Dirichlet spectrum and Green function

TL;DR

This paper explores the relationship between the Dirichlet spectrum, Green functions, and geometric properties of domains, providing new identities, estimates, and explicit formulas for eigenvalues and spectra in various settings.

Contribution

It introduces novel identities and estimates linking spectral properties with geometric and Green function analyses, extending previous results to weighted and extrinsic domains.

Findings

Derived an identity relating radial spectrum and isoperimetric quotient.

Provided bounds for the series of inverse eigenvalues for minimal surface balls.

Explicitly characterized the $L^{1}$-momentum spectrum and first eigenvalue in terms of Green operators.

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 $\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. 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.