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 λ 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.