Suppose that $u:\mathbb R^N \rightarrow \mathbb R$ and that $\Omega \subset \mathbb{R}^N$ is a bounded convex domain. Define the pointwise global Lipschitz constant of $u$ in $\Omega$ by $$ L(x) = \sup_{y \in \Omega} \frac{|u(x) - u(y)|}{|x-y|}. $$
Question: Are there any natural conditions that ensure $L\in L^p(\Omega)$?
If $u\in W^{1,\infty}(\mathbb R ^N)$, then $u$ is locally Lipschitz in $\mathbb R ^N$ so of course we have $L \in L^\infty (\Omega)$. But what if we assume say that $u \in W^{1,p}(\mathbb R ^N) \cap C^\infty(\Omega)$ and that $u$ is bounded? I suspect that this does not imply $L\in L^p (\Omega)$ but so far I didn't find a counter example.