Cooper, Shane; Kamotski, Ilia; (2026) Quantitative homogenisation for differential equations with highly anisotropic partially degenerating coefficients. In: Chatzakou, Marianna and Ruzhansky, Michael and Van Bockstal, Karel, (eds.) Direct and Inverse Problems with Applications. DIPARTCSSMA 2024. (pp. pp. 69-75). Birkhäuser: Cham, Switzerland.

Preview

Text Cooper_2507.23380v1.pdf - Accepted Version Download (162kB) | Preview

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, ε, tends to zero. Our primary interest is in the asymptotic behaviour of the resolvent of this operator in the limit of small ε. 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 https://arxiv.org/abs/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 orderε operator-type error estimates.

Download activity - last month

Export as JSONXMLCSV

Download activity - last 12 months

Export as XMLJSONCSV

Downloads by country - last 12 months

Export as XMLJSONCSV

Archive Staff Only

View Item