# Approximating Sobolev functions in $W^{1,p}(\mathbb{R}_+^n)$

Let $$p \geq 1$$. I know that there exists a continuous and linear extension operator $$E : W^{1,p}(\mathbb{R}_+^n) \to W^{1,p}(\mathbb{R}^n) .$$

I read that from the existence of such an extension one can deduce that $$C_c^{\infty}(\bar{\mathbb{R}^n_+})$$ is dense in $$W^{1,p}(\mathbb{R}_+^n)$$, where for $$C_c^{\infty}(\bar{\mathbb{R}^n_+})$$ I mean smooth functions on $$\mathbb{R}_+^n$$ whose support is contained in $$\bar{\mathbb{R}_+^n}$$(I don't know whether this is a standard notation or not).

I tried to use convolution and use that $$C_c^{\infty}(\mathbb{R}^n)$$ is dense in $$W^{1,p}(\mathbb{R}^n)$$ but I wasn't able to get anything .

• for $C_c^{\infty}(\bar{\mathbb{R}^n_+})$ the support of the functions can contain elements of the boundary of $\mathbb R^n_+$, correct? – supinf Dec 10 '18 at 9:52
• Sorry it was not clear I edited – Tommaso Scognamiglio Dec 10 '18 at 10:14

$$C_c^{\infty}(\bar{\mathbb{R}^n_+})$$ is dense in $$W^{1,p}(\mathbb{R}_+^n)$$,
This is not correct. We choose $$n=1,p=1$$. Lets assume that we can approximate the constant function $$1\in W^{1,p}(\mathbb R^n_+)$$ with functions from $$C_c^\infty (\overline{\mathbb R_+^n})$$.
Let $$\phi\in C_c^\infty (\overline{\mathbb R_+^n})$$. Then $$\phi(0)=0$$.
If $$\phi(x) \leq \frac12$$ for all $$x\in (0,1)$$, then we have $$\| 1- \phi |_{W^{1,1}} \geq \|1-\phi|_{L^1} \geq \frac12.$$ So we can assume that $$\phi(x)>\frac12$$ for some $$x\in (0,1)$$. Then $$\frac12 = \int_0^x \phi'(y) \mathrm dy \leq \int_0^1 | \phi'(y) |\mathrm dy \leq \|\phi'\|_{L^1} \leq \|1-\phi\|_{W^{1,1}}.$$ So in any case, we cannot approximate $$1$$ with $$\phi$$.
• You misunderstood the definition of $C_c^\infty(\overline{\mathbb{R}^n})$ the op is using. It's the space of all smooth functions with compact support in $\mathbb{R}^{n-1}\times [0,\infty)$. In the one-dimensional case this means that they have support in $[0,R]$ for some $R>0$, but not necessarily satisfy $\phi(0)=0$. – MaoWao Dec 10 '18 at 10:46