring which is completion of the ring r by measure λ.

In the case of the measure lλ1 the collection of a sets {A1 × A2 | A1r( ℝ), A2𝒦1} is used instead of the collection 𝒦 on the first step. According to definition the equality lλ1(I1 × Π1) = λ (I1 × Π1) holds for any measurable rectangle I1 × Π1 where I1r( ℝ), Π1𝒦1. Therefore the inequalities λ_(A)lλ1¯(A)lλ1_(A)λ¯(A) hold for any set AE. Hence if a set AE is measurable with respect to the measure λ then it is measurable with respect measure lλ1 and the the extensions of the measures λ and lλ1 coincides on the ring .

On the other hand any set of type {A1 × A2 | A1r( ℝ), A2𝒦1} belongs to the ring and in this case the equality λ (A1 × A2) = (lλ1)(A1 × A2) holds. Since the measure λ : → ℝ is complete then the continuation of the function of a set (lλ1) : {A1 × A2 | A1r( ℝ), A2𝒦1} → ℝ by means of continuation on the minimal ring (steps 2)) and completion (step 3)) coincides with the measure λ.

Lemma 9

The linear space span({χΠ, Π ∈ 𝒦 }) is dense in the Hilbert space L2(E, , λ, ℂ).

Proof

In fact, according to definition the Hilbert space L2(E, , λ, ℂ) is the closure of the linear space span({χA,A})¯ endowed with the norm of the space L2(E, , λ, ℂ). Since the ring is the completion of the ring r by the measure λ then the following equality L2(E,,λ,)=span({χA,Ar})¯ holds. Note that any set Ar is the finite union of disjoint sets A=k=1NBk where Bk is the complement of a measurable rectangle to finite union of measurable rectangles: k1,N¯Bk=Πk,0\j=1mkΠk,j, Πk,i𝒦i0,mk¯. So, χA ∈ span({χΠ, Π ∈ 𝒦 }) for any Ar. Consequently, L2(E,,λ,)=span({χΠ,Π𝒦})¯.

Theorem 10

Morphism ℐ mapping element of L2( ℝ) ⊗ L2(E1, 1, λ1,ℂ), which is limit of fundamental sequence fkvk into limit of sequence fkvk in space L2(E, , λ, ℂ) provides us with canonical isomorphism between these two space.

Proof

Space L2(E, , λ, ℂ) according to lemma 9 is a completion of span({χA, A𝒦 }) with respect to norm ‖ · ‖L2(E,ℛ, λ, ), defined by sesquilinear form β, see (1). So, L2(E, , λ, ℂ) is completion of the space span({χA0×A1, A0𝒦, A1𝒦1}) with respect to norm ‖ · ‖L2 (E,ℛ, λ, ), defined by sesquilinear form β. (Here we define by 𝒦 a set of all bounded intervals of ℝ).

Space L2( ℝ) ⊗ L2(E1, 1, λ1, ℂ) is completion of linear span of elements fv, where fL2( ℝ), vL2(E1, 1, λ1, ℂ) with respect to norm ‖ · ‖, defined by sesquilinear form β on with restriction β (f1v1, f2v2) = (f1, f2)L2( ℝ)(v1, v2)L2(E1,ℛℰ1,λℰ1,ℂ). Note that in space L2( ℝ) linear span 0 of set of characteristic functions from 𝒦 is dense linear submanifold, and in space L2(E1, ℛℰ1, λℰ1, ℂ) according to lemma 9 linear span 1 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{χA0χA1, A0𝒦, A1𝒦ℰ1} with respect to norm ‖ · ‖.

Since for any interval Δ ∈ 𝒦 and any measurable rectangle Π′∈ 𝒦1 holds ‖χΔχΠ′ = ‖χΠL2(E,ℛℰ, λℰ,), where Π = Δ × Π′, then for any sets A0𝒦, A1𝒦1 holds
χA0χA1=χA0×A1L2(E,,λ,).

