2024/07/02 by Alessio Figalli, Yi Ru‐Ya Zhang, Figalli, Alessio +1 · 2 citations
Computer Science · #35N25 #Analysis of PDEs (math.AP) #Computational Geometry and Mesh Generation #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2407.02293
openalex publication_date 2024/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Serrin's overdetermined theorem states that a C2 bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin's theorem to Lipschitz domains remained unresolved. This paper answers this open question affirmatively. Actually, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.