Unbounded periodic solutions to Serrin's overdetermined boundary value problem

2016/03/17 by Fall, Mouhamed Moustapha, Minlend, Ignace Aristide, Weth, Tobias · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.05727

Abstract

We study the existence of nontrivial unbounded domains Ω in ℝ^N such that the overdetermined problem -Δu = 1 in Ω, u=0, ∂_νu=\textrmconst on ∂ Ω admits a solution u. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in ℝ^N. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem.

Cited by

• A contractible Schiffer counterexample on the half-sphere
• New Homogeneous Solutions for the One-Phase Free Boundary Problem

Related

• Serrin's overdetermined problem on the sphere
• Nontrivial solutions to Serrin's problem in annular domains
• Serrin's overdetermined problem in rough domains
• Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains
• Overdetermined elliptic problems in nontrivial contractible domains of the sphere