Convergence Concepts in Variable Banach Spaces

  • Peter I. Kogut
  • Günter R. Leugering
Part of the Systems & Control: Foundations & Applications book series (SCFA)


The aim of this chapter is to provide a systematic exposition of the main properties of weak and strong convergence in variable L p - and W 1,p -spaces for p>1. It will always be assumed that p and q are conjugate indices (i.e., 1=1/p+1/q) and that p>1. The main objects of our consideration are sequences of the types \(\left\{y^{h}_{\varepsilon}\in L^{p}(\Omega,\mathrm{d}\mu^{h}_{\varepsilon})\right\}_{{\varepsilon}>0}\) and \(\left\{y^{h}_{\varepsilon}\in W^{1,p}(\Omega,\mathrm{d}\mu^{h}_{\varepsilon})\right\}_{{\varepsilon}>0}\), where \(\mu^{h}_{\varepsilon}\) is a two-parametric Borel measure related to the geometry of thin periodic structures. Typically, the parameter ε defines the periodicity cell and εh is the thickness of constituting elements of such structures. However, there is a principal difference between perforated domains and thin structures. For perforated domains, the typical case is when the parameter h is either independent of ε or such that lim inf  ε→0 h(ε)=h >0, whereas the principle feature of thin structures is the fact that the parameters ε and h=h(ε) are related by the supposition h(ε)→0 as ε→0. Therefore, our main intension in this chapter is to shed some light on convergence properties in described spaces.


Weak Convergence Strong Convergence Variable Space Borel Measure Periodicity Cell 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and Mechanics, Department of Differential EquationsOles Honchar Dnipropetrovsk National UniversityDnipropetrovskUkraine
  2. 2.Department of Mathematics, Chair of Applied Mathematics IIFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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