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

Max Engelstein, Daniel Restrepo, Zihui Zhao

TL;DR

This work analyzes the one-phase Bernoulli free boundary problem, focusing on solutions modeled by one-homogeneous cones with isolated singularities. By combining Weiss energy methods with a careful linearization around a general one-homogeneous cone and a boundary-analytic extension, the authors prove a uniqueness of blowups under integrability through rotations and a rigidity of blowdowns at infinity under strong integrability, extending Allard–Almgren and Simon–Solomon-type results to non-minimizing free boundary problems. Central to the results is a detailed spectral analysis of the De Silva–Jerison cone, which underpins the quantitative convergence rates and the DSJS foliation-based rigidity statement. The findings yield the first explicit blow-up/blow-down uniqueness in the Bernoulli free boundary context for non-minimizing solutions and illuminate the asymptotic structure of global solutions, with implications for the stability and classification of free boundaries in singular settings.

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 (à la Allard--Almgren) and we prove a rigidity result at infinity (à 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.

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

TL;DR

This work analyzes the one-phase Bernoulli free boundary problem, focusing on solutions modeled by one-homogeneous cones with isolated singularities. By combining Weiss energy methods with a careful linearization around a general one-homogeneous cone and a boundary-analytic extension, the authors prove a uniqueness of blowups under integrability through rotations and a rigidity of blowdowns at infinity under strong integrability, extending Allard–Almgren and Simon–Solomon-type results to non-minimizing free boundary problems. Central to the results is a detailed spectral analysis of the De Silva–Jerison cone, which underpins the quantitative convergence rates and the DSJS foliation-based rigidity statement. The findings yield the first explicit blow-up/blow-down uniqueness in the Bernoulli free boundary context for non-minimizing solutions and illuminate the asymptotic structure of global solutions, with implications for the stability and classification of free boundaries in singular settings.

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 (à la Allard--Almgren) and we prove a rigidity result at infinity (à 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.

Paper Structure

This paper contains 10 sections, 20 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proofs and Comparison with minimal surfaces
  3. Organization of the paper.
  4. Some properties of the linearized operator
  5. Particular solutions of the linearized operator
  6. Spectrum of the De Silva-Jerison cone
  7. Qualitative properties of solutions close to regular cones
  8. Rates of convergence of solutions asymptotic to integrable cones: Proof of Theorem \ref{['thm AA']}
  9. Classification of global solutions asymptotic to strongly integrable cones: Proof of Theorem \ref{['thm main']}
  10. Appendix

Key Result

Theorem 1.1

Let $u: D \subset \mathbb{R}^d\to \mathbb{R}_+$ be a non-degenerate weak solution to the one phase Bernoulli problem (in the sense of Definition d:weaksol). Suppose $0 \in \partial \{u > 0\}$ and assume that there exist $r_j \downarrow 0$ such that $u(r_j x)/r_j \rightarrow U$ in $L^2_{\mathrm{loc}} for all $r < 1$. Similarly, with the same assumptions on $U$ if $u(r_jx)/r_j \rightarrow U$ in $L^2

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2: Theorem 1 of DSJS
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Integrability
  • Remark 2.4
  • Definition 2.5: Strong integrability
  • Lemma 2.6
  • proof
  • ...and 41 more