科研之友
微信
新浪微博
Facebook
分享链接
摘要
We consider the solution of the torsion problem %26lt;br%26gt;-Delta u = N in Omega, u = 0 on partial derivative Omega, %26lt;br%26gt;where Omega is a bounded domain in R-N. %26lt;br%26gt;Serrin%26apos;s celebrated symmetry theorem states that, if the normal derivative u(v) is constant on partial derivative Omega, then Omega must be a ball. In [6], it has been conjectured that Serrin%26apos;s theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate %26lt;br%26gt;r(e) - r(i) %26lt;= Ct (max(Gamma i) u - min(Gamma i) u) %26lt;br%26gt;for some constant C-t depending on t, where r(e) and r(i) are the radii of an annulus containing partial derivative Omega and Gamma i is a surface parallel to partial derivative Omega at distance t and sufficiently close to partial derivative Omega; secondly, if in addition u(v) is constant on partial derivative Omega, show that %26lt;br%26gt;max(Gamma i) u - min(Gamma i) u = o(C-t) as t -%26gt; 0(+). %26lt;br%26gt;The estimate constructed in [6] is not sharp enough to achieve this goal. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains Omega are ellipses.
- 出版日期2014-10
