On the symmetry of minimizers

TL;DR

This paper proves that for many variational problems, minimizers are symmetric if they are continuously differentiable, highlighting a fundamental property of solutions in calculus of variations.

Contribution

It establishes a general symmetry result for minimizers under the condition of being $C^1$, extending previous symmetry theorems.

Findings

Minimizers are symmetric if they are $C^1$ for a broad class of variational problems.

The symmetry property holds under minimal regularity assumptions.

The result applies to various classical and new variational problems.

Abstract

For a large class of variational problems we prove that minimizers are symmetric whenever they are $C^1$.