<abstract xmlns="http://www.w3.org/1999/xhtml"><p>We study the class of finite additive shift invariant measures on the real separable Hilbert space <italic>E</italic>. For any choice of such a measure we consider the Hilbert space <italic>ℋ</italic> of complex-valued functions which are square-integrable with respect to this measure. Some analogs of Sobolev spaces of functions on the space <italic>E</italic> are introduced. The analogue of Gauss theorem is obtained for the simplest domains such as the rectangle in the space <italic>E</italic>. 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.</p></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 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.

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

Moscow Institute of Physics and Technology

Institute of Mathematics with Computer Center of the Ufa Science Center of the Russian Academy of Sciences

Mathematics and Mechanics of the Nizhniy Novgorod State University

Steklov Mathematical Institute of Russian Academy of Sciences

Applied Mathematics and Nonlinear Sciences

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

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...

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

Published Online: Jun 30, 2020

Page range: 329 - 344

Received: Jan 26, 2020

Accepted: Mar 10, 2020

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

Keywords
Shift 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.

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

Published Online: Jun 30, 2020

Page range: 329 - 344

Received: Jan 26, 2020

Accepted: Mar 10, 2020

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

KeywordsShift 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.

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