Dirichlet energy-minimizers with analytic boundary

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 Hölder continuous at the interface.

In this version we add a new section 8 to further analyze the exceptional case and show it is indeed Dir-minimizing. We also add a (trivial) missing case in Proposition 6.1 and include discussions in that regard thereafter