On some properties of elliptic partial differential equation solutions

Abstract

This paper is devoted to the study of properties of second-order elliptic equation solutions. The main content of the paper coincides with the report made by the author at the international conference dedicated to the 75th anniversary of I. V. Volovich. The solution behavior near the boundary and the Dirichlet problem formulation, which is closely related to this issue, are studied. At the end of the paper, we will briefly discuss the results obtained in elegant and extremely important works by E. De Giorgi and J. Nash regarding Hölder continuity of the equation solutions within the considered domain. We present results that combine and complement the belonging of the solution to the Hölder and Sobolev spaces. Note that all the concepts and statements under consideration are united by a common approach and are formulated in close terms.