A shift between Dirichlet and Neumann spectrum for generalized linear elasticity

Article type: Research Article

Authors: Anné, Colette

Affiliations: Département de Mathématiques, UMR 6629, Faculté des Sciences, 2, rue de la Houssinière, F‐44322 Nantes Cedex 03, France E‐mail: [email protected]‐nantes.fr

Abstract: We consider the equation of linear elasticity on a general Riemannian manifold with boundary, and prove a formula relating the counting functions of the Neumann and the Dirichlet problem to the counting function of the Dirichlet‐to‐Neumann operator. Namely, the difference of the two counting functions at \alpha equals the number of negative eigenvalues of the Dirichlet‐to‐Neumann operator related to the resolvent at \alpha. We then apply this formula to bounded domains of Riemannian symmetric spaces of non‐compact type in the homogeneous case of elasticity (i.e., when the Lamé functions \lambda, \mu are constant). The conclusion is that the difference of the two counting functions is greater or equal to 1 under one of the following hypothesis: either the rank of the symmetric space is greater or equal to 2, or the rank is 1 but the dimension of the nilpotent part is smaller than {8\mu}/({\lambda+2\mu}). The Euclidean space is an example of the first case, but even in that situation the conclusion we draw is new.

Keywords: Elasticity, boundary conditions, spectrum, symmetric spaces

Journal: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 297-316, 1999