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

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.