Minimizers of convex functionals with small degeneracy set

We study the question whether Lipschitz minimizers of integral F(del u) dx in R-n are C-1 when F is strictly convex. Building on work of De Silva-Savin, we confirm the C-1 regularity when (DF)-F-2 is positive and bounded away from finitely many points that lie in a 2-plane. We then construct a counterexample in R-4, where F is strictly convex but (DF)-F-2 degenerates on the Clifford torus. Finally we highlight a connection between the case n = 3 and a result of Alexandrov in classical differential geometry, and we make a conjecture about this case.