First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof

Let λ ε be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a( x/ε)∇ ) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·( A∇) . We characterise the possible limit points of the ratio (λ ε –λ)/ε as ε→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.