2017/11/29 by Carlo Nitsch, Cristina Trombetti, Nitsch, C. +1 · 1 citation
Computer Science · Mathematics · #26D15 #35B50 #35N25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1711.10787
openalex publication_date 2017/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.