Interpolating estimates with applications to some quantitative symmetry results

TL;DR

This paper develops interpolating estimates linking a function's oscillation to its gradient's L^p norms, with applications to stability results in geometric analysis and boundary value problems.

Contribution

It introduces explicit, general interpolating estimates based on Riesz potentials, applicable to Lipschitz domains, and demonstrates their use in stability proofs for classical geometric theorems.

Findings

Explicit bounds for function oscillation in terms of gradient norms

Application to stability of Alexandrov's Soap Bubble Theorem

Application to Serrin's overdetermined boundary value problem

Abstract

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $L^p$ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.