Smooth approximation of bounded function belonging to some Sobolev space

Question

I need your help to answer this questions:

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$, and let $u\in W^{1,p}_{0} (\Omega)\cap L^{\infty}(\Omega)$. There exists $u_{n}\in C^{\infty}_{0}(\Omega)\cap L^{\infty}(\Omega)$ converging to $u$ strongly in $ W^{1,p}_{0} (\Omega)$?

If $u\in W^{1,p}_{0} (\Omega)$, with $u\geq 0$. There exists a sequence of nonnegative smooth functions $\{u_{n}\}$ converging to $u$ strongly in $ W^{1,p}_{0} (\Omega)$?

What kind of a domain is $\Omega$? What have you already tried? — Jonas Lenz, Sep 01 '18 at 16:35
Isn't the definition of $W^{1,p}_0$ the closure of $C^\infty_0$ in the $W^{1,p}$ norm? If not, then what is your definition? — Ian, Sep 01 '18 at 17:38

score 0 · Accepted Answer · answered Sep 01 '18 at 18:56

0

For your first question, any function in $C^\infty_0(\Omega)$ is in $L^\infty(\Omega)$ (continuous function on compact sets are bounded). Do you want the sequence to be bounded? In that case, see this question.

For your second question: this has been answered here.

answered Sep 01 '18 at 18:56

Kusma

3,979

Thanks so much Kusma. – Said Sep 01 '18 at 21:42

Smooth approximation of bounded function belonging to some Sobolev space

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