论文标题
Steklov-Dirichlet Spectrum：特征值的稳定性，优化和连续性
Steklov-Dirichlet spectrum: stability, optimization and continuity of eigenvalues
论文作者
论文摘要
在本文中，我们研究了steklov-dirichlet eigenvalues $λ_k（ω，γ_s）$，其中$ω\ subset \ mathbb {r}^d $是一个域，$γ_s\ subset \ subset \ subset \ subset \ partialω$是边界的子集。在首先讨论了steklov-dirichlet本征函数的规律性特性之后，我们获得了特征值的稳定性结果。我们研究了对集合$γ_S$的度量约束下的优化问题，我们证明了最小化器的存在和最大化器的不存在。在飞机上，我们证明了特征值对序列$γ_{s，n} $的连接组件的数量施加的连续性结果，以这种方式获得了steklov-dirichlet eigenvalues的V. sverak的著名结果。使用此结果，我们证明了在相同的拓扑约束和度量约束下的最大化器的存在。
In this paper we study the Steklov-Dirichlet eigenvalues $λ_k(Ω,Γ_S)$, where $Ω\subset \mathbb{R}^d$ is a domain and $Γ_S\subset \partial Ω$ is the subset of the boundary in which we impose the Steklov conditions. After a first discussion about the regularity properties of the Steklov-Dirichlet eigenfunctions we obtain a stability result for the eigenvalues. We study the optimization problem under a measure constraint on the set $Γ_S$, we prove the existence of a minimizer and the non-existence of a maximizer. In the plane we prove a continuity result for the eigenvalues imposing a bound on the number of connected components of the sequence $Γ_{S,n}$, obtaining in this way a version of the famous result of V. Sverak for the Steklov-Dirichlet eigenvalues. Using this result we prove the existence of a maximizer under the same topological constraint and the measure constraint.
