H\"older 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 $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $\Omega$ (for instance, if $\Omega$ is convex), we show that $\partial\Omega$ is contained in a spherical annulus of radii $r_i<r_e$, where $r_e-r_i\leq C\,[u_\nu]_{\partial\Omega}^\alpha$ for some constants $C>0$ and $\alpha\in (0,1]$. Here, $[u_\nu]_{\partial\Omega}$ is the Lipschitz seminorm on $\partial\Omega$ 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 ($f\equiv 1$) by means of integral identities. The proof hinges on ideas contained in [1] and uses Carleson-type estimates and improved Harnack inequalities in cones.