论文标题
Neumann特征值的最大化
Maximization of Neumann eigenvalues
论文作者
论文摘要
本文是由$ {\ Mathbb r}^n $的neumann边界条件的$ k $ th特征值的最大化，并具有规定的度量。我们将（可能是退化）密度（\ Mathbb r}^n $带有规定的质量的（可能是退化）密度的类别的问题，并证明存在最佳密度。对于$ k = 1,2 $，这两个问题是等效的，最大化器分别是一个和两个相等的球。对于$ k \ ge 3 $，此问题仍然打开，除了在空间的一个维度上，我们证明最大密度对应于$ k $相等的段的结合。该结果为Sturm-Liouville特征值提供了尖锐的上限，并证明了Pólya猜想在$ {\ Mathbb r} $中的密度类别中的有效性。基于放松的公式，我们提供了$ k = 1，\ dots，$ {\ mathbb r}^2 $的最佳密度的数值近似值。
This paper is motivated by the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ${\mathbb R}^N$ with prescribed mass and prove the existence of an optimal density. For $k=1,2$ the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For $k \ge 3$ this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union of $k$ equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the Pólya conjecture in the class of densities in ${\mathbb R}$. Based on the relaxed formulation, we provide numerical approximations of optimal densities for $k=1, \dots, 8$ in ${\mathbb R}^2$.
