Minimal graphs and differential inclusions

Abstract

In this paper, we study the differential inclusion associated with 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 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.

Keywords:

• Area functional
• differential inclusions
• polyconvexity
• regularity in two dimensions
• stationary points

AMS Subject Classification (2010)::

• 35B20
• 35B65
• 49Q05
• 58E12

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.

Additional information

Funding

The author has been partially supported by the SNF Grant 182565.