Hölder stability for Serrin’s overdetermined problem

In a bounded domain Omega, we consider a positive solution of the problem Delta u + f (u) = 0 in Omega, u = 0 on partial derivative Omega, where f : R -> R is a locally Lipschitz continuous function. Under sufficient conditions on Omega (for instance, if Omega is convex), we show that partial derivative Omega is contained in a spherical annulus of radii r(i) < r(e), where r(e) - r(i) <= C [u(nu)](partial derivative Omega)(t) for some constants C > 0 and tau is an element of (0, 1]. Here, [u(nu)](partial derivative Omega) is the Lipschitz seminorm on partial derivative Omega of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907-932, 1999) for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the Holder estimate obtained in Brandolini et al. (J Differ Equ 245: 1566-1583, 2008) for the case of torsional rigidity (f equivalent to 1) by means of integral identities. The proof hinges on ideas contained in Aftalion et al. (1999) and uses Carleson-type estimates and improved Harnack inequalities in cones.