Optimal Control Problems for Partial Differential Equations on Reticulated Domains pp 161-215 | Cite as

# Convergence Concepts in Variable Banach Spaces

## 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.

## Keywords

Weak Convergence Strong Convergence Variable Space Borel Measure Periodicity Cell## Preview

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