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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2015, Volume 19, Number 1, Pages 19–43 (Mi vsgtu1383)

Differential Equations and Mathematical Physics

On the Dirichlet problem for an elliptic equation

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991, Russian Federation

Abstract: It is well known that the concept of a generalized solution from the Sobolev space $W_2 ^ 1$ of the Dirichlet problem for a second order elliptic equation is not a generalization of the classical solution sensu stricto: not every continuous function on the domain boundary is a trace of some function from $W_2 ^ 1$. The present work is dedicated to the memory of Valentin Petrovich Mikhailov, who proposed a generalization of both these concepts. In the Mikhailov's definition the boundary values of the solution are taken from the $L_2$; this definition extends naturally to the case of boundary functions from $L_p$, $p> 1$. Subsequently, the author of this work has shown that solutions have the property $(n-1)$-dimensional continuity; $n$ is a dimension of the space in which we consider the problem. This property is similar to the classical definition of uniform continuity, but traces of this function on the measures from a special class should be considered instead of values of the function at points. This class is a little more narrow than the class of Carleson measures. The trace of function on the measure is an element of $L_p$ with respect to this measure. The property $(n-1)$-dimensional continuity makes it possible to give another definition of the solution of the Dirichlet problem (a definition of $(n-1)$-dimensionally continuous solution), which is in the form close to the classical one. This definition does not require smoothness of the boundary. The Dirichlet problem in the Mikhailov's formulation and especially for the $(n-1)$-dimensionally continuous solution was studied insufficiently (in contrast to the cases of classical and generalized solutions). First of all, it refers to conditions on the right side of the equation, in which the Dirichlet problem is solvable. In this article the new results in this direction are presented. In addition, we discuss the conditions on the coefficients of the equation and the conditions on the boundary of a domain in which the problem is considered. The results about the solvability and about the boundary behavior of solutions are compared with the analogous theorems for classical and generalized solutions. Some unsolved problems arising from such comparison are discussed.

Keywords: elliptic equation, Dirichlet problem, function space

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00065-à This work has been supported by the Russian Foundation for Basic Research (project no. 13–01–00065-a).

DOI: https://doi.org/10.14498/vsgtu1383

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Bibliographic databases:

UDC: 517.956.223
MSC: 35J25
Original article submitted 19/XII/2014
revision submitted – 05/II/2014

Citation: A. K. Gushchin, “On the Dirichlet problem for an elliptic equation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:1 (2015), 19–43

Citation in format AMSBIB
\Bibitem{Gus15} \by A.~K.~Gushchin \paper On the Dirichlet problem for an elliptic equation \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2015 \vol 19 \issue 1 \pages 19--43 \mathnet{http://mi.mathnet.ru/vsgtu1383} \crossref{https://doi.org/10.14498/vsgtu1383} \zmath{https://zbmath.org/?q=an:06968946} \elib{http://elibrary.ru/item.asp?id=23681740}