Smooth approximation of a function belonging to Sobolev spaces

Question

I need your help to answer this question:

Let $\psi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ and $u\in W_{0}^{1,p}(\Omega)$, with $u\geq\psi$ a.e. in $\Omega$. There exits a sequence $u_{n}\in C_{0}^{\infty}(\Omega)$ such that $u_{n}\geq\psi$ a.e. in $\Omega$, and $u_{n}\to u$ strongly in $W_{0}^{1,p}(\Omega)$?

score 1 · Accepted Answer · answered Aug 30 '18 at 15:32

1

This is not possible.

Consider $\Omega=(0,1)$ and $u = \psi=x(1-x)$.

Then there is no function $u_n\in C_0^\infty(\Omega)$ with $u_n\geq \psi$ a.e., because $u_n$ needs to have a compact support and has to be equal to $0$ on a set of positive measure.

answered Aug 30 '18 at 15:32

supinf

13,433

Thanks supinf for you help. What about the case $\psi=0$? – Said Aug 30 '18 at 16:06
@Said then i would guess that it is possible, but i am not sure why. – supinf Aug 30 '18 at 17:21
ok, thanks so much supinf – Said Aug 30 '18 at 18:02

Smooth approximation of a function belonging to Sobolev spaces

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