Book Description

We consider two-valued solutions to elliptic problems, which arise from the study branched minimal submanifolds. Simon and Wickramasekera constructed examples of two-valued solutions to the Dirichlet problem for the minimal surface equation on the cylinder $\mathcal{C} = \breve{B}_1^2(0) \times \mathbb{R}^{n-2}$ with Holder continuity estimates on the gradient assuming the boundary data satisfies a symmetry condition. However, their method was specific to the minimal surface equation. We generalize Simon and Wickramasekera's result to an existence theorems for a more general class elliptic equations and for a class of elliptic systems with small data. In particular, we extend Simon and Wickramasekera's result to the minimal surface system. Our approach uses techniques for elliptic differential equations such as the Leray-Schauder theory and contraction mapping principle, which have the advantage of applying in more general contexts than codimension 1 minimal surfaces. We also show that for two-valued solutions to elliptic equations with real analytic data, the branch set of their graphs are real analytic $(n-2)$-dimensional submanifolds. This is a consequence of using the Schauder estimate for two-valued functions and a technique involving majorants due to Friedman to inductively get estimates on the derivatives of the two-valued solutions.