Potential Theory on Minimal Hypersurfaces I: Singularities as Martin Boundaries

TL;DR

This paper develops potential theory for elliptic operators on singular minimal hypersurfaces, establishing boundary Harnack inequalities and Martin theory to address boundary value, eigenvalue, and variational problems on these complex spaces.

Contribution

It introduces a novel potential theory framework for singular minimal hypersurfaces using hyperbolic unfoldings, enabling analysis of boundary behavior and classical problems.

Findings

Boundary Harnack inequalities along singular sets

Martin boundary theory for singular hypersurfaces

Solutions to boundary value and eigenvalue problems on singular spaces

Abstract

This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.