Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
type: Item
instance of: scholarly article
title: Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
zbMATH Open document ID: 0906.35064
author: R. S. Laugesen
published in: Illinois Journal of Mathematics
publication date: 1998-10-25T00:00:00Z
review text: The prototypical result of this paper says, roughly, that if \(f(z)= \sum_{j\in\mathbb{Z}} a_jz^j\) is a conformal map of an annulus \(A\) onto a doubly connected plane domain \(\Omega\) with \(| a_1|=1\), then \[ \sum^\infty_{j=1} {1\over \lambda_j (\Omega)^s} \geq\sum^\infty_{j= 1} {1\over \lambda_ j(A)^s} \quad \text{for all }s>1, \] where \(\lambda_j (\Omega)\) is the \(j\)-th eigenvalue of the Laplacian on \(\Omega\) under Dirichlet boundary conditions on the outer boundary of \(\Omega\) and Neumann conditions on the inner boundary, and similarly for \(\lambda_j (A)\). That is, the zeta function of the Laplacian is at least as big for \(\Omega\) as it is for the annulus \(A\). Analogous sharp inequalities are proved for domains on cones and cylinders, where the ``Laplacian'' has the form \(| z|^p \Delta\), for domains on the hyperbolic punctured disk and on surfaces of variables curvature. Furthermore, the zeta function for the Laplace-Beltrami operator \(\Delta_{wg}\), on a 2-dimensional Riemannian manifold with metric \(g\), is shown to be convex as a function of the conformal factor \(w\). This paper builds upon methods developed for purely Dirichlet boundary conditions by the author and \textit{C. Morpurgo} [J. Funct. Anal. (to appear)]. Both these papers actually apply to the class of functionals \(\sum_{j=1}^n \Phi (\Gamma/ \lambda_j)\), where \(\Phi(x)\) is convex and increasing and \(n\) is a positive integer or \(+\infty\). Taking \(\Phi(x) =x^s\) and \(n=+\infty\) yields the zeta function. Taking \(\Phi(x) =e^{-t/x}\) (which is convex for \(0<x\leq t/2)\) yields the trace of the heat kernel.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1190784
zbMATH Keywords: curvature bound
zbMATH Keywords: conformal class
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: zeta function of the Laplacian
zbMATH Keywords: sharp inequalities
zbMATH Keywords: trace of the heat kernel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1190784