Homogenisation and the weak operator topology. (English) Zbl 1423.35024
Summary: This article surveys results that relate homogenisation problems for partial differential equations and convergence in the weak operator topology of a suitable choice of linear operators. More precisely, well-known notions like \(G\)-convergence, \(H\)-convergence as well as the recent notion of nonlocal \(H\)-convergence are discussed and characterised by certain convergence statements under the weak operator topology. Having introduced and described these notions predominantly made for static or variational type problems, we further study these convergences in the context of dynamic equations like the heat equation, the wave equation or Maxwell’s equations. The survey is intended to clarify the ideas and highlight the operator theoretic aspects of homogenisation theory in the autonomous case.
MSC:
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 74Q10 | Homogenization and oscillations in dynamical problems of solid mechanics |
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
| 35L04 | Initial-boundary value problems for first-order hyperbolic equations |
| 35Q61 | Maxwell equations |
Keywords:
homogenisation; evolutionary equations; \(G\)-convergence; \(H\)-convergence; nonlocal \(H\)-convergence; heat conduction; wave equation; Maxwell’s equationsReferences:
| [1] | Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, vol. 5. North-Holland Publishing Co., Amsterdam (1978) · Zbl 0404.35001 |
| [2] | Cherednichenko, K., Waurick, M.: Resolvent estimates in homogenisation of periodic problems of fractional elasticity. J. Differ. Equ. 264(6), 3811-3835 (2018) · Zbl 1387.35028 · doi:10.1016/j.jde.2017.11.038 |
| [3] | Cherednichenko, K.D., Cooper, S.: Resolvent estimates for high-contrast elliptic problems with periodic coefficients. Arch. Ration. Mech. Anal. 219(3), 1061-1086 (2016) · Zbl 1334.35027 · doi:10.1007/s00205-015-0916-4 |
| [4] | Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications, vol. 17. The Clarendon Press, Oxford University Press, New York (1999) · Zbl 0939.35001 |
| [5] | Cooper, S., Waurick, M.: Fibre homogenisation. J. Funct. Anal. 276(11), 3363-3405 (2019) · Zbl 1414.35021 · doi:10.1016/j.jfa.2019.03.004 |
| [6] | Engel, K.-J., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt (2000) · Zbl 0952.47036 |
| [7] | Fourès, Y., Segal, I.E.: Causality and analyticity. Trans. Am. Math. Soc. 78, 385-405 (1955) · Zbl 0064.36805 · doi:10.2307/1993070 |
| [8] | Franz, S., Waurick, M.: Homogenisation of parabolic/hyperbolic media. Technical report BAIL conference, TU Dresden, University of Strathclyde. arXiv:1810.01234 (2018) |
| [9] | Franz, S., Waurick, M.: Resolvent estimates and numerical implementation for the homogenisation of one-dimensional periodic mixed type problems. Zeitschrift für Angewandte Mathematik und Mechanik 98(7), 1284-1294 (2018) · Zbl 1538.35230 · doi:10.1002/zamm.201700329 |
| [10] | Kalauch, A., Picard, R., Siegmund, S., Trostorff, S., Waurick, M.: A Hilbert space perspective on ordinary differential equations with memory term. J. Dyn. Differ. Equ. 26(2), 369-399 (2014) · Zbl 1307.34103 · doi:10.1007/s10884-014-9353-6 |
| [11] | Murat, F., Tartar, L.: \[HH\]-convergence. In: Cherkaev, A., Kohn, R. (eds.) Topics in the Mathematical Modelling of Composite Materials, vol. 31 of Progr. Nonlinear Differential Equations Appl., pp. 21-43. Birkhäuser, Boston (1997) · Zbl 0870.00018 |
| [12] | Pauly, D.: A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A \[_0^*0\]∗-A1-lemma in Hilbert spaces. Technical report, University of Duisburg-Essen. arXiv:1707.00019 (2017) |
| [13] | Pauly, D., Zulehner, W.: The divDiv-complex and applications to Biharmonic equations. Appl. Anal. (2019) https://doi.org/10.1080/00036811.2018.1542685 · Zbl 1459.35126 |
| [14] | Picard, R.: On the boundary value problems of electro- and magnetostatics. Proc. R. Soc. Edinb. Sect. A 92(1-2), 165-174 (1982) · Zbl 0516.35023 · doi:10.1017/S0308210500020023 |
| [15] | Picard, R.: Ein Hodge-Satz für Manningfaltigkeiten mit nicht-glattem Rand. Math. Methods Appl. Sci. 5(2), 153-161 (1983) · Zbl 0513.58004 · doi:10.1002/mma.1670050111 |
| [16] | Picard, R.: An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math. Z. 187(2), 151-164 (1984) · Zbl 0527.58038 · doi:10.1007/BF01161700 |
| [17] | Picard, R.: Some decomposition theorems and their application to nonlinear potential theory and Hodge theory. Math. Methods Appl. Sci. 12(1), 35-52 (1990) · Zbl 0702.35212 · doi:10.1002/mma.1670120103 |
| [18] | Picard, R.: A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci. 32, 1768-1803 (2009) · Zbl 1200.35050 · doi:10.1002/mma.1110 |
