Search results for "Serrin’s problem"
showing 2 items of 2 documents
Hölder stability for Serrin’s overdetermined problem
2015
In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–93…
A note on Serrin's overdetermined problem
2014
We consider the solution of the torsion problem $$−Δu = N \quad\mathrm{in}\quad Ω,\quad u = 0\quad\mathrm{on}\quad ∂Ω,$$ where Ω is a bounded domain in RN. ¶ Serrin's celebrated symmetry theorem states that, if the normal derivative uν is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate $$r_e − r_i ≤ C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant Ct depending on t, where re and ri are the radii of an annulus containing ∂Ω and Γt is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly…