Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
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.
scientific article; zbMATH DE number 1190784