Communications in Analysis and Geometry

Volume 27 (2019)

Number 4

Existence and regularity of multivalued solutions to elliptic equations and systems

Pages: 877 – 935



Brian Krummel (Department of Mathematics, University of California at Berkeley)


We construct $C^{1,\mu}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the Laplace equation. This extends work of Simon and Wickramasekera in which they construct a large class of $C^{1,\mu}$ multivalued solutions to the minimal surface 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 are real analytic submanifolds by inductively using Schauder estimates.

Received 12 July 2013

Accepted 25 April 2017

Published 8 October 2019