Min-max construction of minimal surfaces with a fixed angle at the boundary

TL;DR

This paper establishes the existence of minimal surfaces within convex regions in three-dimensional space that meet the boundary at a fixed contact angle, using a min-max approach inspired by Almgren-Pitts theory.

Contribution

It introduces a novel min-max construction method for minimal surfaces with prescribed boundary contact angles in convex domains.

Findings

Proves existence of minimal surfaces with fixed boundary contact angle

Develops a min-max construction method for capillarity problems

Extends Almgren-Pitts framework to capillarity functional

Abstract

We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.