The classical overdetermined Serrin problem

TL;DR

This survey reviews various methods for solving the classical overdetermined Serrin problem, highlighting historical approaches, recent alternatives, and their respective advantages and limitations.

Contribution

It provides a comprehensive comparison of multiple approaches to the Serrin problem, including classical, elementary, duality, shape derivative, and integral methods.

Findings

Classical proof uses Alexandrov's moving plane method.

Alternative approaches avoid maximum principles.

Different methods have unique advantages and generalizations.

Abstract

In this survey we consider the classical overdetermined problem which was studied by Serrin in 1971. The original proof relies on Alexandrov's moving plane method, maximum principles, and a refinement of Hopf's boundary point Lemma. Since then other approaches to the same problem have been devised. Among them we consider the one due to Weinberger which strikes for the elementary arguments used and became very popular. Then we discuss also a duality approach involving harmonic functions, a shape derivative approach and a purely integral approach, all of them not relying on maximum principle. For each one we consider pros and cons as well as some generalizations.