Maximal $m$-convex functions and the Dirichlet problem in the class of $m$-convex functions

In this talk we consider one of the key concepts of potential theory — extremal (maximal) $m$-convex functions and the related Dirichlet problem. The formulation and solvability of the Dirichlet problem in non-strictly $m$-convex domains are studied, and conditions for the existence of a solution are found. The concept of Hessians in the class of bounded $m$-convex functions is introduced, which are considered as Borel measures. It is shown that the corresponding measures for maximal $m$-convex functions are trivial.