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 C1 boundary interface are H\"older continuous at the interface.

To appear in Indiana University Mathematics Journal