Summary: a shift between dirichlet and neumann spectrum
for generalized linear elasticity
Presentation : defining the equation of linear elasticity on a general rie
mannian manifold with boundary, we prove a formula relating the counting
functions of the Neumann and the Dirichlet problem to the counting func
tion of the Dirichlet2Neumann operator. Namely the difference of the two
counting functions at ff equals the number of negative eigenvalues of the
Dirichlet2Neumann operator related to the resolvant at ff.
With this formula we can show that this difference is always bigger than
one in the homogeneous case (i.e. when the Lam'e functions –; ¯ are con
stant) for bounded domains of symmetric spaces of non compact type and
rank bigger than 2, for instance the euclidean space, and for rank 1 if the
dimension of the nilpotent part is less than 8¯
Key words : elasticity, boundary conditions, spectrum, symmetric spaces.
Class. Math. : 58G25, 73C35, 35P15.
Aknowledge : special thanks to Georgi Popov and Jean Giroire who introduced
me to this subject.