Stability analysis of an overdetermined fourth order boundary value problem via an integral identity
Stability analysis of an overdetermined fourth order boundary value problem via an integral identity
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基于积分恒等式的超定四阶边值问题的稳定性分析
DOI：
10.1016/j.jde.2021.08.017
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发表时间：
2021
影响因子：
2.4
通讯作者：
Onodera Michiaki
中科院分区：
文献类型：
--
作者：
Okamoto Yuya;Onodera Michiaki
考虑一类超定四阶边值问题，其中解的拉普拉斯算子的边值除了齐次Dirichlet边界条件外，还被给定。众所周知，在给定的边界值是一个常数的情况下，这个超定问题有一个解决方案，当且仅当所考虑的区域是一个球。本文研究了当给定的边值从一个常数作微小扰动时，超定问题的解存在的区域的形状。我们得到了一个积分恒等式的四阶Dirichlet问题和一个非线性加权迹不等式，和它们的组合的结果在一个定量的稳定性估计，测量的偏差从一个球的域的边值的扰动。
We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.