Soap bubbles and convex cones: optimal quantitative rigidity

Abstract

We consider a class of recent rigidity results in a convex cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N . These include overdetermined Serrin-type problems for a mixed boundary value problem relative to Σ \Sigma , Alexandrov’s soap bubble-type results relative to Σ \Sigma , and Heintze-Karcher’s inequality relative to Σ \Sigma . Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L 2 L^2 -pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case Σ = R N \Sigma = \mathbb {R}^N .

Stability bounds in terms of the Hausdorff distance are also provided.

Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone Σ \Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in ∂ Σ \partial \Sigma .

We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N , which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case Σ = R N \Sigma =\mathbb {R}^N , these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).