×

zbMATH — the first resource for mathematics

Density of smooth functions in \(W^{k,p(x)}({\Omega{}})\). (English) Zbl 0779.46027
Stimulated by boundary-value problems for quasi-linear partial differential equations with lower-order terms having variable growth Kováčik and J. Rákosník [Czech. Math. J. 41, 592-618 (1991)] studied the spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\). These are the analogues of the Lebesgue spaces \(L^ p(\Omega)\) and Sobolev spaces \(W^{k,p}(\Omega)\) which arise when \(p\) is allowed to be a function on the underlying space domain \(\Omega\subset\mathbb{R}^ m\) which takes its values in the interval \([1,+\infty]\); they can be considered as particular cases of Orlicz and Orlicz-Sobolev spaces respectively.
The present paper continues the work of Kováčik and Rákosník and describes a class of functions \(p(x)\) for which the set \(C^ \infty(\Omega)\cap W^{k,p(x)}(\Omega)\) is dense in \(W^{k,p(x)}(\Omega)\). This enables the authors to show that for such functions \(p(x)\), the conditions \(f\in W^{k,p(x)}(\Omega)\), \(d^{|\alpha|-k}D^ \alpha f\in L^{p(x)}(\Omega)\) for all \(\alpha\) with \(|\alpha|\leq k\) (where \(d(x)\equiv\text{dist}(x,\partial\Omega))\) imply that \(f\in W_ 0^{k,p(x)}(\Omega)\), the closure of \(C^ \infty_ 0(\Omega)\) in \(W^{k,p(x)}(\Omega)\).

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDF BibTeX XML Cite
Full Text: DOI