Stability inequalities for one-phase cones

TL;DR

This paper establishes strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem, especially in dimension 7 and above, with implications for regularity and eigenvalue bounds.

Contribution

It introduces new strict stability inequalities for one-phase Bernoulli solutions and characterizes stability of cohomogeneity one solutions with bi-orthogonal symmetry in high dimensions.

Findings

Cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable in dimension 7 and above.

Derived bounds on the first eigenvalue of the stability operator.

Established decay rates for Jacobi fields related to the problem.

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.