The classical overdetermined Serrin problem

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.

Keywords:

• Overdetermined problems
• moving planes
• p-functions
• maximum principle

AMS Subject Classifications:

• 35N25
• 35B50
• 26D15

Acknowledgements

The authors sincerely thank the reviewer for constructive criticisms and valuable comments, which were of great help in revising the manuscript.

Notes

No potential conflict of interest was reported by the authors.