Since linear span span({χA0×A1, A0𝒦, A1𝒦1}) is dense in space L2(E, , λ, ℂ), and linear span span({χA0χA1, A0𝒦, A1𝒦1}) is dense in space L2( ℝ) ⊗ L2(E1, 1, λ1, ℂ), then (dee equation 2) spaces L2(E, , λ, ℂ) and L2( ℝ) ⊗ L2(E1, 1λ1, ℂ) are isometrically isomorphic and L2(E, , λ, ℂ) = (L2( ℝ) ⊗ L2(E1, 1λ1, 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 1 with respect to the first coordinate is defined on the space L2(E, , λ, ℂ) = L2( ℝ) ⊗ L2(E1, 1, λ1,ℂ). The partial Fourier transform 1 is defined on the linear hull of the elements of type u1(x1)u2(x), u1L2( ℝ), u2L2(E1, 1, λ1, ℂ) by the equality
1(k=1nu1,ku2,k)=k=1nu^1,ku2,
where û1,kL2( ℝ) is Fourier transform of the function u1,kL2( ℝ). Since the linear hull of the elements of type u1(x1)u2(ξ), u1L2( ℝ), u2L2(E1, 1, λ1, ℂ) is dense in the space L2(E, , λ, ℂ) then the partial Fourier transform 1 has the unique continuation up to the unitary transform of the space L2(E, , λ, ℂ) into itself.

Analogously, partial Fourier transform n with respect to first n coordinates is defined on the space L2(E, , λ, 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, , λ, ℂ) with respect to first n coordinates is unitary mapping of the space L2(E, , λ, 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 . It also be used further in the studying of generators of diffusion semigroups and its fraction powers.

3 Sobolev spaces and spaces of smooth functions

3.1 Averaging of random shifts and space of smooth functions

Let DB(E) be a nonnegative trace class operator with the orthonormal basis 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 hE is defined on the space = L2(E, , ℂ) by the equality
Shu(x)=u(xh).
It is obvious that for any hH operator Sh belongs to the Banach space B() of bounded linear operators in the space endowed with the operator norm; moreover Sh is the unitary operator in the space . Let h be a random vector of the space E whose distribution is given by the measure ν. Then the mean value UB() of random shift operator Sh is given by the Pettis integral
EShdν(h)=U(Uf,g)=E(Shf,g)dν(h)f,g.

According to the paper [12] (see also [15]) the following statement holds.

Theorem 11
LetDB(E) be a nonnegative trace class operator with the orthonormal basis ℰ of eigenvectors. Then one-parametric family of operators
Ut=EShdνtD(h),t0,
is a one-parametric semigroup of self-adjoint contractions in the space ℋ. The semigroupUt, t ≥ 0, is strong continuous in the space ℋ if and only ifD12is trace class operator.
Definition 2

A function u′j is called the derivative of the function u in the direction of a unite vector ej if the following equality holds lims01s(Ssejuu)uj=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 operatorDand𝒰D(t)=ESthdνD(h), t ≥ 0, u then for any l ∈ ℕ; there is the number cl > 0 such that for any u, j ∈ ℕ;, t > 0 there is the derivative of the power ljl𝒰D(t)uand the following estimates take place
jl𝒰D(t)ucl(tdj)lu.

Let D be a positive trace class operator in the space E. Let CD(E) be a linear hull of the following system of elements {𝒰D(t)u, t > 0, u}. The linear manifold CD(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 W2,Da1(E) notes the linear space of elements u of the space such that the following two condition hold

1) for any j ∈ ℕ; there is the derivative ujxju with respect to the direction of eigenvector ej of operator D;
2)j=1ndjauj2<+.

The space W2,Da1(E) endowed with the norm uW2,Da1(E)=(u2+j=1ndjauj2)12 is the Hilbert space which is continuously embedded into the space (see [3, 11]).

For any numbers a > 0 and any l ∈ ℕ; the symbol W2,Da1(E) notes the linear space of elements u of the space such that the following two conditions hold:

1) for any j ∈ ℕ; there is the l-order derivative 2ejlu;
2)j=1ndjalejlu2<+.
Analogously, the space W2,Da1(E) endowed with the norm
uW2,Dal(E)=(u2+j=1ndjalejlu2)12
is the Hilbert space which is continuously embedded into the space (see [3, 11]).
Theorem 13

[see [11], lemmas 7.1, 7.2]. Let u. LetDbe a positive trace class operator in the space E such thatD12is the trace class operator. Then for any t > 0 the inclusion𝒰D12(t)uW2,D1(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, +∞) thenCDα(E)W2,Dbl(E).

