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 Lp. 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 Lp 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.
Citation:
A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439
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\paper Solvability of the Dirichlet problem for an~inhomogeneous second-order elliptic equation
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\pages 1410--1439
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This publication is cited in the following 20 articles:
A. K. Gushchin, “On Dirichlet problem”, Theoret. and Math. Phys., 218:1 (2024), 51–67
A. K. Gushchin, “On some properties of elliptic partial differential equation solutions”, Int. J. Mod. Phys. A, 37:20n21 (2022)
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
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
A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Sb. Math., 211:11 (2020), 1551–1567
L. M. Kozhevnikova, “Renormalized solutions of elliptic equations with variable exponents and general measure data”, Sb. Math., 211:12 (2020), 1737–1776
A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
A. K. Gushchin, “On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
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
A. K. Gushchin, “A criterion for the existence of Lp boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64
M. O. Katanaev, “Chern–Simons action and disclinations”, Proc. Steklov Inst. Math., 301 (2018), 114–133
Yu. N. Drozhzhinov, “Asymptotically homogeneous generalized functions and some of their applications”, Proc. Steklov Inst. Math., 301 (2018), 65–81
V. V. Zharinov, “Analysis in algebras and modules”, Proc. Steklov Inst. Math., 301 (2018), 98–108
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
A. S. Trushechkin, “Finding stationary solutions of the Lindblad equation by analyzing the entropy production functional”, Proc. Steklov Inst. Math., 301 (2018), 262–271
V. V. Zharinov, “Analysis in differential algebras and modules”, Theoret. and Math. Phys., 196:1 (2018), 939–956
M. O. Katanaev, “Description of disclinations and dislocations by the Chern–Simons action for SO(3) connection”, Phys. Part. Nuclei, 49:5 (2018), 890–893
M. O. Katanaev, “Cosmological models with homogeneous and isotropic spatial sections”, Theoret. and Math. Phys., 191:2 (2017), 661–668
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
A. K. Gushchin, “Lp-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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