Genericity Results on Bloch Spectrum of Periodic Elliptic Operators and Applications to Homogenization
Sivaji Ganesh Sista, Vivek Tewary · arXiv (Cornell University) · 2018
We consider a second-order elliptic operator in divergence form with periodic coefficients. The spectrum of such operators is completely described by Bloch eigenvalues. We study the structure and regularity properties of Bloch eigenvalues near the edge of a spectral gap. We show that a Bloch eigenvalue can be made simple at a point, and hence in a neighborhood of that point. Using this result, we prove that under small fiberwise perturbation of coefficients, a Bloch eigenvalue can be made simple for all parameter values. Further, under different regularity assumptions on the coefficients of the periodic operator, we obtain simplicity of a spectral edge under small perturbations of the coefficients. These spectral tools are used to establish Bloch wave homogenization at an internal edge in the presence of multiplicity.