A free boundary problem with facets

TL;DR

This paper investigates a lattice-based free boundary problem with a discontinuous Hamiltonian, providing explicit formulas, proving solution uniqueness, and demonstrating the presence of facets in all rational directions, motivated by physical experiments.

Contribution

It introduces a new free boundary problem with a discontinuous Hamiltonian, derives an explicit formula, and develops a novel uniqueness proof for such problems.

Findings

Explicit formula for the Hamiltonian.

Solutions are unique.

Facets appear in every rational direction.

Abstract

We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove the solutions are unique, and prove that the limiting free boundary has a facets in every rational direction. Our choice of problem presents difficulties that require the development of a new uniqueness proof for certain free boundary problems. The problem is motivated by physical experiments involving liquid drops on patterned solid surfaces.