Solutions to a two-dimensional, Neumann free boundary problem

TL;DR

This paper investigates the regularity of solutions to a two-dimensional two-phase elliptic free boundary problem with Neumann boundary conditions, proving Lipschitz continuity up to the fixed boundary and exploring boundary interactions numerically.

Contribution

It establishes Lipschitz regularity of solutions near the Neumann boundary and provides a numerical analysis of free and fixed boundary interactions.

Findings

Solutions are Lipschitz continuous up to the Neumann boundary.

Numerical results illustrate the interaction between free and fixed boundaries.

The study advances understanding of boundary regularity in free boundary problems.

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.