Approximation of Sobolev Function in $W^{k,p}(\Omega)$ by Continuous Functions of limited Regularity

Question

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

Answer 1

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 $W^{k,p}(\Omega)$

Answer 2

To approximate a Sobolev function $f$ in $W^{k,p} (\Omega)$ by continuous functions on $\Omega$, one can use the following approach:

Choose a smooth function $g$ that approximates $f$ in the $L^p$ norm. This can be done using the approximation property of smooth functions, which states that for any $\epsilon>0$, there exists a smooth function $g$ such that $|f-g|_p < \epsilon$.

Choose a smooth function $h$ that approximates $g$ in the $W^{k,p}$ norm. This can be done using the density of smooth functions in the Sobolev space $W^{k,p} (\Omega)$, which states that for any $\epsilon>0$, there exists a smooth function $h$ such that $|g-h|_{W^k_p} < \epsilon$.

The function $h$ is a continuous function on $\Omega$ that approximates $f$ in the $W^{k,p}$ norm.

Alternatively, one can use the fact that the Sobolev space $W^{k,p} (\Omega)$ is compactly embedded in the space $C^0(\Omega)$ of continuous functions on $\Omega$, which means that any sequence of functions in $W^{k,p} (\Omega)$ has a convergent subsequence in $C^0(\Omega)$. This allows us to use a diagonalization argument to construct a sequence of continuous functions that converges to $f$ in the $W^{k,p}$ norm.