Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model
Kirill Cherednichenko, Alexander V. Kiselev · Communications in Mathematical Physics · 2016
We prove operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the M-matrix of an associated boundary triple (“Krein resolvent formula”). The resulting asymptotic behaviour is shown to be described, up to a unitary transformation, by a non-standard version of the Kronig–Penney model on $${\mathbb{R}}$$ .