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.

arXiv comment26 pages. Various typos (including bibliographical entries) corrected from previous version. Moreover we have added new results, collected in the new Section 7