# On a free boundary problem and minimal surfaces

## TL;DR

This paper constructs new solutions for a free boundary problem inspired by minimal surfaces, demonstrating optimality of Savin's theorem in dimension 8 through variational methods.

## Contribution

It introduces a refined Lyapunov-Schmidt reduction approach to generate solutions with two-component free boundaries and proves their minimality in dimension 8.

## Key findings

- New solutions with two-component free boundaries derived from minimal surfaces.
- Solutions in dimension 8 are proven to be global energy minimizers.
- Savin's theorem is shown to be optimal in this context.

## Abstract

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.