Abstract:
We consider a Dirichlet problem in which the boundary value of a solution is understood as the Lp-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 W1p,loc) 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.
\Bibitem{Gus12}
\by A.~K.~Gushchin
\paper The Dirichlet problem for a~second-order elliptic equation with an $L_p$ boundary function
\jour Sb. Math.
\yr 2012
\vol 203
\issue 1
\pages 1--27
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This publication is cited in the following 25 articles:
Omar Benslimane, Ahmed Aberqi, Jaouad Bennouna, “On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space”, AJMS, 29:1 (2023), 29
Omar Benslimane, Ahmed Aberqi, Jaouad Bennouna, “Study of some nonlinear elliptic equation with non-polynomial anisotropic growth”, Adv. Oper. Theory, 7:3 (2022)
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, 70:3 (2021), 1579–1608
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
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
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
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
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
V. I. Vlasov, “Hardy spaces, approximation issues and boundary value problems”, Eurasian Math. J., 9:3 (2018), 85–94
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
A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43
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
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
A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439
Dumanyan V.Zh., “On solvability of the Dirichlet problem with the boundary function in L2 for a second-order elliptic equation”, J. Contemp. Math. Anal., 50:4 (2015), 153–166
V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931
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