About smooth approximation in a Sobolev space

Question

I want to prove the following fact :

Consider $\Omega \subset R^n$ a bounded and open set. Let $v \in H^{1}_{0}(\Omega)$ a nonnegative function. Then exists a sequence $v_m$ in $C^{\infty}_{0}(\Omega)$ of nongative functions converging to $v$ in $H^{1}( \Omega)$.

I know how to prove this fact (I think this fact can help): If $u_m$ is a sequence converging to $u$ in $H^1(\Omega)$ , then ${u^{+}_m} \rightarrow u^+$.

Someone can give me a hint ?

Thanks in advance

Answer 1

By your known fact, if the definition of $u^+$ is $$ u^+ = \begin{cases} u \quad \text{ when }u\geq 0, \\ 0 \quad \text{ when }u< 0.\end{cases} $$ Then for any non-negative $v\in H^1_0(\Omega)$, we can find a sequence of $u_m \in C^{\infty}_0(\Omega)$ converging to $v$. Hence by the proposition in your question: $$ u_m^+\to v^+ = v \quad \text{ in }\;H^1(\Omega). $$ Now we can construct a smoothing sequence $w_{m,k} \to u_m^+$ for any $m$ using mollifier $\phi_k$ compactedly supported near 0: $$w_{m,k} = \int_{\Omega} \phi_k(x-y)u_m^+(y)dy \geq 0.$$ Now let $v_m =w_{m,m}$ which is non-negative and in $C^{\infty}_0(\Omega)$, then we have: $$ \|v - v_m\|_{H^1}\leq \|v - u_m^+\|_{H^1}+ \|u_m^+ - w_{m,m}\|_{H^1} \to 0. $$

Answer 2

Since $v+1/n\to v$, it suffices to approximate $v+1/n$. As in your previous question Sobolev spaces - about smooth aproximation you can use a smooth threshhold function $\phi:\mathbb R\to\mathbb R$ such that

• $\phi(x)=x$ when $x\ge 1/n$
• $\phi\ge 0$ everywhere
• $0\le \phi'\le 1$ everywhere

The proof stays the same: pick a smooth approximation $u_m\to v+1/n$, compose with $\phi$, argue that a subsequence of $\phi\circ u_m$ converges to $v+1/n$.