Hölder stability for Serrin’s overdetermined problem

In a bounded domain Ω, we consider a positive solution of the problem Δu+f(u)=0 in Ω, u=0 on ∂Ω, where f:ℝ→ℝ is a locally Lipschitz continuous function. Under sufficient conditions on Ω (for instance, if Ω is convex), we show that ∂Ω is contained in a spherical annulus of radii ri<re, where re−ri≤C[uν]α∂Ω for some constants C>0 and α∈(0,1]. Here, [uν]∂Ω is the Lipschitz seminorm on ∂Ω of the normal derivative of u. This result improves to H\"older stability the logarithmic estimate obtained in [1] for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the H\"older estimate obtained in [6] for the case of torsional rigidity (f≡1) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones.