Eigenvalue convergence on perturbed Lipschitz domains for elliptic systems with mixed general decompositions of the boundary

Abstract

We consider eigenvalues of an elliptic operatorLu=−∂∂xj(Aij∂u∂xi) where u=(u1,...,um)T is a vector valued function and the coefficients Aij are m×m matrices whose elements aijαβ are bounded and symmetric. We perturb our domain Ω0 by adding a set of small measure, Tε to form the domain Ωε. We prescribe mixed boundary conditions on quite general decompositions of the boundary and look at the behavior of the eigenvalues of Ωε as Tε shrinks to zero. We look at systems which satisfy either a strong ellipticity condition, a Legendre–Hadamard condition, or in particular, the system of linear elasticity.