H\uf6lder stability for Serrin\u2019s overdetermined problem

In a bounded domain \u3a9, we consider a positive solution of the problem \u394u+f(u)=0 in \u3a9, u=0 on 02\u3a9, where f:\u211d\u2192\u211d is a locally Lipschitz continuous function. Under sufficient conditions on \u3a9 (for instance, if \u3a9 is convex), we show that 02\u3a9 is contained in a spherical annulus of radii ri0 and \u3b1 08(0,1]. Here, [u\u3bd] 02\u3a9 is the Lipschitz seminorm on 02\u3a9 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 611) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones