How to prove $C_{c}^{\infty}(R^{n}_{+})$ is not dense in $H^{m,p}(R_{+}^{n})$ when $m\geq 1$?

Question

I want to prove $C_{c}^{\infty}(R^{n}_{+})$ is not dense in $H^{m,p}(R_{+}^{n})$ when $m\geq 1$.

Here $R^{n}_{+}$ is the upper half space and $H^{m,p}(\Omega)=\{f:D^{\alpha}f\in L^{p}(\Omega)\}$ is Sobolev space with the canonical norm.

But I cannot find a $u\in H^{m,p}$ with no series in $C_{c}^{\infty}(R^{n}_{+})$ convergent to it.

Any help will be thanked.

Answer 1

Use a trace argument: if $u_n \in C_{c}^{\infty}(\mathbb{R}^{n}_{+})$ is such that $u_n \to u$ in the $H^{m,p}(\mathbb{R}_{+}^{n})$-norm then $Trace(u)=\lim\limits_{n \to +\infty}Trace (u_n)=0$ but it is not true that every $u \in H^{m,p}(\mathbb{R}_{+}^{n})$ is equal to $0$ at the boundary of $\mathbb{R}^n_+$.

Remark 1: in my argument I used the trace for the following reason. If $pm>n$ then all $u \in H^{m,p}(\mathbb{R}_{+}^{n})$ are continuous to the boundary of the domain. If not we must use the generalization of $u_{|\partial \mathbb{R}^n_+}$ which is the trace.

Remark 2: the closure of $C_{c}^{\infty}(\mathbb{R}^{n}_{+})$ in $H^{m,p}(\mathbb{R}_{+}^{n})$ is defined as $H_0^{m,p}(\mathbb{R}_{+}^{n})$.

Hope this is useful for you.