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Linearized equation and generic regularity in the Alt-Caffarelli problem

Xavier Fernández-Real, Hui Yu

TL;DR

This work analyzes the Alt-Caffarelli free boundary problem through a sharp linearization around homogeneous minimizers. By establishing a uniform lower bound on the principal eigenvalue of the Jacobi operator on spheres and a Harnack inequality for positive Jacobi fields, the authors derive quantitative decay that enables a refined separation of singularities. Leveraging a framework of regularity scales and a superlinear cleaning estimate, they prove that generic boundary data yield analytic free boundaries in dimension six and improve generic bounds on the singular set in higher dimensions. The results advance the conjecture on the critical dimension by showing that, under generic perturbations, the free boundary regularity improves substantially, with implications for Bernstein-type results and broader free boundary theory.

Abstract

For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In general, we improve previous generic Hausdorff dimensions of the singular sets. To achieve this, we analyze positive solutions to the linearized equation around homogeneous minimizers (possibly with singular sections on the sphere). For this equation, we prove a Harnack inequality and establish a dimensional lower bound for its principal eigenvalue.

Linearized equation and generic regularity in the Alt-Caffarelli problem

TL;DR

This work analyzes the Alt-Caffarelli free boundary problem through a sharp linearization around homogeneous minimizers. By establishing a uniform lower bound on the principal eigenvalue of the Jacobi operator on spheres and a Harnack inequality for positive Jacobi fields, the authors derive quantitative decay that enables a refined separation of singularities. Leveraging a framework of regularity scales and a superlinear cleaning estimate, they prove that generic boundary data yield analytic free boundaries in dimension six and improve generic bounds on the singular set in higher dimensions. The results advance the conjecture on the critical dimension by showing that, under generic perturbations, the free boundary regularity improves substantially, with implications for Bernstein-type results and broader free boundary theory.

Abstract

For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In general, we improve previous generic Hausdorff dimensions of the singular sets. To achieve this, we analyze positive solutions to the linearized equation around homogeneous minimizers (possibly with singular sections on the sphere). For this equation, we prove a Harnack inequality and establish a dimensional lower bound for its principal eigenvalue.
Paper Structure (16 sections, 43 theorems, 249 equations)

This paper contains 16 sections, 43 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Generic regularity for the Alt-Caffarelli problem
  3. Principal eigenvalue of the Jacobi equation on the sphere
  4. Harnack inequality for the Jacobi equation
  5. Preliminaries
  6. Minimizers of the Alt-Caffarelli energy
  7. Jacobi equation around global minimizers
  8. Tools and previous results on generic regularity
  9. Regularity scales and Harnack inequality for the Jacobi equation
  10. Definition and basic properties of regularity scales
  11. Harnack inequality for the Jacobi equation
  12. Principal eigenvalue of the Jacobi equation
  13. Lower bound on the principal eigenvalue
  14. Decay of positive Jacobi fields
  15. Separation between ordered minimizers
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{g_t\}_{t\in(-1,1)}$ be an admissible family of boundary dataThe family is required to be continuous for each time and increasing on $\partial B_1$ with respect to $t$ at a linear rate FeG. on $\partial B_1\subset\mathbb{R}^d$. For each $t$, let $u_t$ be a minimizer of EqnAC with $u_t|_{\parti

Theorems & Definitions (75)

  • Conjecture 1
  • Theorem 1.1: Theorem 1.5 in FeY, Theorem 1.4 in FeG
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1: ACV
  • ...and 65 more