Overdetermined elliptic problems in nontrivial contractible domains of the sphere.

In this paper, we prove the existence of nontrivial contractible domains Ω ⊂ S d , d ≥ 2 , such that the overdetermined elliptic problem { − ε Δ g u + u − u p = 0 in Ω, u > 0 in Ω, u = 0 on ∂Ω, ∂ ν u = constant on ∂Ω, admits a positive solution. Here Δ g is the Laplace-Beltrami operator in the unit sphere S d with respect to the canonical round metric g , ε > 0 is a small real parameter and 1 < p < d + 2 d − 2 (p > 1 if d = 2). These domains are perturbations of S d ∖ D , where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains. [ABSTRACT FROM AUTHOR]