# Green functions and the Dirichlet spectrum

### G. Pacelli Bessa

Universidade Federal do Ceará, Fortaleza, Brazil### Vicent Gimeno

Universitat Jaume I, Castelló, Spain### Luquesio Jorge

Universidade Federal do Ceará, Fortaleza, Brazil

## Abstract

This article has results of four types. We show that the first eigenvalue $λ_{1}(Ω)$ 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 $λ_{1}(Ω)=lim_{k→∞}∥G_{k}(f)∥_{L_{2}}/∥G_{k+1}(f)∥_{L_{2}}$ for any $f∈L_{2}(Ω,μ)$, $f>0$. Then, we study the $L_{1}(Ω,μ)$-moment spectrum of $Ω$ 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 $L_{1}(Ω,μ)$-moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum $σ_{rad}(B_{h}(o,r))$ of rotationally invariant geodesic balls $B_{h}(o,r)$ of model manifolds. We prove an identity relating the radial eigenvalues of $σ_{rad}(B_{h}(o,r))$ to an isoperimetric quotient, i.e., $∑1/λ_{i}=∫V(s)/S(s)ds$, $V(s)=vol(B_{h}(o,s))$ and $S(s)=vol(∂B_{h}(o,s))$. We then consider a proper minimal surface $M⊂R_{3}$ and the extrinsic ball $Ω=M∩B_{R_{3}}(o,r)$. We obtain upper and lower estimates for the series $∑λ_{i}(Ω)$ in terms of the volume $vol(Ω)$ and the radius $r$ of the extrinsic ball $Ω$.

## Cite this article

G. Pacelli Bessa, Vicent Gimeno, Luquesio Jorge, Green functions and the Dirichlet spectrum. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 1–36

DOI 10.4171/RMI/1119