Minimizers of convex functionals with small degeneracy set

Abstract

We study the question whether Lipschitz minimizers of $\int F(\nabla u)dx$ in ${\mathbb{R}}^n$ are $C^1$ when $F$ is strictly convex. Building on work of De Silva-Savin, we confirm the $C^1$ regularity when $D^{2}F$ is positive and bounded away from finitely many points that lie in a $2$-plane. We then construct a counterexample in ${\mathbb{R}}^4$, where $F$ is strictly convex but $D^{2}F$ degenerates on the intersection of a Simons cone with $S^3$. 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.

arXiv comment18 pages, 3 figures, references added, typos corrected