Dirichlet energy-minimizers with analytic boundary

TL;DR

This paper studies multi-valued energy-minimizing graphs with real analytic interfaces, proving the discreteness of singularities in 2D and regularity at the boundary in higher dimensions, advancing understanding of area minimizing currents.

Contribution

It establishes the discreteness of singularities for Dirichlet energy-minimizers with real analytic boundaries in 2D, confirming a conjecture by B. White.

Findings

Singular set is discrete in 2D

Energy-minimizers are Hölder continuous at the boundary in any dimension

Supports a conjecture on finite singularities for area minimizing currents

Abstract

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and provides a first step to) a conjecture by B. White \cite{White97} that area minimizing $2$-dimensional currents with real analytic boundaries have a finite number of singularities. We also show that, in any dimension, Dirichlet energy-minimizers with a $C^1$ boundary interface are H\"older continuous at the interface.