A local symmetry result for linear elliptic problems with solutions changing sign
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, p. 551-564

We prove that the only domain Ω such that there exists a solution to the following problem $\Delta u+{\omega }^{2}u=-1$ in Ω, $u=0$ on ∂Ω, and $\frac{1}{|\partial \Omega |}{\int }_{\partial \Omega }{\partial }_{𝐧}u=c$, for a given constant c, is the unit ball ${B}_{1}$, if we assume that Ω lies in an appropriate class of Lipschitz domains.

@article{AIHPC_2011__28_4_551_0,
author = {Canuto, B.},
title = {A local symmetry result for linear elliptic problems with solutions changing sign},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {28},
number = {4},
year = {2011},
pages = {551-564},
doi = {10.1016/j.anihpc.2011.03.005},
zbl = {1242.35182},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2011__28_4_551_0}
}

Canuto, B. A local symmetry result for linear elliptic problems with solutions changing sign. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, pp. 551-564. doi : 10.1016/j.anihpc.2011.03.005. http://www.numdam.org/item/AIHPC_2011__28_4_551_0/

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