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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:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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