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

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