The structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$

TL;DR

This paper characterizes the structure of finite Morse index solutions to two phase transition free boundary problems in the plane, showing they have finitely many ends and exponential convergence, with a key curvature decay estimate.

Contribution

It provides a detailed description of finite Morse index solutions in  0, including their asymptotic behavior and curvature decay, advancing understanding of free boundary problems.

Findings

Solutions have finitely many ends

Solutions converge exponentially to their ends

A quadratic decay estimate for free boundary curvature is established

Abstract

We give a description of the structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$. These free boundary problems are models of phase transition and they are closely related to minimal hypersurfaces. We show that these finite Morse index solutions have finitely many ends and they converge exponentially to these ends at infinity. As an important tool in the proof, a quadratic decay estimate for the curvature of free boundaries is established.