Stability inequalities for one-phase cones

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.