Quantitative stability for the nonlocal overdetermined Serrin problem

TL;DR

This paper develops a new quantitative stability estimate for the nonlocal Serrin overdetermined problem using the moving planes method, introducing innovative tools like an antisymmetric barrier and nonlocal maximum principles.

Contribution

It introduces a novel antisymmetric barrier and a unified approach to the moving planes method, improving stability estimates for the nonlocal problem beyond classical results.

Findings

Stability estimate surpasses classical Laplacian results

New antisymmetric barrier construction

Quantitative nonlocal maximum principles

Abstract

We establish quantitative stability for the nonlocal Serrin overdetermined problem, via the method of the moving planes. Interestingly, our stability estimate is even better than those obtained so far in the classical setting (i.e., for the classical Laplacian) via the method of the moving planes. A crucial ingredient is the construction of a new antisymmetric barrier, which allows a unified treatment of the moving planes method. This strategy allows us to establish a new general quantitative nonlocal maximum principle for antisymmetric functions, leading to new quantitative nonlocal versions of both the Hopf lemma and the Serrin corner point lemma. All these tools -- i.e., the new antisymmetric barrier, the general quantitative nonlocal maximum principle, and the quantitative nonlocal versions of both the Hopf lemma and the Serrin corner point lemma -- are of independent interest.