Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities

TL;DR

This paper introduces new integral identities linking Serrin's overdetermined problem and Alexandrov's Soap Bubble Theorem, leading to improved stability estimates and revealing a strong analogy between the two classical geometric problems.

Contribution

The paper develops novel integral identities that enhance quantitative stability estimates for both Serrin's problem and Alexandrov's theorem, improving upon existing results.

Findings

Derived new integral identities connecting the two problems.

Achieved better quantitative stability estimates, sometimes optimal.

Highlighted the analogy between Serrin's problem and the Soap Bubble Theorem.

Abstract

We consider Serrin's overdetermined problem for the torsional rigidity and Alexandrov's Soap Bubble Theorem. We present new integral identities, that show a strong analogy between the two problems and help to obtain better (in some cases optimal) quantitative estimates for the radially symmetric configuration. The estimates for the Soap Bubble Theorem benefit from those of Serrin's problem.