H\"older stability for Serrin's overdetermined problem

In a bounded domain $\mathrm{\Omega }$, we consider a positive solution of the problem $\mathrm{\Delta }u+f(u)=0$ in $\mathrm{\Omega }$, $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$, where $f:\mathbb{R}\to \mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $\mathrm{\Omega }$ (for instance, if $\mathrm{\Omega }$ is convex), we show that $\mathrm{\partial }\mathrm{\Omega }$ is contained in a spherical annulus of radii $r_i<r_e$, where $r_e-r_i\le C[u_{\nu }{]}_{\mathrm{\partial }\mathrm{\Omega }}^{\alpha }$ for some constants $C>0$ and $\alpha \in (0,1]$. Here, $[u_{\nu }{]}_{\mathrm{\partial }\mathrm{\Omega }}$ is the Lipschitz seminorm on $\mathrm{\partial }\mathrm{\Omega }$ 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.