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Mat. Sb., 2012, Volume 203, Number 1, Pages 3–30 (Mi msb7825)  

This article is cited in 20 scientific papers (total in 20 papers)

The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function

A. K. Gushchin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider a Dirichlet problem in which the boundary value of a solution is understood as the $L_p$-limit, $p>1$, of traces of this solution on surfaces ‘parallel’ to the boundary. We suggest a setting of this problem which (in contrast to the notion of solution in $W_{p,\operatorname{loc}}^1$) enables us to study the solvability of the problem without making smoothness assumptions on the coefficients inside the domain. In particular, for an equation in selfadjoint form without lower-order terms, under the same conditions as those used for $p=2$, we prove unique solvability and establish a bound for an analogue of the area integral.
Bibliography: 37 titles.

Keywords: elliptic equation, Dirichlet problem, boundary value.

Funding Agency Grant Number
Russian Foundation for Basic Research 10-01-00178-а
Ministry of Education and Science of the Russian Federation НШ-7675.2010.1


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English version:
Sbornik: Mathematics, 2012, 203:1, 1–27

Bibliographic databases:

UDC: 517.956.223
MSC: 35J25
Received: 25.11.2010 and 07.04.2011

Citation: A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Mat. Sb., 203:1 (2012), 3–30; Sb. Math., 203:1 (2012), 1–27

Citation in format AMSBIB
\by A.~K.~Gushchin
\paper The Dirichlet problem for a~second-order elliptic equation with an $L_p$ boundary function
\jour Mat. Sb.
\yr 2012
\vol 203
\issue 1
\pages 3--30
\jour Sb. Math.
\yr 2012
\vol 203
\issue 1
\pages 1--27

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    This publication is cited in the following articles:
    1. Benslimane O., Aberqi A., Bennouna J., “Existence and Uniqueness of Entropy Solution of a Nonlinear Elliptic Equation in Anisotropic Sobolev-Orlicz Space”, Rend. Circ. Mat. Palermo  crossref  isi
    2. Gushchin A.K., “Estimates of the nontangential maximal function for solutions of a second-order elliptic equation”, Dokl. Math., 86:2 (2012), 667–669  mathnet  crossref  zmath  isi  elib  elib  scopus
    3. A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69  mathnet  crossref
    4. A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    6. A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43  mathnet  crossref  zmath  elib
    7. A. K. Gushchin, “V.A. Steklov's work on equations of mathematical physics and development of his results in this field”, Proc. Steklov Inst. Math., 289 (2015), 134–151  mathnet  crossref  crossref  isi  elib
    8. L. M. Kozhevnikova, A. A. Khadzhi, “Existence of solutions of anisotropic elliptic equations with nonpolynomial nonlinearities in unbounded domains”, Sb. Math., 206:8 (2015), 1123–1149  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Dumanyan V.Zh., “On solvability of the Dirichlet problem with the boundary function in $L^2$ for a second-order elliptic equation”, J. Contemp. Math. Anal., 50:4 (2015), 153–166  crossref  mathscinet  zmath  isi  scopus
    11. A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. 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  mathscinet  adsnasa  isi  elib
    13. 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  mathscinet  isi  elib  elib
    14. 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  mathscinet  isi  elib  elib
    15. V. I. Vlasov, “Hardy spaces, approximation issues and boundary value problems”, Eurasian Math. J., 9:3 (2018), 85–94  mathnet  crossref
    16. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    17. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    18. F. Kh. Mukminov, “Existence of a Renormalized Solution to an Anisotropic Parabolic Problem for an Equation with Diffuse Measure”, Proc. Steklov Inst. Math., 306 (2019), 178–195  mathnet  crossref  crossref  mathscinet  isi  elib
    19. A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Sb. Math., 211:11 (2020), 1551–1567  mathnet  crossref  crossref  mathscinet  isi  elib
    20. V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, “On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation”, Sb. Math., 212:6 (2021), 745–781  mathnet  crossref  crossref  isi  elib
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