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

This article is cited in 10 scientific papers (total in 10 papers)

$L_p$-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

Full text: PDF file (919 kB)
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English version:
Sbornik: Mathematics, 2016, 207:10, 1384–1409

Bibliographic databases:

UDC: 517.956.223
MSC: Primary 35J25; Secondary 35J67
Received: 11.03.2016 and 21.06.2016

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. V. V. Zharinov, “Lie–Poisson structures over differential algebras”, Theoret. and Math. Phys., 192:3 (2017), 1337–1349  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    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  mathnet  crossref  crossref  isi  elib  elib
    6. M. O. Katanaev, “Chern–Simons action and disclinations”, Proc. Steklov Inst. Math., 301 (2018), 114–133  mathnet  crossref  crossref  isi  elib  elib
    7. V. V. Zharinov, “Analysis in algebras and modules”, Proc. Steklov Inst. Math., 301 (2018), 98–108  mathnet  crossref  crossref  isi  elib  elib
    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  mathnet  crossref  crossref  isi  elib  elib
    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  mathnet  crossref  crossref  isi  elib  elib
    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  crossref  isi  scopus
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