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.

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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. 1–36

DOI 10.4171/RMI/1119