Abstract:
A test is established for the existence of a boundary value for the solution of the elliptic second order equation
−n∑i,j=1(aij(x)uxi(x))xj=f(x)−divF(x),x∈Q.
In this connection, it is proved that the solution has a property (u∈Cn−1(¯Q)) similar to continuity with respect to all variables in ¯Q, and that its boundary value u|∂Q∈L2(∂Q) is the limit in L2 of the traces of the solution on surfaces in a large class (which are not necessarily “parallel” to the boundary).
Citation:
A. K. Gushchin, V. P. Mikhailov, “On the existence of boundary values of solutions of an elliptic equation”, Math. USSR-Sb., 73:1 (1992), 171–194
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\by A.~K.~Gushchin, V.~P.~Mikhailov
\paper On~the existence of boundary values of solutions of an elliptic equation
\jour Math. USSR-Sb.
\yr 1992
\vol 73
\issue 1
\pages 171--194
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Linking options:
https://www.mathnet.ru/eng/sm1324
https://doi.org/10.1070/SM1992v073n01ABEH002540
https://www.mathnet.ru/eng/sm/v182/i6/p787
This publication is cited in the following 38 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, 37:20 (2022), 2243002–9
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 $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
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
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
Vlasov I V., “Hardy Spaces, Approximation Issues and Boundary Value Problems”, Eurasian Math. J., 9:3 (2018), 85–94
Petrushko I.M. Petrushko M.I., “On the First Mixed Problem in l-P, P > 1, For the Degenerating on the Boundary Parabolic Equations of Second Order”, AIP Conference Proceedings, 2048, ed. Pasheva V. Popivanov N. Venkov G., Amer Inst Physics, 2018, 040006
Petrushko I.M., “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. 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
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 l (2) For a Second-Order Elliptic Equation”, J. Contemp. Math. Anal.-Armen. Aca., 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
V. P. Mikhailov, “O suschestvovanii granichnykh znachenii u reshenii ellipticheskikh uravnenii”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 97–105
A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27
A. K. Guschin, “Otsenki resheniya zadachi Dirikhle s granichnoi funktsiei iz $L_p$”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 53–67
V. Zh. Dumanyan, “Solvability of the Dirichlet problem for a general second-order elliptic equation”, Sb. Math., 202:7 (2011), 1001–1020
Dumanyan V.Zh., “Solvability of the Dirichlet Problem for the General Second-Order Elliptic Equation”, Dokl. Math., 83:1 (2011), 30–33
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