Hyperbolic solutions to Bernoulli’s free boundary problem
Hyperbolic solutions to Bernoulli’s free boundary problem
复制标题
伯努利自由边界问题的双曲解
DOI：
10.1007/s00205-021-01620-z
复制
发表时间：
2021
影响因子：
2.5
通讯作者：
M. Onodera
中科院分区：
文献类型：
--
作者：
A. Henrot;M. Onodera
伯努利的自由边界问题是一个超定问题，其中人们寻求一个环形域，使得电容势满足额外的边界条件。存在两种不同类型的解，称为椭圆解和双曲解。椭圆解是“稳定”解，可以通过超解法和子解法、变分法和纳什-莫泽隐函数定理来处理，而双曲解是“不稳定”解，其定性行为鲜为人知。我们引入了一个基于抛物线最大正则性的新隐函数定理，该定理适用于导数损失问题。在该方法中，叶状双曲解和椭圆解的存在性被简化为非局部几何流的可解性，并且后者是通过调和分析阐明线性化算子的谱结构来建立的。
Bernoulli’s free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are “stable” solutions and tractable by the super and subsolution method, variational methods and the implicit function theorem of Nash–Moser, while hyperbolic solutions are “unstable” solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. In this approach, the existence of foliated hyperbolic solutions as well as elliptic solutions is reduced to the solvability of a non-local geometric flow, and the latter is established by clarifying the spectral structure of the linearized operator by harmonic analysis.