Extremals for Eigenvalues of Laplacians under Conformal Mapping

G. Pólya and G. Szegő showed in 1951 that for simply connected plane domains, the first eigenvalue of the Laplacian (with Dirichlet boundary conditions) is maximal for a disk, under a conformal mapping normalization. That is, iff(z) is a conformal map of a diskDonto a bounded, simply connected plane domainΩ, normalized by |f′(0)|=1, thenλ1(Ω)⩽λ1(D). Later, Pólya and M. Schiffer showed that actually∑j=1n 1λj(Ω)⩾∑j=1n 1λj(D), for each n=1, 2, 3, ….This paper shows that for every convex increasing functionΦ,∑j=1n Φ 1λj(Ω)⩾∑j=1n Φ 1λj(D), for each n=1, 2, 3, ….In particular, takingΦ(a)=asfor a fixeds>1 gives that the zeta function of the eigenvalues of the Laplacian is minimal for the disk, under Dirichlet boundary conditions. The bulk of the paper addresses similar questions for simply and doubly connected domains on cones and cylinders and on surfaces of variable curvature, extending the work of C. Bandle, T. Gasser, and J. Hersch. Also, letMgbe anN-dim. Riemannian manifold with boundary and for each smooth functionwon the closure ofM, writeλj(w) for thejth eigenvalue of the operatorw−1Δg, under Dirichlet boundary conditions. Note that in two dimensions,w−1Δgis the Laplace–Beltrami operator of the metricwg. It is proved here that the “zeta-type” functional ∑nj=1 Φ(λj(w)−1) is convex with respect to the mass densityw.