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Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).
We study the stability of Gorenstein preenvelopes and precovers in the cases of$H$-extensions and smash products with$H$, where$H$is a Hopf algebra. We use these to define Gorenstein dimensions and give new examples of the so-called Gorenstein categories.