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 Mat. Sb., 2016, Volume 207, Number 10, Pages 28–55 (Mi msb8698)

$L_p$-estimates for the nontangential maximal function of the solution 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 properties of the solution to a Dirichlet problem for a homogeneous second-order elliptic equation with $L_p$-boundary function, $p>1$. The same conditions are imposed on the coefficients of the equation and the boundary of the bounded domain as were used to establish the solvability of this problem. The $L_p$-norm of the nontangential maximal function is estimated in terms of the $L_p$-norm of the boundary value. This result depends on a new estimate, proved below, for the nontangential maximal function in terms of an analogue of the Lusin area integral.
Bibliography: 31 titles.

Keywords: elliptic equation, Dirichlet problem, nontangential maximal function.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work was supported by the Russian Science Foundation (project no. 14-50-00005).

DOI: https://doi.org/10.4213/sm8698

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English version:
Sbornik: Mathematics, 2016, 207:10, 1384–1409

Bibliographic databases:

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

Citation: A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Mat. Sb., 207:10 (2016), 28–55; Sb. Math., 207:10 (2016), 1384–1409

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8698
• https://doi.org/10.4213/sm8698
• http://mi.mathnet.ru/eng/msb/v207/i10/p28

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This publication is cited in the following articles:
1. M. O. Katanaev, “Cosmological models with homogeneous and isotropic spatial sections”, Theoret. and Math. Phys., 191:2 (2017), 661–668
2. V. V. Zharinov, “Lie–Poisson structures over differential algebras”, Theoret. and Math. Phys., 192:3 (2017), 1337–1349
3. F. Kh. Mukminov, “Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents”, Sb. Math., 209:5 (2018), 714–738
4. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839
5. 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
6. M. O. Katanaev, “Chern–Simons action and disclinations”, Proc. Steklov Inst. Math., 301 (2018), 114–133
7. V. V. Zharinov, “Analysis in algebras and modules”, Proc. Steklov Inst. Math., 301 (2018), 98–108
8. N. A. Gusev, “On the definitions of boundary values of generalized solutions to an elliptic-type equation”, Proc. Steklov Inst. Math., 301 (2018), 39–43
9. A. S. Trushechkin, “Finding stationary solutions of the Lindblad equation by analyzing the entropy production functional”, Proc. Steklov Inst. Math., 301 (2018), 262–271
10. M. O. Katanaev, “Description of disclinations and dislocations by the Chern–Simons action for $\mathbb{SO}(3)$ connection”, Phys. Part. Nuclei, 49:5 (2018), 890–893
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