Dirichlet spectrum and Green function

Abstract

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/{\lambda }_i^{\mathrm{rad}}=\int V(s)/S(s)ds$. 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.

Replacement: We removed few misprints. Comments are welcome