Generic simplicity of spectral edges and applications to homogenization

Article type: Research Article

Authors: Sivaji Ganesh, Sista^{; *} | Tewary, Vivek

Affiliations: Mathematics Department, Indian Institute of Technology Bombay, Mumbai, 400076, India. E-mails: sivaji.ganesh@iitb.ac.in, vivekt@iitb.ac.in

Correspondence: [*] Corresponding author. Tel.: (+91-22) 2576 7476 (Office); Fax: (+91-22) 2572 3480; E-mail: sivaji.ganesh@iitb.ac.in.

Abstract: We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.

Keywords: Genericity, Bloch eigenvalues, periodic operators, homogenization

DOI: 10.3233/ASY-191542

Journal: Asymptotic Analysis, vol. 116, no. 3-4, pp. 219-248, 2020