一类多扰动Serrin问题的稳定性

Stability of a class of a Serrin′s problem with multiple perturbations

本文研究了一个具有未知子域及边界条件有扰动的Serrin超定问题的稳定性.根据度量域与球的偏差,从而建立了一个定量的稳定性估计.利用Rellich-Pohozaev-type型积分等式,证明当未知子域的勒贝格测度小且边界上的法向导数趋于一个常数时,域在几何上接近于一个球.

In this paper, we investigate the stability of a Serrin-type overdetermined problem with perturbations in the unknown subdomain and boundary conditions. A quantitative stability estimate is established by measuring the deviation between the domain and a sphere. We prove that when the subdomain has a small Lebesgue measure and the outward normal derivative on the boundary approaches a constant, the domain is geometrically close to a sphere. The proof is based on a RellichPohozaev-type integral identity.