Eigenvalues of Laplacians with mixed boundary conditions, under conformal mapping
Richard Snyder Laugesen · Illinois Journal of Mathematics · 1998
IntroductionThe prototypical result of this paper says, roughly, that if f(z) Yjz aJ zj is a conformal map of an annulus A onto a doubly connected plane domain g2 with lail 1, then .j(f2),Zj(A)s for all s > 1, where j (')is the j-th eigenvalue of the Laplacian on under Dirichlet boundary conditions on the outer boundary of g2 and Neumann conditions on the inner boundary, and similarly for ,ky (A).That is, the zeta function of the Laplacian is at least as big for as it is for the annulus A.This introduction provides some historical context; then in Section 2 the results are all stated precisely.For similar results but under purely Dirichlet boundary conditions, see the earlier paper 13], written with C. Morpurgo.This present work draws heavily on the arguments and intuition in 13], and is best read in conjunction with that paper.The eigenvalues of the Laplacian have many physical interpretations, for example as the frequencies of vibration of a membrane, as rates of decay for the heat (or mass diffusion) equation, and as cut-off frequencies for waveguides.However, the eigen- values of doubly connected regions can be calculated exactly only for a few special regions, most notably for annuli, and while numerical methods are sophisticated and successful [11 ], they can only estimate finitely many of the eigenvalues.This paper will give sharp estimates involving all the eigenvalues.Incidentally, the mixed boundary conditions employed in this paper have drawn increasing attention in recent years (see for example [4], [17] and the references therein).G. P61ya and G. Szeg6 [16] proved by conformal transplantation an upper bound on the first eigenvalue of a simply connected plane domain under Dirichlet boundary conditions: if f (z) is a conformal map of the open unit disk D onto a bounded, simply connected plane domain f2 and if [f'(0)l 1, then .; (f2) < X(D).In [13, Cor.3], the author and C. Morpurgo proved a direct analogue of this for doubly connected