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

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
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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\jour Sb. Math.
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\vol 210
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\pages 1724--1752
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  • http://mi.mathnet.ru/eng/msb/v210/i12/p67

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