First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof
Shari Moskow, Michael S. Vogelius · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1997
Let λε 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 (λε–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.