On the Dirichlet problem for an elliptic equation

A. K. Gushchin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 1, pp. 19-43

Résumé

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.

Détail

Mots-clés : elliptic equation
Keywords: Dirichlet problem, function space.