Smooth approximation of non-negative Sobolev function

Question

I need a help to solve this question:

Let $u\in H^{1}_{0}(\Omega)$ with $u\geq 0$. Can I find a sequence of smooth non-negative functions converging to $u$ in $H^{1}_{0}(\Omega)$?

Thank you in advance.

Answer 1

Let $\{u_i\}$ be a sequence of compactly supported smooth functions approximating $u$ in $W^{1,2}$. By Chebyshev, given $\epsilon>0$, $$|\{|u-u_i|> \epsilon\}|\leq \delta_i/\epsilon^2.$$ Now pick $i$ large enough so that $\|\partial^{\alpha}u\|_{L^p(A)} < \epsilon$, whenever $|A|<\delta_i/\epsilon^2$, $|\alpha|=1$. Consider next the standard mollifications $v_i^{t}$ of $\max\{u_i,-\epsilon\}$ for $t$ small enough. Then $v_i^t \geq -\epsilon,$ and is smooth and compactly supported. Also $v_i^t+\epsilon$ is arbitrarily close to $u$ (in norm) for $\epsilon$ and $t$ small enough and $i$ large enough.