Minimal graphs and differential inclusions

Abstract

In this paper, we study the differential inclusion associated to the minimal surface system for two-dimensional graphs in $\mathbb{R}^{2 + n}$. We prove regularity of $W^{1,2}$ solutions and a compactness result for approximate solutions of this differential inclusion in $W^{1,p}$. Moreover, we make a perturbation argument to infer that for every $R > 0$ there exists $\alpha(R) >0$ such that $R$-Lipschitz stationary points for functionals $\alpha$-close in the $C^2$ norm to the area functional are always regular. We also use a counterexample of \cite{KIRK} to show the existence of irregular critical points to inner variations of the area functional.