Existence and regularity of multivalued solutions to elliptic equations and systems

Abstract

We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,\mu}$ multivalued solutions to the minimal surface equation, to produce $C^{1,\mu}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the Laplace equation. We use methods for differential equations, which are more general than the specific minimal submanifold approach adopted by Simon and Wickramasekera. We also prove the branch set of the graphs of the solutions constructed by Simon and Wickramasekera are real analytic submanifolds by inductively using Schauder estimates.