Hölder stability for Serrin's overdetermined problem

In a bounded domain $Om$, we consider a positive solution of the problem $De u+f(u)=0$ in $Om$, $u=0$ on $paOm$, where $f:RR oRR$ is a locally Lipschitz continuous function. Under sufficient conditions on $Om$ (for instance, if $Om$ is convex), we show that $paOm$ is contained in a spherical annulus of radii $r_i0$ and $ auin (0,1]$. Here, $[u_ u]_{paOm}$ is the Lipschitz seminorm on $paOm$ of the normal derivative of $u$. This result improves to H"older stability the logarithmic estimate obtained in cite{ABR} for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the H"older estimate obtained in cite{BNST2} for the case of torsional rigidity ($fequiv 1$) by means of integral identities. The proof hinges on ideas contained in cite{ABR} and uses Carleson-type estimates and improved Harnack inequalities in cones.