In a bounded domain $��$, we consider a positive solution of the problem $��u+f(u)=0$ in $��$, $u=0$ on $\partial��$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $��$ (for instance, if $��$ is convex), we show that $\partial��$ is contained in a spherical annulus of radii $r_i0$ and $��\in (0,1]$. Here, $[u_��]_{\partial��}$ is the Lipschitz seminorm on $\partial��$ of the normal derivative of $u$. This result improves to H��lder stability the logarithmic estimate obtained in [1] for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the H��lder estimate obtained in [6] for the case of torsional rigidity ($f\equiv 1$) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones.

14 pages, 2 figures