JournalsrmiVol. 36, No. 1pp. 1–36

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
Green functions and the Dirichlet spectrum cover

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This article has results of four types. We show that the first eigenvalue λ1(Ω)\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 λ1(Ω)=limkGk(f)L2/Gk+1(f)L2\lambda_{1}(\Omega)=\lim_{k\to \infty} \Vert G^{k}(f)\Vert_{L^2}/\Vert G^{k+1}(f)\Vert_{L^2} for any fL2(Ω,μ)f\in L^{2}(\Omega, \mu), f>0f > 0. Then, we study the L1(Ω,μ)L^{1}(\Omega, \mu)-moment spectrum of Ω\Omega in terms of iterates of the Green operator GG, 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 L1(Ω,μ)L^{1}(\Omega, \mu)-moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum σrad(Bh(o,r))\sigma^{\rm rad}(B_{h}(o,r)) of rotationally invariant geodesic balls Bh(o,r)B_{h}(o,r) of model manifolds. We prove an identity relating the radial eigenvalues of σrad(Bh(o,r))\sigma^{\rm rad}(B_{h}(o,r)) to an isoperimetric quotient, i.e., 1/λirad=V(s)/S(s)ds\sum 1/\lambda_{i}^{\rm rad} = \int V(s)/S(s) ds, V(s)=vol(Bh(o,s))V(s)={\rm vol}(B_{h}(o,s)) and S(s)=vol(Bh(o,s))S(s)={\rm vol}(\partial B_{h}(o,s)). We then consider a proper minimal surface MR3M\subset \mathbb{R}^{3} and the extrinsic ball Ω=MBR3(o,r)\Omega=M\cap B_{\mathbb{R}^{3}}(o,r). We obtain upper and lower estimates for the series λi2(Ω)\sum \lambda_i^{-2}(\Omega) in terms of the volume vol(Ω){\rm vol}(\Omega) and the radius rr of the extrinsic ball Ω\Omega.

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