Nontrivial solutions to Serrin's problem in annular domains

Kamburov, Nikola; Sciaraffia, Luciano

doi:10.1016/j.anihpc.2020.05.001

Kamburov, Nikola ¹ ; Sciaraffia, Luciano ¹

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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Résumé

We construct nontrivial bounded, real analytic domains $Ω \subseteq R^{n}$ of the form $Ω_{0} ∖ {\overline{Ω}}_{1}$ , bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem

{\begin{matrix} - Δ u = 1, u > 0 & in Ω, \\ u = 0, \partial_{ν} u = const & on \partial Ω_{0}, \\ u = const, \partial_{ν} u = const & on \partial Ω_{1}, \end{matrix}

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

\partial_{ν} u \leq 0

\partial Ω_{1}

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.

MR Zbl

DOI : 10.1016/j.anihpc.2020.05.001

Classification : 35N25, 37G25, 47A75, 49Q10
Mots-clés : Overdetermined elliptic problems, Bifurcation methods, Eigenvalues, Cheeger problem

Affiliations des auteurs :

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://www.numdam.org/articles/10.1016/j.anihpc.2020.05.001/

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