Resolvent estimates in homogenisation of periodic problems of fractional elasticity

TL;DR

This paper establishes operator-norm convergence estimates for solutions to a one-dimensional fractional elasticity problem with periodic coefficients, advancing homogenisation theory for viscoelastic composites.

Contribution

It provides uniform operator-norm estimates for fractional elasticity with periodic coefficients using Fourier--Laplace and Gelfand transforms, a novel approach in this context.

Findings

Operator-norm convergence estimates for solutions

Uniform estimates on each fibre in the decomposition

Convergence estimates for applied forces with smooth densities

Abstract

We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier--Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomposition and in the period of oscillations of the coefficients as well as quadratic with respect to the spectral variable. On the basis of these uniform estimates we derive operator-norm-type convergence estimates for the original fractional elasticity problem, for a class of sufficiently smooth densities of applied forces.