Abstract
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.
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© 2011 Springer Science+Business Media, LLC
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Kogut, P.I., Leugering, G.R. (2011). Convergence Concepts in Variable Banach Spaces. In: Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Systems & Control: Foundations & Applications. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8149-4_6
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DOI: https://doi.org/10.1007/978-0-8176-8149-4_6
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-8148-7
Online ISBN: 978-0-8176-8149-4
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