Abstract
In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in We prove regularity of solutions and a compactness result for approximate solutions of this differential inclusion in Moreover, we make a perturbation argument to infer that for every R > 0, there exists such that R-Lipschitz stationary points for functionals α-close in the C2 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.
Acknowledgments
I would like to thank my advisor, Camillo De Lellis, for posing this question and for helpful discussions. I would also like to thank Guido De Philippis, for his interest in this problem and for his suggestions, Antonio De Rosa and Maria Strazzullo, for helping me check some computations, and Yash Jhaveri, for discussions about Monge-Ampère equation.