Serrin's overdetermined problem on the sphere

2016/12/12 by Fall, Mouhamed Moustapha, Minlend, Ignace Aristide, Weth, Tobias · 4 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1612.03717

Abstract

We study Serrin's overdetermined boundary value problem -Δ_S^N u=1 in Ω, u=0, ∂_ηu=\textrmconst on ∂ Ω in subdomains Ω of the round unit sphere S^N ⊂ ℝ^N+1, where Δ_S^N denotes the Laplace-Beltrami operator on S^N. A subdomain Ω of S^N is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in S^N, N ≥ 2 which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in S^N which are not bounded by geodesic spheres.

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