If, moreover, α ≥ 2γ then the linear manifoldCDα(E)is dense in the spaceW2,Dbl(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 be an orthonormal basis in the space E. Let e1 and 1 = \{e1}. Let E1 = (span(e1)) and 1 = L2(E1, 1, λ1, ℂ). Let ( 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 = L2( ℝ, ( ℝ), λL, 1)) where L2( ℝ, ( ℝ), λL, 1)) is the space of integrable in Bokhner sense with respect to Lebesgue measure λL mappings ℝ → 1.

We define a linear space W21(,(),λL,1)={u:e1u} endowed with the Sobolev norm
|uW21(,(),λL,1))=(u2+e1u2)12.
According to the definition of partial Fourier transform
uW21(,(),λL,1)2=(1+ξ2)u^1(ξ)12dξ
for any uW21(,(),λL,1) where û = 1(u). Hence the space W21(,(),λL,1)) endoved with the norm (6) is the Hilbert space.
Theorem 15
IfuW21(,(),λL,1)then the equivalence class u contains the continuous function ũC( ℝ, 1). Moreover, there is the constant C > 0 such that
u˜C(,1)CuW21(,(),λL,1).
Proof

In the case of separable space 1 the proof of this theorem is given in the monograph [6]. In the case under consideration the space 1 is not separable. But according to the condition uW21(,(),λL,1) there is the separable subspace of the space 1 containing the values of the mapping u : ℝ → 1. Therefore the proof of the theorem 3.1 by [6] can be applied to the obtaining of the statement of theorem 15.

If uW2,D1(E) then the function u can be considered as the function of the space W21(,(),λL,1). Therefore the function uW2,D1(E) can be considered as the continuous maping ũ : ℝ → 1 according to theorem 15. Hence we can use the following definition of the trace of function.

Definition 3

The trace of the function uW2,D1(E) at the hyperplane x1 = t0, t0 ∈ ℝ, is the value of function ũC( ℝ, 1) at the point t0.

Corollary 16

IfuW21(,(),λL,1)then for any t0 ∈ ℝ the estimateu˜(t0)1CuW21(,(),λL,1)holds.

Corollary 17
IfuW21(,(),λL,1)andddsSse1u=vL2(,(),λL,1), then for any t1, t2 ∈ ℝ the equality holds
u(t2)u(t1)=t1t2v(s)ds.

4 The analog of Gauss theorem for rectangle

Let be the ONB of positive trace-class covariation operator D of Gaussian measure γ. Let Πa,b𝒦 () be a measurable rectangle. For any j ∈ ℕ; the equality E = ℝ ⊕ Ej where ℝ = span(ej) and Ej = (span(ej)). Let j = {e1,...,ej−1, ej+1,...}. Therefore the following equality Πa,b=[aj,bj)×Π^a^j,b^j holds where Π^a^j,b^j𝒦(j) is the measurable rectangle in the space Ej.

Let uW2,D2(E). Then for any j ∈ ℕ; and any aR there is the trace u|xj=aℰj of the function u according to the theorem 3.4. Since for any j ∈ ℕ; the function ju has the derivative j2u in the direction ej, then according to the theorem 3.4 there is the trace (ju)|xj=aj.

