The Dirichlet problem for discontinuous perturbations of the mean curvature operator in Minkowski space

TL;DR

This paper proves the existence of solutions for a discontinuous Dirichlet problem involving a mean curvature operator in Minkowski space, using advanced critical point theory for nonsmooth functionals.

Contribution

It introduces a novel application of critical point theory to solve a discontinuous mean curvature problem in Minkowski space.

Findings

Existence of solutions established for the problem.

Application of convex, lower semicontinuous perturbations.

Extension of critical point methods to discontinuous operators.

Abstract

Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.