$L^p-$bound of the gradient

Question

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. If $u\in W^{1,p}(\Omega)$, is there some kind of "inverse" Poincaré inequality such that we can control the $L^p(\Omega)-$norm of $\nabla u$ using some norm for $u$?

Answer 1

No, take a weakly differentiable function that oscillates wildly around $0$, but is bounded by some $\epsilon>0$.

To be precise: Let $\Omega =(0,1)$. Assume we find $C>0$ such that for all $u \in W^{1,p}(\Omega)$ we have $$||\nabla{u}||_{L^{p}} \le C ||u||_{L^{p}}$$

Then let $u_{n}$ be the piecewise linear function of $n$ bumps of height $\frac{1}{n}$ and width $\frac{1}{n}$. Then the $L^{p}$-norm is bounded by $\frac{1}{n}$, but since $|u'|=1$ almost everywhere (weakly), we have $||u'||_{L^{p}} = 1$

Letting $n \rightarrow \infty$, we have a contradiction to existence of such $C$.