The Dirichlet problem in the class of m-convex functions

Abstract

The well-known classical Dirichlet problem states that if $D\subset {\mathbb R}^{n}$ is a regular domain, then for any continuous function $\varphi (\xi )\in C(\partial D)$, there exists a unique harmonic function $\omega (x)\in C(\overline{D})$, such that $\omega |_{\partial D} =\varphi $. In the work of Sadullaev-Sharipov, under an additional condition of strict $m$-convexity of the domain $D\subset {\mathbb R}^{n} $, an analogous result for $m$-convex $(m-cv)$ functions has been proven.