On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\) (Q5955043)

A measurable function \(u:\Omega\to \mathbb{R}\) belongs to \(L^{p(x)} (\Omega)\), by definition, if \(\lim_{\lambda \downarrow 0}\int_\Omega |\lambda u(x)|^{p(x)} dx=0\). The authors study the properties of the space \(L^{p(x)} (\Omega)\), equipped with some kind of Luxemburg norm. They also consider a parallel construction for the Sobolev space \(W^{m,p(x)} (\Omega)\). Such constructions are motivated by certain elliptic or variational problems.