Quantitative homogenisation for differential equations with highly anisotropic partially degenerating coefficients

TL;DR

This paper analyzes the asymptotic behavior of solutions to highly anisotropic, partially degenerating elliptic differential equations with periodic coefficients, providing resolvent asymptotics and error estimates for small periods.

Contribution

It offers a new asymptotic description and operator error estimates for the resolvent of anisotropic, partially degenerating elliptic operators, extending previous two-scale convergence results.

Findings

Established order-$$ operator-type error estimates.

Provided an asymptotic description of the resolvent.

Addressed directional ellipticity loss and boundary layer effects.

Abstract

We consider a non-uniformly elliptic second-order differential operator with periodic coefficients that models composite media consisting of highly anisotropic cylindrical fibres periodically distributed in an isotropic background. The degree of anisotropy is related to the period of the coefficients via a `critical' high-contrast scaling. In particular, ellipticity is lost in certain directions as the period, $\epsilon$, tends to zero. Our primary interest is in the asymptotic behaviour of the resolvent of this operator in the limit of small $\epsilon$. Two-scale resolvent convergence results were established for such operators in Cherednichenko, Smyshlyaev and Zhikov (Proceedings of The Royal Society of Edinburgh:Seciton A Mathematics. 136(1), 87--114(2006)). In this work, we provide an asymptotic description of the resolvent and establish operator-type error estimates. Our approach adopts the general scheme of Cooper, Kamotski and Smyshlyaev (preprint available at arXiv:2307.13151). However, we face new challenges such as a directional dependence on the loss of ellipticity in addition to a key `spectral gap' assumption of the above article only holding in a weaker sense. This results in an additional `interfacial' boundary layer analysis in the vicinity of each fibre to arrive at order-$\epsilon$ operator-type error estimates.