Theorem 18
LetDbe a positive trace class operator in Hilbert space E. LetuW2,D2(E), let Πa,b𝒦 () be a measurable rectangle. Then the equality
Πa,bΔDuφdλ=Πa,b(u,Dφ)Edλ+Πa,b(n,Du)φds,
holds for any functionφW2,D1(E). Here
Πa,b(n,Du)φds=j=1djΠ^a^j,b^j[(φju)|xj=bj(φju)|x=aj]dλj.
Proof
Since φW2,D1(E) then the condition ϕ|xj=a| holds for any j ∈ ℕ; and any aR according to the theorem 15, moreover
φxj=ajC(φL2(,(),λL,j)2+jφL2(,(),λL,j)2)12
according to theorem 15 and corollary (16). Therefore
|Π^a^j,b^j[(φju)|xj=b(φju)|xj=a]dλj|=|(φ|x=b,ju|x=b)j(φ|x=a,ju|x=a)j|2juC(,j)φC(,j)
for any j ∈ ℕ;. Hence the estimates
|Π^a^j,b^j[(φju)|xj=b(φju)|xj=a]dλj|2C2(φ2+jφ2)12(ju2+j2u2)12C2(φ2+jφ2+ju2+j2u2)
hold for any j ∈ ℕ;. Let us note that
j=1dj(φ2+jφ2)=(TrD1)φ2+φW2,D1(E)2.
Since partial Fourier transform with respect to coordinate xj is unitary operator in the space and according to the inequality k2 ≤ 1 + k4, k ∈ ℝ, we obtain
j=1djju2j=1dj(u2+j2u2)
Hence
j=1dj(ju2+j2u2)2uW2,D2(E)2+(TrD2)u2.

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

Since
j(φdjju)=djjφju+djj2uφ,
then according to the corollary 17 the equality
Πa,b(n,ejdjju)φds=Πa,bdjj2uφdλ+Πa,bdjjujφdλ.
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 W2,D2(E). Hence the series in right hand side of (10) absolutely converges and the equality (8) takes place.

5 Dirichlet problem for Poisson equation

Let us consider the unique rectangle Π0,1𝒦. Let symbol L20,1) notes the subspace of the space with the support in the set Π0,1. Let us note that f = χΠ0,1f for any fL20,1).

Let symbol W˙2,D1(Π0,1) notes the space of function W2,D1(E)L2(Π0,1) with the support in the set Π0,1. It should be note that for any uW˙2,D1(Π0,1) the following statement holds:
u|Π0,1=0;xkuL2(Π0,1)k.
We also introduce the space W˙2,D2(Π0,1) of functions uW˙2,D1(Π0,1) such that
gjkL2(Π0,1):(Dku,jφ)=(gjk,φ)φW˙2,D1(Π0,1),j,k;j=1gjj2<+.
We pose the following problem. For a given function fL20,1) and a given number a ≥ 0 we should find a function uW˙2,D2(Π0,1) such that
ΔDu=au+f,
u|Π0,1=0.

To investigate the above problem we apply the variation approach (see [7]). At first we study the space W˙2,D1(Π0,1).

Let us introduce the trapezoid-like function ψδ : ℝ → ℝ which is given by the equalites ψδ(x)=0,x(,0][1,+); ψδ (x) = 1, x ∈ [δ, 1 − δ]; ψδ(x)=1δx,x(0,δ); ψδ(x)=1δ(x1),x(1δ,1) for any δ(0,12). Then ψδW˙21([0,1]), ‖ψδL2([0,1]) ∈ [1 − 2δ, 1], xψL2([0,1])=2/δ.

Let D a the nonnegative trace-class operator such that D12 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 {δk}:(0,12) the function Ψ{δ}(x)=Πj=1ψδj(xj) is defined.

Lemma 19

If{δk}:(0,12)andj=1δj<+then Ψ{δ} and the estimatesΠk=1(12δk)ΨδL21hold.

Proof
The inclusion Ψ{δ} for the nonnegative function Ψ{δ} is equivalent to the condition
ε>0g,G:gΨ{δ}GandGg<ε.
Let us define
Gn(x)=Πk=1nψδk(xk)Πk=n+1χ[0,1](xk)
and gn(x)=Πk=1nψδk(xk)Πk=n+1χ[δk,1δk](xk) for some n ∈ ℕ;. Then gn ≤ Ψ{δ}Gn and Gngn<1exp(k=n+1ln(12δk)) for any n ∈ ℕ;. Since limnk=n+1δk=0 then limnGngn=0. Therefore Ψ{δ}.

Let us note that 0 ≤ χΠδ,1−δ ≤ Ψδ ≤ χΠ0,1. Hence λ(Πδ,1δ)Ψδ2λ(Π0,1). Therefore the statements of the 19 are proved.

Note that λ(Πδ,1δ)=exp(k=1ln(12δk)). Since δk(0,12) for any k ∈ ℕ; then the series k=1ln(12δk) converges if and only if the series k=1δk converges. In this case λδ,1−δ) > 0. In the other case λδ,1−δ) = 0.

