This article has results of four types. We show that the first eigenvalue 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 for any , . Then, we study the -moment spectrum of in terms of iterates of the Green operator , 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 -moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum of rotationally invariant geodesic balls of model manifolds. We prove an identity relating the radial eigenvalues of to an isoperimetric quotient, i.e., , and . We then consider a proper minimal surface and the extrinsic ball . We obtain upper and lower estimates for the series in terms of the volume and the radius of the extrinsic ball .
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G. Pacelli Bessa, Vicent Gimeno, Luquesio Jorge, Green functions and the Dirichlet spectrum. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 1–36DOI 10.4171/RMI/1119