On the partial regularity of energy-minimizing, area-preserving maps

We prove a partial regularity assertion for a Lipschitz continuous mapping u in the plane that minimizes an appropriate convex (or quasiconvex) energy functional, under the “hard” constraint that det Du = 1 a.e. The primary technical assumption is that u be nondegenerate, meaning that, locally, at least one of its partial derivatives is bounded away from zero a.e. The method of proof is to convert to a related minimization problem for a generating function w, the advantage being that we now have the “soft” constraint \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $w_{y_1y_2}> 0$\end{document}.