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Mat. Sb., 2019, Volume 210, Number 12, Pages 67–97 (Mi msb9274)  

This article is cited in 1 scientific paper (total in 1 paper)

The boundary values of solutions of an elliptic equation

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: The paper is devoted to the study of the boundary behaviour of solutions of a second-order elliptic equation. Criteria are established for the existence of a boundary value of a solution of the homogeneous equation under the same conditions on the coefficients of the equation as were used to establish that the Dirichlet problem with a boundary function in $L_p$, $p>1$, has a unique solution. In particular, an analogue of Riesz's well-known theorem (on the boundary values of an analytic function) is proved: if a family of norms in the space $L_p$ of the traces of a solution on surfaces ‘parallel’ to the boundary is bounded, then this family of traces converges in $L_p$. This means that the solution of the equation under consideration is a solution of the Dirichlet problem with a certain boundary value in $L_p$. Estimates of the nontangential maximal function and of an analogue of the Luzin area integral hold for such a solution, which make it possible to claim that the boundary value is taken in a substantially stronger sense.
Bibliography: 57 titles.

Keywords: elliptic equation, boundary value, Dirichlet problem.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614
This work was supported from a grant to the Steklov International Mathematical Center in the framework of the national project “Science” of the Russian Federation.


DOI: https://doi.org/10.4213/sm9274

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English version:
Sbornik: Mathematics, 2019, 210:12, 1724–1752

Bibliographic databases:

UDC: 517.956.223
MSC: Primary 35J67; Secondary 35J25
Received: 30.04.2019 and 12.11.2019

Citation: A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Mat. Sb., 210:12 (2019), 67–97; Sb. Math., 210:12 (2019), 1724–1752

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, “On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation”, Sb. Math., 212:6 (2021), 745–781  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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