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Weak Lower Semicontinuity by Means of Anisotropic Parametrized Measures

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Trends in Applications of Mathematics to Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 27))

Abstract

It is well known that besides oscillations, sequences bounded only in L 1 can also develop concentrations, and if the latter occurs, we can at most hope for weak∗ convergence in the sense of measures. Here we derive a new tool to handle mutual interferences of an oscillating and concentrating sequence with another weakly converging sequence. We introduce a couple of explicit examples showing a variety of possible kinds of behavior and outline some applications in Sobolev spaces.

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Notes

  1. 1.

    In [16] it is assumed that the compactification of the entire space \({\mathbb R}^m\times {\mathbb R}^{m\times n}\) is a subset in \({\mathbb R}^N\) for some \(N\in {\mathbb N}\). This however is not required for the proof in [16] which only uses separability of the compactification.

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Acknowledgements

This work was partly done during MK’s visiting Giovanni-Prodi professorship at the University of Würzburg, Germany. The hospitality and support of the Institute of Mathematics is gratefully acknowledged. This work was also supported by GAČR through projects 16-34894L and 17-04301S. The work of AK was supported by NCN grant 2011/03/N/ST1/00111.

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Agnieszka Kałamajska

  2. The Czech Academy of Sciences, Institute of Information Theory and Automation, Praha 8, Czech Republic

    Stefan Krömer & Martin Kružík

Authors
  1. Agnieszka Kałamajska
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  2. Stefan Krömer
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  3. Martin Kružík
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Corresponding author

Correspondence to Martin Kružík .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Augusto Visintin

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Kałamajska, A., Krömer, S., Kružík, M. (2018). Weak Lower Semicontinuity by Means of Anisotropic Parametrized Measures. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_2

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