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

In the talk we introduce the notion of maximal $m$-convex $(m-cv)$ functions and for strictly $m$-convex domains $D\subset {{\mathbb{R}}^{n}}$ we solve the Dirichlet problem with a given continuous boundary function. We prove that for solution of the Dirichlet problem in the class $m-cv$ of functions, its Hessian $H_{\omega }^{n-m+1}=0$ in the domain $D.$