Calculus of Variations and Geometric Measure Theory
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C. De Lellis - Z. Zhao

Dirichlet energy-minimizers with analytic boundary

created by delellis on 27 Jun 2019



Inserted: 27 jun 2019
Last Updated: 27 jun 2019

Year: 2019


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.


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