Resumen de Green functions and the Dirichlet spectrum

G. Pacelli Bessa, Vicent Gimeno Garcia, Jorge Luquesio

• This article has results of four types. We show that the first eigenvalue λ1(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 λ1(Ω)=limk→∞∥Gk(f)∥L2/∥Gk+1(f)∥L2 for any f∈L2(Ω,μ), f>0. Then, we study the L1(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 L1(Ω,μ)-moment spectrum, generalizing the work of Hurtado–Markvorsen–Palmer. Finally, we study the radial spectrum σrad(Bh(o,r)) of rotationally invariant geodesic balls Bh(o,r) of model manifolds. We prove an identity relating the radial eigenvalues of σrad(Bh(o,r)) to an isoperimetric quotient, i.e., ∑1/λradi=∫V(s)/S(s)ds, V(s)=vol(Bh(o,s)) and S(s)=vol(∂Bh(o,s)). We then consider a proper minimal surface M⊂R3 and the extrinsic ball Ω=M∩BR3(o,r). We obtain upper and lower estimates for the series ∑λ−2i(Ω) in terms of the volume vol(Ω) and the radius r of the extrinsic ball Ω.