Is $C^{k+1}(\Omega)\cap W^{k,p}(\Omega) $ dense in $W^{k,p} (\Omega)$? Assuming $\Omega$ has a $C^1$ boundary (or Lipschitz continuous boundary, if allowed?). I know the standard results on smooth compactly supported functions being dense in Sobolev spaces on $\mathcal{R}^n$. Also, on a bounded domain what is the minimum regularity/conditions needed on continuous functions, intersection with $W^{k,p}$ taken as needed, to generate a density result for $W^{k,p}$. Thanks, Sandy
Approximation of Sobolev Function in $W^{k,p}(\Omega)$ by Continuous Functions of limited Regularity
analysisfunctional-analysisfunctionsreal-analysissobolev-spaces
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Yes, this is the case. You can show that for any $\Omega \subset \mathbb R^n$ open and any $k \in \mathbb N$, $1 \leq p \leq \infty$ the inclusion $C_c^\infty \subset W^{k,p}$ holds true. First acknowledge that the case $k=0$ is trivial. Then convince yourself that the case $k=1$ contains all the difficulty. Now to see this for $k=1$ let $u,\varphi \in C_c^\infty$, pick $i = 1,...,n$, you want to find $v \in L^p$ such that $$ \int_\Omega u \frac{\partial \varphi}{\partial x_i} = - \int_\Omega v \varphi. $$ If life was good then $v = \frac{\partial u}{\partial x_i}$ the classical partial derivative would be ok. And indeed that's the case, you can first observe $$ \int_\Omega u \frac{\partial \varphi}{\partial x_i} + \int_\Omega \frac{\partial u}{\partial x_i} \varphi = \int_\Omega \frac{\partial (u\varphi)}{\partial x_i}. $$ At this point either you know integration by part (Gauss Green formula, divergence theorem or Stokes formula) or you can show by basic real analysis that $$ \forall \psi \in C^1_c(\Omega),\quad \int_\Omega \frac{\partial \psi}{\partial x_i} = 0. $$
When one speaks of an element of a Sobolev space $u\in W^{k,p}(\Omega)$ being continuous, this is typically meant to mean that there exists an element of the equivalence class $u_c \in u$ such that $u_c$ is a continuous function on $\Omega$ with finite $k,p$ Sobolev norm. Therefore, the set inclusion $u\in C^\infty_0(\Omega)$ reads "the equivalence class $u\in W^{k,p}(\Omega)$ contains an element $u_c$ such that $u_c\in C^\infty_c(\Omega).$"
Since the Sobolev space only cares about function up to a set of measure zero, we could ask questions about whether functions in the space are continuous, strongly differentiable, etc., but those questions are not invariant under modifications on a set of measure zero, so they can only be answered by seeing if there are sufficiently smooth elements of the equivalence class for which these properties apply. Once you have established that such a smooth function $u_c$ exists within an equivalence class $u$, you can then consider pointwise values of $u_c$ and perform classical oeprations on them, like evaluation and strong differentiation.
For the last question about the trace operator, we say that $u\in W^{1,p}(\Omega )\cap C(\overline{\Omega})$ if $u\in W^{1,p}(\Omega)$ contains an element $u_c\in u$ such that $u_c\in C(\overline{\Omega})$. For these functions, $u|_{\partial\Omega}$ is defined to be $u_c|_{\partial\Omega}$ and this defines the trace operator $Tu := u_c|{\partial\Omega}$. However, for a general function $u\in W^{1,p}(\Omega)$ that may not have such a continuous element, $u|_{\partial\Omega}$ is defined to be the limit $\lim_{m\to\infty}Tu_m$, where $u_m\in C^\infty (\overline{\Omega})$ and $\{u_m\}_{m=1}^\infty$ converges to $u$ in $W^{1,p}(\Omega)$. One can then show that $\{Tu_m\}_{m=1}^\infty$ is a Cauchy sequence in $L^p(\partial\Omega)$, so $Tu\in L^p(\partial\Omega)$, which are equivalence classes of functions on the boundary. Here instead of starting with the $L^p$ space and selecting continuous elements, we had to complete the space of continuous functions on the boundary formed by the elements of the Cauchy sequence $\{Tu_m\}_{m=1}^\infty$ so that our function space on the boundary contained their limit point, which necessitated the use of equivalence classes.
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please do not tag me when you post a new question. Classical questions about Sobolev spaces are discussed in the three books of Brézis, Evans, Leoni, which should encompass all the information you need.
Here are some density results for Sobolev functions, of course I skipped $p = \infty$ (can you tell why?)
$W_0^{k,p}(\Omega)$ is defined as the closure of $C_c^\infty(\Omega)$ for the $W^{k,p}(\Omega)$ norm, hence the density. There might be a touchy point in this definition when $p = \infty$ (can you tell why?).
On a bounded non empty $\Omega$, $C_c^\infty(\Omega)$ is never dense in $W^{1,p}(\Omega)$. This can be shown by the Poincaré inequality.
$C_c^\infty(\mathbb R^d)$ is dense in $W^{k,p}(\mathbb R^d)$ (can you show it?)
The Meyer Serrin density theorem: $C^\infty(\Omega) \cap W^{k,p}(\Omega)$ dense in $W^{k,p}(\Omega)$.
The Friedrich density theorem: If $u \in W^{k,p}(\Omega)$ there is $(u_j)$ a sequence of $C_c^\infty(\mathbb R^d)$ such that $u_j \rightarrow u$ in $W^{k-1,p}(\Omega)$ and forall $\omega \subset \subset \Omega$, forall $|\alpha | \leq k$, $\partial ^\alpha u_j \rightarrow \partial ^\alpha u$ in $L^p(\omega)$.
Density up to extension : if $\Omega$ is bounded, $\partial \Omega$ is Lipschitz, then $C_c^\infty(\mathbb R^d)$ is dense in $W^{k,p}(\Omega)$ (can you show it?). Actually you can weaken the regularity of $\Omega$ to be open with boundary of class $C^0$ (see Leoni, theorem 11.35).
Observe that the Meyer Serrin's theorem shows that $C(\Omega) \cap W^{k,p}(\Omega)$ is dense in $W^{k,p}(\Omega)$. Now a good question is to know when $C(\overline \Omega) \cap W^{k,p}(\Omega)$ is dense in $W^{k,p}(\Omega)$. The result in Leoni's book tells us that a sufficent condition is $\partial \Omega$ to be continuous. In fact the exercise 11.48 tells us that it's pretty sharp: there exists $\Omega \subset \mathbb R^2$ open bounded non empty for which $C(\overline \Omega) \cap W^{k,p}(\Omega)$ is never dense in $W^{k,p}(\Omega)$