Existence and qualitative properties of isoperimetric sets in periodic media

TL;DR

This paper reviews and extends recent findings on the existence and properties of isoperimetric sets in periodic media, focusing on minimal surfaces, homogenized surface tension, and their mathematical characteristics.

Contribution

It provides new insights into the existence, strict convexity, and differentiability of the homogenized surface tension in anisotropic periodic media.

Findings

Existence of minimal surfaces in periodic media.

Homogenized surface tension is strictly convex.

Differentiability properties of the stable norm.

Abstract

We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties.