Convergence Concepts in Variable Banach Spaces

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations