Confirmed answer to the Schiffer conjecture and the Berenstein conjecture

2025/01/06 by Dai, Guowei
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.02724

Abstract

Let Ω be a bounded domain in ℝ^N+1 with a connected C^2,ε (ε∈(0,1)) boundary. We show that, if the following overdetermined elliptic problem -Δu=αu in Ω, u=0 on ∂Ω, (∂ u)/(∂ n) =c on ∂Ωhas a nontrivial solution, then Ω is a ball, which is exactly the affirmative answer to the Berenstein conjecture. Similarly, we show that, if Ω has a Lipschitz connected boundary and the following overdetermined elliptic problem -Δu=αu in Ω, (∂ u)/(∂ n)=0 on ∂Ω, u =c on ∂Ωhas a nontrivial solution, then Ω is also a ball, which is exactly the affirmative answer to the Schiffer conjecture.

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