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 Mat. Sb., 2015, Volume 206, Number 10, Pages 71–102 (Mi msb8560)

Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We consider a statement of the Dirichlet problem which generalizes the notions of classical and weak solutions, in which a solution belongs to the space of $(n-1)$-dimensionally continuous functions with values in the space $L_p$. The property of $(n-1)$-dimensional continuity is similar to the classical definition of uniform continuity; however, instead of the value of a function at a point, it looks at the trace of the function on measures in a special class, that is, elements of the space $L_p$ with respect to these measures. Up to now, the problem in the statement under consideration has not been studied in sufficient detail. This relates first to the question of conditions on the right-hand side of the equation which ensure the solvability of the problem. The main results of the paper are devoted to just this question. We discuss the terms in which these conditions can be expressed. In addition, the way the behaviour of a solution near the boundary depends on the right-hand side is investigated.
Bibliography: 47 titles.

Keywords: elliptic equation, Dirichlet problem, boundary value.

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

DOI: https://doi.org/10.4213/sm8560

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English version:
Sbornik: Mathematics, 2015, 206:10, 1410–1439

Bibliographic databases:

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

Citation: A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Mat. Sb., 206:10 (2015), 71–102; Sb. Math., 206:10 (2015), 1410–1439

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8560
• https://doi.org/10.4213/sm8560
• http://mi.mathnet.ru/eng/msb/v206/i10/p71

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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, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409
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3. I. M. Petrushko, “On boundary and initial values of solutions of a second-order parabolic equation that degenerate on the domain boundary”, Dokl. Math., 96:3 (2017), 568–570
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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. Yu. N. Drozhzhinov, “Asymptotically homogeneous generalized functions and some of their applications”, Proc. Steklov Inst. Math., 301 (2018), 65–81
8. V. V. Zharinov, “Analysis in algebras and modules”, Proc. Steklov Inst. Math., 301 (2018), 98–108
9. 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
10. A. S. Trushechkin, “Finding stationary solutions of the Lindblad equation by analyzing the entropy production functional”, Proc. Steklov Inst. Math., 301 (2018), 262–271
11. V. V. Zharinov, “Analysis in differential algebras and modules”, Theoret. and Math. Phys., 196:1 (2018), 939–956
12. 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
13. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
14. 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
15. A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Sb. Math., 211:11 (2020), 1551–1567
16. L. M. Kozhevnikova, “Renormalized solutions of elliptic equations with variable exponents and general measure data”, Sb. Math., 211:12 (2020), 1737–1776
17. Lan H.-y., Nieto J.J., “Solvability of Second-Order Uniformly Elliptic Inequalities Involving Demicontinuous Psi-Dissipative Operators and Applications to Generalized Population Models”, Eur. Phys. J. Plus, 136:2 (2021), 258
18. 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
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