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

Let lambda(epsilon) be a Dirichlet eigenvalue of the 'periodically, rapidly oscillating' elliptic operator -del.(a(x/epsilon)del) and let lambda be a corresponding (simple) eigenvalue of the homogenised operator -del.(A del). We characterise the possible limit points of the ratio (lambda(epsilon)-lambda)/epsilon as epsilon-->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.