Estimates of Eigenvalues and Eigenfunctions in Periodic Homogenization
  For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an $O(\epsilon)$ estimate in $H^1$ for solutions with Dirichlet condition. 
Carlos E. Kenig, Fanghua Lin, and Zhongwei Shen
math.AP