Hölder stability for Serrin’s overdetermined problem

Abstract

In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) 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 (\(f\equiv 1\)) 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.