| [19] | Picard, R., McGhee, D.: Partial Differential Equations: A Unified Hilbert Space Approach. Expositions in Mathematics, vol. 55. DeGruyter, Berlin (2011) · Zbl 1275.35002 · doi:10.1515/9783110250275 |
| [20] | Picard, R., Trostorff, S., Waurick, M.: Well-posedness via monotonicity: an overview. In: Arendt, W., Chill, R., Tomilov, Y. (eds.) Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics. Operator Theory: Advances and Applications, vol. 250, pp. 397-452 (2015) · Zbl 1334.35015 |
| [21] | Picard, R., Trostorff, S., Waurick, M.: On the well-posedness of a class of non-autonomous SPDEs: an operator-theoretical perspective. GAMM-Mitteilungen. Appl. Oper. Thoor. Part II 41(4), e201800014 (2018) · Zbl 1536.60055 |
| [22] | Spagnolo, S.: Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Ann. Scuola Norm. Sup. Pisa 3(21), 657-699 (1967) · Zbl 0153.42103 |
| [23] | Spagnolo, S.: Convergence in energy for elliptic operators. In: Numerical Solution of Partial Differential Equations III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), pp. 469-498 (1976) · Zbl 0347.65034 |
| [24] | Süß, A., Waurick, M.: A solution theory for a general class of SPDEs. Stoch. Partial Differ. Equ. Anal. Comput. 5(2), 278-318 (2017) · Zbl 1375.60110 |
| [25] | Tartar, L.: The General Theory of Homogenization. Lecture Notes of the Unione Matematica Italiana, vol. 7. Springer, Berlin (2009). (UMI, Bologna. A personalized introduction) · Zbl 1188.35004 |
| [26] | Elst, A., Gorden, G., Waurick, M.: The Dirichlet-to-Neumann operator for divergence form problems. Annali di Matematica Pura ed Applicata 198(1), 177-203 (2019) · Zbl 1412.35094 · doi:10.1007/s10231-018-0768-2 |
| [27] | Trostorff, S.: Exponential stability for linear evolutionary equations. Asymptot. Anal. 85(3-4), 179-197 (2013) · Zbl 1393.47021 |
| [28] | Trostorff, S., Waurick, M.: A note on elliptic type boundary value problems with maximal monotone relations. Mathematische Nachrichten 287(13), 1545-1558 (2014) · Zbl 1316.35124 · doi:10.1002/mana.201200242 |
| [29] | Waurick, M.: Limiting processes in evolutionary equations—a Hilbert space approach to homogenization. Dissertation, TU Dresden. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-67442 (2011) |
| [30] | Waurick, M.: A Hilbert space approach to homogenization of linear ordinary differential equations including delay and memory terms. Math. Methods Appl. Sci. 35(9), 1067-1077 (2012) · Zbl 1252.35039 · doi:10.1002/mma.2515 |
| [31] | Waurick, M.: Homogenization of a class of linear partial differential equations. Asymptot. Anal. 82, 271-294 (2013) · Zbl 1280.35152 |
| [32] | Waurick, M.: G-convergence of linear differential equations. J. Anal. Appl. 33(4), 385-415 (2014) · Zbl 1302.47026 |
| [33] | Waurick, M.: Homogenization in fractional elasticity. SIAM J. Math. Anal. 46(2), 1551-1576 (2014) · Zbl 1348.74291 · doi:10.1137/130941596 |
| [34] | Waurick, M.: G-convergence and the weak operator topology. PAMM 16, 521-522 (2016) · doi:10.1002/pamm.201610430 |
| [35] | Waurick, M.: On the continuous dependence on the coefficients of evolutionary equations. Habilitation, TU Dresden. arXiv:1606.07731 (2016) · Zbl 1346.74160 |
| [36] | Waurick, M.: On the homogenization of partial integro-differential-algebraic equations. Oper. Matrices 10(2), 247-283 (2016) · Zbl 1346.74160 · doi:10.7153/oam-10-15 |
| [37] | Waurick, M.: Stabilization via homogenization. Appl. Math. Lett. 60, 101-107 (2016) · Zbl 1339.35198 · doi:10.1016/j.aml.2016.04.004 |
| [38] | Waurick, M.: Continuous dependence on the coefficients for a class of non-autonomous evolutionary equations. In: Proceedings of the Special Semester 2016 in RICAM in Linz (2017) (Accepted) · Zbl 1366.34085 |
| [39] | Waurick, M.: A functional analytic perspective to the div-curl lemma. J. Oper. Theory 80(1), 95-111 (2018) · Zbl 1413.35020 · doi:10.7900/jot.2017jun09.2154 |
| [40] | Waurick, M.: Nonlocal \[HH\]-convergence. Calc. Var Partial Differ. Equ. 57(6), 46 (2018) · Zbl 1406.35037 |
| [41] | Waurick, M.: On operator norm convergence in time-dependent homogenisation problems. PAMM (2018). https://doi.org/10.1002/pamm.201800009 · doi:10.1002/pamm.201800009 |
| [42] | Weiss, G.: Representation of shift-invariant operators on \[L^2\] L2 by \[H^\infty H\]∞ transfer functions: an elementary proof, a generalization to \[L^p,\] Lp, and a counterexample for \[L^\infty\] L∞. Math. Control Signals Syst. 4(2), 193-203 (1991) · Zbl 0724.93021 |
| [43] | Zhikov, V., Kozlov, S., Oleinik, O., Ngoan, K.T.: Averaging and G-convergence of differential operators. Russ. Math. Surv. 34(5), 69-147 (1979) · Zbl 0445.35096 · doi:10.1070/RM1979v034n05ABEH003898 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
