LEADER 04025nam 22005775 450 001 9911031629403321 005 20251003131008.0 010 $a3-032-01928-1 024 7 $a10.1007/978-3-032-01928-8 035 $a(MiAaPQ)EBC32327767 035 $a(Au-PeEL)EBL32327767 035 $a(CKB)41543273000041 035 $a(DE-He213)978-3-032-01928-8 035 $a(EXLCZ)9941543273000041 100 $a20251003d2025 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aHomogenisation of Laminated Metamaterials and the Inner Spectrum /$fby Marcus Waurick 205 $a1st ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2025. 215 $a1 online resource (181 pages) 225 1 $aSpringerBriefs in Mathematics,$x2191-8201 311 08$a3-032-01927-3 311 08$a3-032-01930-3 327 $aChapter 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. 330 $aThis book investigates homogenisation problems for divergence form equations with rapidly sign-changing coefficients. Focusing on problems with piecewise constant, scalar coefficients in a (d-dimensional) crosswalk type shape, we will provide a limit procedure in order to understand potentially ill-posed and non-coercive settings. Depending on the integral mean of the coefficient and its inverse, the limits can either satisfy the usual homogenisation formula for stratified media, be entirely degenerate or be a non-local differential operator of 4th order. In order to mark the drastic change of nature, we introduce the ?inner spectrum? for conductivities. We show that even though 0 is contained in the inner spectrum for all strictly positive periods, the limit inner spectrum can be empty. Furthermore, even though the spectrum was confined in a bounded set uniformly for all strictly positive periods and not containing 0, the limit inner spectrum might have 0 as an essential spectral point and accumulate at ? or even be the whole of C. This is in stark contrast to the classical situation, where it is possible to derive upper and lower bounds in terms of the values assumed by the coefficients in the pre-asymptotics. Along the way, we also develop a theory for Sturm?Liouville type operators with indefinite weights, reduce the question on solvability of the associated Sturm?Liouville operator to understanding zeros of a certain explicit polynomial and show that generic real perturbations of piecewise constant coefficients lead to continuously invertible Sturm?Liouville expressions. 410 0$aSpringerBriefs in Mathematics,$x2191-8201 606 $aDifferential equations 606 $aMathematical physics 606 $aOperator theory 606 $aDifferential Equations 606 $aMathematical Physics 606 $aOperator Theory 615 0$aDifferential equations. 615 0$aMathematical physics. 615 0$aOperator theory. 615 14$aDifferential Equations. 615 24$aMathematical Physics. 615 24$aOperator Theory. 676 $a515.35 700 $aWaurick$b Marcus$0721074 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9911031629403321 996 $aHomogenisation of Laminated Metamaterials and the Inner Spectrum$94444842 997 $aUNINA