Hölder stability for Serrin's overdetermined problem

Abstract

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 $r_e-r_i\le C[u_{\nu }{]}_{\partial \Omega }^{\alpha }$ 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 Hölder stability the logarithmic estimate obtained in [1] for Serrin's overdetermined problem. It also extends to a large class of semilinear equations the Hölder 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

Cited by in corpus (2)

• Symmetry and linear stability in Serrin's overdetermined problem via the stability of the parallel surface problem
• Stability analysis of an overdetermined fourth order boundary value problem via an integral identity