Lemma 20

The conditionk=1dkδk<+is necessary and sufficient to the inclusionΨ{δ}W˙2,D1(Π0,1).

Proof
We should proof that ∂jΨ{δ} for any j ∈ ℕ; and the series
j=1djjΨ{δ}2
converges. Note that Ψ{δ}(x)=ψδj(xj)Ψ{δ^}(x^),xE, xE, for any j ∈ ℕ; where x^={x1,...,xj1,xj+1,...} and δ^={δ1,...,δj1,δj+1,...}. Therefore for any j ∈ ℕ; the following equality holds
jΨ{δ}(x)=jψδj(xj)Ψ{δ^}(x^),xE.
Hence jΨ{δ} for any j ∈ ℕ; and
jΨ{δ}=jψδjL2(R)Ψ{δ^}Ej(2/δj)12.

Therefore the series j=1djjΨ{δ}2 converges if and only if the series j=1djδj converges.

Note 21.LetDbe a positive operator in the space E such thatDis trace class operator. Then there is the sequence{δk}:(0,12)such thatj=1δj<+and the conditionk=1dkδk<+satisfies. For example,δk=dk, k∈ ℕ.

Lemma 22

LetfW2,D1(E). Let the sequence{δk}:(0,12)satisfies the conditionk=1dkδk<+. ThenΨ{δ}fW˙21(Π0,1).

Proof
In fact, ‖Ψ{δ}f ≤ ‖ f according to Cauchy inequality and lemma 1. Since ∂j{δ}f) = ∂jΨ{δ}) f + Ψ{δ}jf then
j(Ψ{δ}f)22supxE|Ψ{δ}(x)|fW2,D12+1δj(0δj+21δj1)E^j|f(x)|2dλj(x^j)dxj2fW2,D12+2δjf2.
Therefore
j=1djj(Ψ{δ}f)22[Tr(D)fW2,D12+j=1djδjf2].

Thus we obtain that Ψ{δ}fW2,D1(E). Since (Ψ{δ}f)|Π0,1 = Ψ{δ}|Π0,1f|Π0,1 then (Ψ{δ}f)|Π0,1 = 0 and Ψ{δ}fW˙2,D1(E).

Lemma 23

LetfW2,D1(E)andk=1dkδk<+. Thenlimε+0fΨε{δ}fL2(Π0,1)=0.

Proof
In fact,
f(1Ψε{δ})L2(Π0,1)2=Π0,1|f(x)|2(1Ψε{δ})2dλΠ0,1\Πε{δ},1{δ}|f(x)|2dλ.

Since {δ} ∈ l1 then limε0λ(Π0,1\Πε{δ},1ε{δ})=limε0λ(Π0,1\Π0,12ε{δ})=0 according to the theorem ? in [10]. Since f then for any σ > 0 there is the simple function gS2(E, , λ, C) such that ‖ fg < σ.

Since the function g has the finite number of values then M=supxE|g(x)|[0,+). Therefore
Π0,1\Πε{δ},1ε{δ}|f(x)|2dλfg2+Π0,1\Πε{δ},1ε{δ}|g(x)|2dλfg2+M2λ(Π0,1\Πε{δ},1ε{δ}).

Hence for any σ > 0 there is a number ε0 > 0 such that Π0,1\Πɛ{δ},1ɛ{δ}|f(x)|2dλ2σ for any ε ∈ (0, ε0). Therefore limε+0fΨε{δ}fL2(Π0,1)=0.

The consequence of the lemma 4 is the following statement.

Theorem 24
LetDbe a nonnegative operator in the space E such thatDis trace class operator. Then the set of functions
S1={Ψε{δ}f,ffW2,D1(E),δ(0,12),{δ}:k=1dkδk<+}
is dense in the space L20,1).
Let a ≥ 0 and fL20,1). Let the functional Jf:W˙2,D1(Π0,1) be defined by the equality
Ja,f(u)=12Π0,1[(u¯,Du)E+a|u|2+u¯f+uf¯]dλ,uW2,D1(Π0,1).
Theorem 25

Let a ≥ 0 and fL20,1). LetuW˙2,D2(Π0,1)be a stationary point of the functional Ja,f. Then u is the solution of Dirichlet problem (11), (12).

