Overdetermined elliptic problems in nontrivial contractible domains of the sphere

In this paper, we prove the existence of nontrivial contractible domains Omega subset of S-d, d >= 2, such that the overdetermined elliptic problem {-epsilon Delta(g)u + u - u(p) = 0 in Omega, u > 0 in Omega, u = 0 on partial derivative Omega, partial derivative(nu)u = constant on partial derivative Omega, admits a positive solution. Here Delta(g) is the Laplace-Beltrami operator in the unit sphere S-d with respect to the canonical round metric g, epsilon > 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. (c) 2023 Elsevier Masson SAS. All rights reserved.