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Stability inequalities for one-phase cones

Benjy Firester, Raphael Tsiamis, Yipeng Wang

TL;DR

This work establishes strict stability for the homogeneous one-phase Bernoulli cones with $O(n-k)\times O(k)$ symmetry in dimensions $n\ge7$, showing that all axisymmetric cones $U_{n,k}$ are strictly stable and deriving a bound on the first eigenvalue together with Jacobi-field decay rates. The analysis reduces stability to a one-dimensional Robin eigenvalue problem on the spherical link and exploits a detailed hypergeometric-function framework for the cone profiles $f_{n,k}$, complemented by Riccati-type subsolution constructions to produce global barriers. Consequently, the results yield implications for generic regularity of the Alt-Caffarelli one-phase problem and related free boundary models (capillary, fractional, Alt-Phillips), linking stability to minimization and rigidity phenomena. Furthermore, the paper provides concrete estimates for indicial roots and first-eigenvalue behavior across the $U_{n,k}$ family, and lays groundwork for a conjecture that the maximal eigenvalue is attained at $k=n-2$, with further validation forthcoming in related work.

Abstract

We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension $7$ and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence, we obtain a bound on the first eigenvalue and the decay rates of Jacobi fields, with applications to the generic regularity of the one-phase problem.

Stability inequalities for one-phase cones

TL;DR

This work establishes strict stability for the homogeneous one-phase Bernoulli cones with symmetry in dimensions , showing that all axisymmetric cones are strictly stable and deriving a bound on the first eigenvalue together with Jacobi-field decay rates. The analysis reduces stability to a one-dimensional Robin eigenvalue problem on the spherical link and exploits a detailed hypergeometric-function framework for the cone profiles , complemented by Riccati-type subsolution constructions to produce global barriers. Consequently, the results yield implications for generic regularity of the Alt-Caffarelli one-phase problem and related free boundary models (capillary, fractional, Alt-Phillips), linking stability to minimization and rigidity phenomena. Furthermore, the paper provides concrete estimates for indicial roots and first-eigenvalue behavior across the family, and lays groundwork for a conjecture that the maximal eigenvalue is attained at , with further validation forthcoming in related work.

Abstract

We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence, we obtain a bound on the first eigenvalue and the decay rates of Jacobi fields, with applications to the generic regularity of the one-phase problem.
Paper Structure (15 sections, 17 theorems, 217 equations, 1 table)

This paper contains 15 sections, 17 theorems, 217 equations, 1 table.

Table of Contents

  1. Introduction
  2. Generic regularity
  3. Applications to other problems
  4. Preliminaries
  5. Stability of solutions
  6. O(n-k)xO(k)-invariant one-phase cones
  7. Hypergeometric functions
  8. Analysis of the linearized problem
  9. Propagation of positivity
  10. Estimates on the roots
  11. The admissible interval
  12. Near-endpoint k
  13. Intermediate k
  14. Large k
  15. Proof of the Main Results

Key Result

Theorem 1.1

For every $n \geq 3$ and $1 \leq k \leq n-2$, there exists a unique homogeneous $O(n-k) \times O(k)$-invariant solution of the one-phase problem, given by with free boundary given by the cone $\Gamma_{n,k} = \{ (x,y) : |y| \leq t_{n,k} \sqrt{|x|^2+ |y|^2} \}$. For every $n \leq 6$ and $1 \leq k \leq n-2$, this solution is unstable; for every $n \geq 7$ and $1 \leq k \leq n-2$, it is strictly stab

Theorems & Definitions (35)

  • Theorem 1.1
  • Proposition 1.2
  • Conjecture 1.3
  • Proposition 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 25 more