Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients. Issue 1 (25th January 2019)

• Record Type: Journal Article
• Authors: Kamotski, I. V.; Smyshlyaev, V. P.
• Is Part Of: Applicable analysis. Volume 98:Issue 1/2(2019)
• Page Start: 64
• Page End: 90
• Publication Date: 2019-01-25
• Subjects: High-contrast homogenization -- two-scale convergence -- operator convergence -- resolvent convergence -- convergence of semigroups -- homogenization of high-contrast evolution problems; 35B27 -- 47A10; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1441994
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: For two-scale homogenization of a general class of asymptotically degenerating strongly elliptic symmetric PDE systems with a critically scaled high contrast periodic coefficients of a small periodε, we derive a two-scale limit resolvent problem under a single generic decomposition assumption for the 'stiff' part. We show that this key assumption does hold for a large number of examples with a high contrast, both studied before and some recent ones, including those in linear elasticity and electromagnetism. Following ideas of V. V. Zhikov, under very mild restrictions on the regularity of the domainΩ we prove that the limit resolvent problem is well-posed and turns out to be a pseudo-resolvent problem for a well-defined non-negative self-adjoint two-scale limit operator. A key novel technical ingredient here is a proof that the linear span of product test functions in the functional spaces corresponding to the degeneracies is dense in associated two-scale energy space for a general coupling between the scales. As a result, we establish (both weak and strong) two-scale resolvent convergence, as well as some of its further implications for the spectral convergence and for convergence of parabolic and hyperbolic semigroups and of associated time-dependent initial boundary value problems. Some of the results of this work were announced in Kamotski IV, Smyshlyaev VP. Two-scale homogenization for a class of partially degenerating PDE systems. arXiv:1309.4579v1. 2013 (https://arxiv.org/abs/1309.4579v1 ).