Minimal graphs and differential inclusions

In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in (Formula presented.) We prove regularity of (Formula presented.) solutions and a compactness result for approximate solutions of this differential inclusion in (Formula presented.) Moreover, we make a perturbation argument to infer that for every R > 0, there exists (Formula presented.) such that R-Lipschitz stationary points for functionals α-close in the C 2 norm to the area functional are always regular. We also use a counterexample of B. Kirchhem (2003) to show the existence of irregular critical points to inner variations of the area functional.