I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of $\Omega$ such that $\| u- u_{\varepsilon}\| < \varepsilon$. I would like to say that $\partial \Omega$ is Lipschitz and $\Omega$ is bounded also.
In particular I am also interested in how to handle the boundary.