Alexandrov, Serrin, Weinberger, Reilly: simmetry and stability by integral identities

TL;DR

This paper reviews classical and recent proofs of symmetry and stability in overdetermined boundary value problems, highlighting their implications and the role of integral identities in establishing these results.

Contribution

It provides an overview of influential proofs related to symmetry and stability, emphasizing the development of new methods using integral identities.

Findings

Classical proofs of the Soap Bubble Theorem and radial symmetry are summarized.

Recent proofs reveal pathways to stability results in boundary value problems.

Integral identities are key tools in establishing symmetry and stability.

Abstract

The distinguished names in the title have to do with influential proofs of the celebrated Soap Bubble Theorem and of radial symmetry in certain overdetermined boundary value problems. We shall give an overeview of those results and indicate some of their ramifications. We will also show how more recent proofs uncover the path to some stability results for the relevant problems.