A shift between Dirichlet and Neumann spectrum for generalized linear elasticity

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.