# Minimizers of convex functionals with small degeneracy set

## TL;DR

This paper investigates the regularity of Lipschitz minimizers of convex functionals, confirming $C^1$ regularity under certain conditions and providing a counterexample in higher dimensions, with connections to classical geometry.

## Contribution

It extends regularity results for convex functionals with small degeneracy sets and constructs a counterexample in four dimensions, linking geometric analysis and differential geometry.

## Key findings

- $C^1$ regularity confirmed when $D^2F$ is positive away from finitely many points in a 2-plane
- Counterexample in $ extbf{R}^4$ with degeneracy on a Simons cone intersection
- Connection established between 3D case and Alexandrov's classical differential geometry

## 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 $D^2F$ 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 $D^2F$ 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.