Riesz平均的微分不等式与特征值的Weyl型界
Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues
摘要：
得到了Dirichlet Laplacian特征值Riesz平均的微分不等式和差分不等式，R_σ（z）：= \sum_k{(z -λ_k)_+^σ} .这里${λ_k}_{k=1}^{\infty}$是有界域$Ω\subset \R^d$上拉普拉斯算子的有序特征值，$x_+ := \max(0, x)$表示量$x$的正部分。作为这些不等式的推论，我们推导出Weyl型的$λ_k$上的界限，平均值，如$\bar{λ_k} := {\frac 1 k}\sum_{\ell \le k}λ_\ell$，和特征值计数功能。例如，我们证明对于所有域和所有$k \ge j \frac{1+\frac d 2}{1+\frac d 4}$，{ \bar{λ_{k}} }/{ \bar{λ_{j}} } \le 2（\frac{1+\frac d 4}{1+\frac d 2}）^{1+ \frac 2 d}（{ \frac k j}）^{ \frac 2 d}。
英文摘要：
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_σ(z) := \sum_k{(z -λ_k)_+^σ}. Here ${λ_k}_{k=1}^{\infty}$ are the ordered eigenvalues of the Laplacian on a bounded domain $Ω\subset \R^d$, and $x_+ := \max(0, x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on $λ_k$, on averages such as $\bar{λ_k} := {\frac 1 k}\sum_{\ell \le k}λ_\ell$, and on the eigenvalue counting function. For example, we prove that for all domains and all $k \ge j \frac{1+\frac d 2}{1+\frac d 4}$, {\bar{λ_{k}}}/{\bar{λ_{j}}} \le 2 (\frac{1+\frac d 4}{1+\frac d 2})^{1+\frac 2 d}({\frac k j})^{\frac 2 d}.
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