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 Tr. Mat. Inst. Steklova, 2018, Volume 301, Pages 53–73 (Mi tm3871)

A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: The paper is devoted to the study of the boundary behavior of solutions to a second-order elliptic equation. A criterion is established for the existence in $L_p$, $p>1$, of a boundary value of a solution to a homogeneous equation in the self-adjoint form without lower order terms. Under the conditions of this criterion, the solution belongs to the space of $(n-1)$-dimensionally continuous functions; thus, the boundary value is taken in a much stronger sense. Moreover, for such a solution to the Dirichlet problem, estimates for the nontangential maximal function and for an analog of the Lusin area integral hold.

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01 This work is supported by the Program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and Its Applications” under grant PRAS-18-01.

DOI: https://doi.org/10.1134/S037196851802005X

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 301, 44–64

Bibliographic databases:

UDC: 517.956.223

Citation: A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Complex analysis, mathematical physics, and applications, Collected papers, Tr. Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 53–73; Proc. Steklov Inst. Math., 301 (2018), 44–64

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3871
• https://doi.org/10.1134/S037196851802005X
• http://mi.mathnet.ru/eng/tm/v301/p53

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This publication is cited in the following articles:
1. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
2. A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
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