Green functions and the Dirichlet spectrum

Abstract

This article has results of four types. We show that the first eigenvalue ${\lambda }_1(\Omega )$ of the weighted Laplacian of a bounded domain with smooth boundary can be obtained by S. Sato's iteration scheme of the Green operator, taking the limit ${\lambda }_1(\Omega )={\lim }_{k\to \infty }\Vert G^k(f){\Vert }_{L^2}/\Vert G^{k+1}(f){\Vert }_{L^2}$ for any $f\in L^2(\Omega ,\mu )$, $f>0$. Then, we study the $L^1(\Omega ,\mu )$-moment spectrum of $\Omega$ in terms of iterates of the Green operator $G$, extending the work of McDonald–Meyers to the weighted setting. As corollary, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^1(\Omega ,\mu )$-moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum ${\sigma }^{\mathrm{rad}}(B_h(o,r))$ of rotationally invariant geodesic balls $B_h(o,r)$ of model manifolds. We prove an identity relating the radial eigenvalues of ${\sigma }^{\mathrm{rad}}(B_h(o,r))$ to an isoperimetric quotient, i.e., $\sum 1/{\lambda }_i^{\mathrm{rad}}=\int V(s)/S(s)ds$, $V(s)=\mathrm{vol}(B_h(o,s))$ and $S(s)=\mathrm{vol}(\partial B_h(o,s))$. We then consider a proper minimal surface $M\subset {\mathbb{R}}^3$ and the extrinsic ball $\Omega =M\cap B_{{\mathbb{R}}^3}(o,r)$. We obtain upper and lower estimates for the series $\sum {\lambda }_i^{-2}(\Omega )$ in terms of the volume $\mathrm{vol}(\Omega )$ and the radius $r$ of the extrinsic ball $\Omega$.