On the asymptotic properties of solutions to one-phase free boundary problems

TL;DR

This paper investigates the structure of solutions to the one-phase Bernoulli free boundary problem, establishing uniqueness and rigidity results for solutions modeled by one-homogeneous solutions with isolated singularities, under symmetry and linearized operator conditions.

Contribution

It provides the first known uniqueness of blow-up and blow-down results at singular points for non-minimizing solutions in the one-phase free boundary problem.

Findings

Proves uniqueness of blowups under symmetry conditions.

Establishes rigidity at infinity under linearized operator constraints.

First results of their kind for non-minimizing solutions.

Abstract

In this article we study the structure of solutions to the one-phase Bernoulli problem that are modeled either infinitesimally or at infinity by one-homogeneous solutions with an isolated singularity. In particular, we prove a uniqueness of blowups result under a natural symmetry condition on the one-homogeneous solution (\`a la Allard--Almgren) and we prove a rigidity result at infinity (\`a la Simon--Solomon) under additional constraints on the linearized operator around the one-homogeneous solution (which are satisfied by the only known examples of minimizing one-homogeneous solutions). We believe these are the first uniqueness of blow-up/blow-down results at singular points for non-minimizing solutions to the one-phase problem.