Proof
Let uW˙2,D2(Π0,1) be a stationary point of the functional (13). Then the function Ja,f (u + ), t ∈ ℝ satisfies the equality ddtJa,f(u+tφ)=0 for any ϕS1. Therefore
Π0,1[(φ¯,Du)E+aφ¯u+φ¯f]dλ+Π0,1[(u¯,Dφ)E+au¯φ+f¯φ]dλ=0
for any ϕS1. Hence Π0,1[φ¯(ΔDufau)]dλ=0 for any ϕS1 according to the theorem 24. Since the set S1 is dense in the space then the function u satisfies Poisson equation (11). Since uW˙2,D2(Π0,1) then the equality (12) is satisfied.
Theorem 26

LetuW˙22(Π0,1)be the solution of Dirichlet problem (11), (12). Then it is the critical point of the functional (13).

Proof
Let φW˙2,D1(Π0,1). Then the function Ja,f (u + tϕ), t ∈ ℝ, has the derivative
dJa,f(u+tφ)dt|t=0=Π0,1[(φ¯,Du)E+aφ¯u+φ¯f]dλ+Π0,1[(u¯,Dφ)E+au¯φ+f¯φ]dλ.
Then according to the theorem 18
ddtJa,f(u+tφ)|t=0=Π0,1[φ¯(ΔDuauf)]dλΠ0,1[φ(ΔDu¯au¯f¯)]dλ.

Since uW˙22(Π0,1) be the soluion of Dirichlet problem (11), (12) then the equality ddtJa,f(u+tφ)|t=0=0 holds for any φW˙2,D1(Π0,1) and the function u is the stationary point of the functional Ja,f.

The inequality uuW˙21 holds according to the definition of the space W˙21. Let fH. Then for any uW˙21 the inequality |(f,u)|cuW˙21 take place where c ≤ ‖ f. Then according to R theorem there is the element vW˙21 such that (f,u)=(v,u)W˙21uW˙21.

Let us endow the space W˙2,D1(Π0,1) with the equivalent norm
uW˙2,D,a1=(au2+k=1dkku2)12
for arbitrary a > 0. The space W˙2,D1(Π0,1) endowed with the equivalent norm (15) is noted by W˙2,D,a1(Π0,1). The inequalities
11+auW˙2,D,a12uW˙2,D12(1+1a)uW˙2,D,a12
holds according to the definition of the space W˙2,D1(Π0,1). Therefore the space W˙2,D,a1(Π0,1) is the Hilbert space.
Theorem 27

Let a > 0 and fL20,1). Then the functional (13) has the unique point of the minimum in the spaceW˙2,D1(Π0,1).

Proof

Let f. Then for any uW˙2,D,a1(Π0,1) the inequality |(f,u)|cuW˙2,D,a1 take place where c ≤ ‖fℋℰ. Then according to Riesz theorem for any a > 0 there is the element vaW˙2,D,a1(Π0,1) such that (f,u)=(v,u)W˙2,D,a1uW˙2,D,a1(Π0,1).

Therefore for any uW˙2,D1(E) the following equality holds
Ja,f(u)=12(u,u)W˙2,D,a112(v,u)W˙2,D,a112(u,v)W˙2,D,a1=12(uv,uv)W˙2,D,a112(v,v)W˙2,D,a1.

Therefore the functional Ja,f has the unique point of the minimum in the space W˙2,D1 which coincides with the element vW˙2,D,a1.

Definition 4
The function vW˙2,D1(Π0,1) is called the generalized solution of the equation (11) with the Dirichlet condition (12) if the equality
(v,φ)W˙2,D,a1(Π0,1)+(a+f,φ)L2(Π0,1)=0
satisfies for any φW˙2,D,a1(Π0,1).
Theorem 28

Let a > 0 and fL20,1). Then the functionuW˙2,D1(Π0,1)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 uW˙2,D1(Π0,1) is point of minimum of the functional (13) then ddtJa,f(u+tφ)|t=0=0 for any φW˙2,D,a1(Π0,1). Hence the equality (16) satisfies for any φW˙2,D,a1(Π0,1) 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 φW˙2,D,a1(Π0,1) the folowing equality holds
Ja,f(u+φ)J(u)=12φW˙2,D,a1(Π0,1)2.

