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 \(W^{k,p(x)}(\Omega)\).


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
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