Geodesic
Nontrivial solutions to Serrin's problem in annular domains
Kamburov, Nikola1 ; Sciaraffia, Luciano1
1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago 7820436, Chile
Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 1-22.

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We construct nontrivial bounded, real analytic domains Ω⊆Rn of the form Ω0∖Ω‾1, bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem

{−Δu=1,u>0 in Ω,u=0,∂νu=const on ∂Ω0,u=const,∂νu=const on ∂Ω1,
where ν stands for the inner unit normal to ∂Ω. From results by Reichel [1] and later by Sirakov [2], it was known that the condition ∂νu≤0 on ∂Ω1 is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, we show that the constructed domains are self-Cheeger.

DOI : 10.1016/j.anihpc.2020.05.001
Classification : 35N25, 37G25, 47A75, 49Q10
Keywords: Overdetermined elliptic problems, Bifurcation methods, Eigenvalues, Cheeger problem

Kamburov, Nikola 1 ; Sciaraffia, Luciano 1

1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago 7820436, Chile 
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Kamburov, Nikola; Sciaraffia, Luciano. Nontrivial solutions to Serrin's problem in annular domains. Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 1-22. doi : 10.1016/j.anihpc.2020.05.001. http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.05.001/

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