Hence the function uW˙2,D1(Π0,1) is point of strong minimum of the functional (13).

6 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 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.

References

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    V.Zh. Sakbaev, Semigroups of operators in the space of function square integrable with respect to traslationary invariant measure on Banach space. Quantum probability. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. VINITI. 151 (2018), 73–90.

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    A.M. Vershik. Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space? Proc. Steklov Inst. Math. 259 (2007), 248–272.

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    D. V. Zavadsky, V. Zh. Sakbaev, Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure. Proc. Steklov Inst. Math., 306 (2019), 102–119.

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    D.V. Zavadsky. Shift-invariant measures on sequence spaces// Proceedings of MIPT. 9 (2017), no. 4., 142–148.

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    D.V. Zavadsky, Analogs of Lebesgue measure on the sequences spaces and the classes of integrable functions. Quantum probability. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. VINITI. 151 (2018), 37–44.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Baker R. “Lebesgue measure” on R. Proceedings of the AMS. 113 (1991), no. 4., 1023–1029.

  • [2]

    V.M. Busovikov. Properties of one finite additive measure on lp invariant to shifts. Proceedings of MIPT. 10 (2018) no. 2. 163–172.

  • [3]

    V.M. Busovikov, V.Zh. Sakbaev. Sobolev spaces of functions on Hilbert space with shift-invariant measure and approximation of semigroups. Izvestiya RAN. Ser. Mathematics. (2020) no. 4.

  • [4]

    Ya. A. Butko. Chernoff approximation of subordinate semigroups. Stoch. Dyn. 1850021 (2017), 19 p., DOI: 10.1142/S0219493718500211.

  • [5]

    N. Dunford, J. Schwartz. Linear operators. General Theory. Moscow, 2004.

  • [6]

    J.L. Lions, E. Magenes. Problems aux limites non homogenes et applications. Dunod, Paris, 1968.

  • [7]

    O.A. Oleynik. Lectures on the partial differential equations. Lomonosov MSU, Moscow, 2015.

  • [8]

    I.D. Remizov, Formulas that represent Cauchy problem solution for momentum and position Schrodinger equation. Potential Anal (2018). https://doi.org/10.1007/s11118-018-9735-1

  • [9]

    I.D. Remizov, Explicit formula for evolution semigroup for diffusion in Hilbert space. Infinite Dimensional Analysis Quantum Probability and Related Topics (2018) Vol. 21, No. 04, 1850025.

    • Crossref
    • Export Citation
  • [10]

    V.Zh. Sakbaev, Averaging of random walks and shift-invariant measures on a Hilbert space. Theoret. and Math. Phys. 191 (2017), no. 3., 886–909.

    • Crossref
    • Export Citation
  • [11]

    V.Zh. Sakbaev, Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations. Differential equations. Mathematical physics. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. VINITI. 140 (2017), 88–118.

  • [12]

    V.Zh. Sakbaev, Semigroups of operators in the space of function square integrable with respect to traslationary invariant measure on Banach space. Quantum probability. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. VINITI. 151 (2018), 73–90.

  • [13]

    N.N. Shamarov, O.G. Smolyanov, Hamiltonian Feynman measures, Kolmogorov integral, and infinite-dimensional pseudodifferential operators Doklady Mathematics 100 (2019) no. 2., 445–449.

    • Crossref
    • Export Citation
  • [14]

    A.M. Vershik. Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space? Proc. Steklov Inst. Math. 259 (2007), 248–272.

    • Crossref
    • Export Citation
  • [15]

    D. V. Zavadsky, V. Zh. Sakbaev, Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure. Proc. Steklov Inst. Math., 306 (2019), 102–119.

    • Crossref
    • Export Citation
  • [16]

    D.V. Zavadsky. Shift-invariant measures on sequence spaces// Proceedings of MIPT. 9 (2017), no. 4., 142–148.

  • [17]

    D.V. Zavadsky, Analogs of Lebesgue measure on the sequences spaces and the classes of integrable functions. Quantum probability. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. VINITI. 151 (2018), 37–44.