An inequality between Dirichlet and Neumann eigenvalues in a centrally symmetric domain

Abstract

We prove an inequality between Dirichlet and Neumann eigenvalues of the Laplacian in a centrally symmetric Euclidean domain. David Jerison and Nikolai Nadirashvili conjectured in [1] (Conjecture 8.6) that, for a centrally symmetric convex domain, A2, the lowest nonzero Neumann eigenvalue of the Laplacian for an eigenfunction that is even with respect to the central symmetry x H-* -x is smaller than AL2, the lowest Dirichlet eigenvalue for an eigenfunction that is odd with respect this symmetry. Notice that this inequality does not follow from the general inequalities between the Dirichlet and the Neumann eigenvalues: it may happen that ,u2 is the second eigenvalue of the Dirichlet Laplacian, and A2 is, say, 100th eigenvalue of the Neumann Laplacian. It turned out that this conjecture, even in somewhat stronger form, has a rather simple proof. In this short paper, I present the proof. The technique is very similar to the one used in [2]. Theorem. Let Q be a centrally symmetric domain in TRn with C2-boundary F Q Secondary 58G25. 0)2000 American Mathematical Society 2057 This content downloaded from 207.46.13.118 on Sun, 11 Sep 2016 06:14:55 UTC All use subject to http://about.jstor.org/terms 2058 LEONID FRIEDLANDER The function v, (x) satisfies the equation AV, (x) + pu2vw (x) = O. so (2) ] Vv,12 dx = A2 ]v 12 dx?+ ]vW)(x) dA where 0/0v is differentiation in the outward normal direction, and dA is the surface element on F. Our goal is to show that (3) j dw jvw(x) v (x)dA < 0. Here dw is the standard measure on S`1 normalized in such a way that the total area of the sphere equals 1. Clearly, (3) implies that iVw(x)v< (x)dA <0 for some value of w, and then (1) would follow from (2). To prove (3), we will explicitly compute the integral on the left. First, on F, Vu = (0u/v so (4) VW (X)= 09U(X)(V a ), X Gr. a!, Here v(x) is the outward normal unit vector to F at x. In a neighborhood of F, we introduce local coordinates x = (Y', Yn), Y' (Y1i... , Yn-i), in such a way that Yn is the distance from a point x to F taken with the positive sign if the point lies outside Q, and taken with the negative sign otherwise; y' are the local coordinates on F of the point that is the closest to x. In this coordinate system, the Euclidean metric tensor equals g 9 /(Y' vYn) ?) where g' is a metric tensor on F that depends on Yn. In the y-coordinates, the Laplacian has the form 02 1&09log g' & ayn 2 9yn 0Yn where A' is the tangent Laplacian, and g' is the determinant of the covariant tensor g'(y', Yn). The function u(x) is an eigenfunction of the Dirichlet Laplacian, so one has Q2U 1 l V'u is a vector field in R' perpendicular to v. Then vw(y) = V'u(y' vy) W + a v W,