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 Mat. Sb., 2018, Volume 209, Number 6, Pages 47–64 (Mi msb8980)

The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: The paper is concerned with the relationship between the nontangential maximal function of the solution to a Dirichlet problem with an $L_p$-boundary function, $p>1$, for a second-order elliptic equation and the Luzin area integral. The equation is considered in the self-adjoint form without lower-degree terms. The $L_p$-norm of the nontangential maximal function of the solution $u$ is estimated from above and below in terms of the squared $L_2(\partial Q)$-norm of the area integral of $v=|u|^{p/2}$. Here the coefficients of the equation need not be smooth in the domain.
Bibliography: 33 titles.

Keywords: elliptic equation, Dirichlet problem, nontangential maximal function, Luzin area integral.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work was supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.4213/sm8980

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English version:
Sbornik: Mathematics, 2018, 209:6, 823–839

Bibliographic databases:

UDC: 517.956.223
MSC: Primary 35J25; Secondary 35J67

Citation: A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Mat. Sb., 209:6 (2018), 47–64; Sb. Math., 209:6 (2018), 823–839

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8980
• https://doi.org/10.4213/sm8980
• http://mi.mathnet.ru/eng/msb/v209/i6/p47

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64
2. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
3. 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
4. A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Sb. Math., 211:11 (2020), 1551–1567
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