Revista Matemática Iberoamericana

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Volume 36, Issue 1, 2020, pp. 1–36
DOI: 10.4171/rmi/1119

Published online: 2019-08-05

Green functions and the Dirichlet spectrum

G. Pacelli Bessa[1], Vicent Gimeno[2] and Luquesio Jorge[3]

(1) Universidade Federal do Ceará, Fortaleza, Brazil
(2) Universitat Jaume I, Castelló, Spain
(3) Universidade Federal do Ceará, Fortaleza, Brazil

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 $\sigma^{\rm rad}(B_{h}(o,r))$ of rotationally invariant geodesic balls $B_{h}(o,r)$ of model manifolds. We prove an identity relating the radial eigenvalues of $\sigma^{\rm rad}(B_{h}(o,r))$ to an isoperimetric quotient, i.e., $\sum 1/\lambda_{i}^{\rm rad} = \int V(s)/S(s) ds$, $V(s)={\rm vol}(B_{h}(o,s))$ and $S(s)={\rm vol}(\partial B_{h}(o,s))$. 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 ${\rm vol}(\Omega)$ and the radius $r$ of the extrinsic ball $\Omega$.

Keywords: Dirichlet spectrum, radial spectrum, Green functions, rotationally invariant balls

Bessa G. Pacelli, Gimeno Vicent, Jorge Luquesio: Green functions and the Dirichlet spectrum. Rev. Mat. Iberoam. 36 (2020), 1-36. doi: 10.4171/rmi/1119