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Let $\Omega \subset \mathbb{R}^n$ be open. Let $1 \leq p < \infty$. I want to prove that the space of bounded functions $f:\Omega \rightarrow \mathbb{R}$ such that $\dfrac{\partial f}{\partial x_i}$ (in the weak sense) is in $L_p(\Omega)$ for every $i = 1, \ldots, n$ and such that $m_n({\rm supp}~ f) < \infty$ ($m_n$ the Lebesgue measure in $\mathbb{R}^n$) is dense in the Sobolev space $W^{1,p}(\Omega)$.

Any hint will be of great value!

guerraufo
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  • What have you tried ? Which density theorems do you already know ? – Stratos supports the strike Mar 03 '23 at 09:14
  • @StratosFair This question is because I am trying to understand lemma 5.2.1/1 in Maz'ya V. (2011) Sobolev spaces with applications to elliptic partial differential equations. Springer. I already know the fact that the distributional gradient coincides with the classical one almost every where in $\Omega$ under these hypotheses. This can be used to prove that certain sequence of bounded functions is dense in $W^{1,p}(\Omega)$, but the condition on the support of $f$ is not clear for me. – guerraufo Mar 03 '23 at 13:42
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    Just multiply by a smooth function with compact support that converges to $1$? – LL 3.14 Mar 03 '23 at 17:45
  • @LL3.14 Yes I had the same idea, but then the text uses a quite complicated argument and it confused me... I tought I was missing something in the original argument. In the text Maz'ya uses a lemma (1.7.1), and the following lines appear unnecessary if we adopt the approach you say... may be there is some subtlety involved... I am not completely sure – guerraufo Mar 03 '23 at 19:52

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