Dirichlet spectrum and Green function
Dirichlet spectrum and Green function
复制标题
狄利克雷谱和格林函数
DOI：
10.4171/rmi/1119
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
L. Jorge
中科院分区：
文献类型：
--
作者：
G. P. Bessa;V. Gimeno;L. Jorge
本文第一部分得到了旋转不变测地球径向谱与等周商$\sum 1/\lambda_(I)^{\rmrad}=\int V(S)/S(S)ds$之间的一个恒等式。我们还得到了序列$\sum_{i}^{-2}(\omega)$的上下界估计，其中$\omega$是$\mathbb{R}^{3}$的真极小曲面的外球。在第二部分中，我们证明了有界域的第一本征值是通过迭代格林算子并取极限$\lambda_{1}(\ommega)=\Lim_{k\to\inty}\Vert G^k(F)\Vert_{2}/\Vert G^{k+1}(F)\Vert_{2}$给出的。在第三部分中，我们用格林算子显式地得到了有界域$\Omega$的$L^{1}(\Omega，\Mu)$-动量谱。特别地，我们利用$L^{1}(欧米伽，µ)动量谱得到了加权有界域的第一本征值，推广了Hurtado-Markvorsen-Palmer关于旋转不变球的第一本征值的工作。
In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum \lambda_{i}^{-2}(\Omega)$ where $\Omega$ is an extrinsic ball of a proper minimal surface of $\mathbb{R}^{3}$. In the second part we show that the first eigenvalue of bounded domains is given by iteration of the Green operator and taking the limit, $\lambda_{1}(\Omega)=\lim_{k\to \infty} \Vert G^k(f)\Vert_{2}/\Vert G^{k+1}(f)\Vert_{2}$ for any function $f>0$. In the third part we obtain explicitly the $L^{1}(\Omega, \mu)$-momentum spectrum of a bounded domain $\Omega$ in terms of its Green operator. In particular, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(\Omega, \mu)$-momentum spectrum, extending the work of Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant balls.