Solutions to a two-dimensional, Neumann free boundary problem

ABSTRACT

We explore regularity properties of solutions to a two-phase elliptic free boundary problem near a Neumann fixed boundary in two dimensions. Consider a function u, which is harmonic where it is not zero and satisfies a gradient jump condition weakly along the free boundary. Our main result is that u is Lipschitz continuous up to the Neumann fixed boundary. We also present a numerical exploration of the way in which the free and fixed boundaries interact.

Keywords:

• Free boundary problems
• elliptic regularity

AMS Subject Classifications:

• 35R35
• 35B65
• 35J20
• 35J60
• 35J05
• 35J25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Simons Foundation [grant number 243686].