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Description
This book investigates homogenisation problems for divergence form equations with rapidly sign-changing coefficients. Focusing on problems with piecewise constant, scalar coefficients in a (d-dimensional) crosswalk type shape, we will provide a limit procedure in order to understand potentially ill-posed and non-coercive settings.
Depending on the integral mean of the coefficient and its inverse, the limits can either satisfy the usual homogenisation formula for stratified media, be entirely degenerate or be a non-local differential operator of 4th order. In order to mark the drastic change of nature, we introduce the inner spectrum for conductivities. We show that even though 0 is contained in the inner spectrum for all strictly positive periods, the limit inner spectrum can be empty. Furthermore, even though the spectrum was confined in a bounded set uniformly for all strictly positive periods and not containing 0, the limit inner spectrum might have 0 as an essential spectral point and accumulate at or even be the whole of C. This is in stark contrast to the classical situation, where it is possible to derive upper and lower bounds in terms of the values assumed by the coefficients in the pre-asymptotics.
Along the way, we also develop a theory for Sturm Liouville type operators with indefinite weights, reduce the question on solvability of the associated Sturm Liouville operator to understanding zeros of a certain explicit polynomial and show that generic real perturbations of piecewise constant coefficients lead to continuously invertible Sturm Liouville expressions.
Chapter 1. Introduction.- Chapter 2. The main theorems.- Chapter 3. Abstract divergence-form operators.- Chapter 4. The one-dimensional problem well-posedness.- Chapter 5. Sturm Liouville problems with indefinite coeffcients.- Chapter 6. The higher-dimensional problem preliminaries.- Chapter 7. The higher dimensional problem well-posedness.- Chapter 8. The inner spectrum in d dimensions.- Chapter 9. Classical G-convergence.- Chapter 10. Holomorphic G-convergence.- Chapter 11. The one-dimensional problem homogenisation.- Chapter 12. The higher-dimensional problem homogenisation.- Chapter 13. Proofs.- Chapter 14. Conclusion.
Marcus Waurick graduated in Mathematics with minor in Physics at TU Dresden in 2009. During his first employment at the Faculty of Civil Engineering, he finished his PhD in 2011 on homogenisation theory and accepted a position at the Institute for Analysis at TU Dresden later that year. In 2015, he took up a research post at the University of Bath, and in 2016, he completed his habilitation thesis. The following year, he became Chancellor s Fellow at the University of Strathclyde where he was honored with the Research Excellence Award in 2018. In 2020, he accepted a position at TU Hamburg as research associate, and in November of the same year, he became deputy professor at TU Bergakademie Freiberg. Since April 2021, he has been a University Professor at TU Bergakademie Freiberg and held the chair for Partial Differential Equations. He has contributed to more than 70 research articles and 3 books, with his work spanning partial differential equations, evolutionary equations, operator theory, numerical analysis, homogenisation, control theory, and functional analysis.
