Volume-constrained minimizers for the prescribed curvature problem in periodic media

TL;DR

This paper proves the existence of volume-constrained minimizers for the prescribed mean curvature problem in periodic media, constructs approximate solutions, and shows their convergence to a Wulff Shape as volume increases.

Contribution

It establishes the existence of minimizers under volume constraints in periodic media and analyzes their asymptotic behavior, which was not previously known.

Findings

Existence of compact minimizers in periodic media.

Construction of approximate solutions to the prescribed mean curvature equation.

Convergence of minimizers to a Wulff Shape as volume tends to infinity.

Abstract

We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a suitable Wulff Shape, when the volume tends to infinity.