Show there does not exist any constant $C>0$ such that, for any $\phi\in C^{\infty}_c(\mathbb{R}^2)$, the inequality $$\lVert\phi\rVert_\infty\leq C\lVert\nabla\phi\rVert_2$$ holds.
Obviously, we can scale to take $\lVert\phi\rVert_\infty=1$. I am thus trying to construct a sequence $\phi_n$ which reaches $1$ at $0$ but for which the $L^2$ norm of the gradient becomes arbitrarily small. Surely some type of bump function will work, but I can't seem to find an explicit form. Or is there a slicker way to approach this question?
(Why just $\mathbb{R}^2$ and not $\mathbb{R}^n$? One can prove that any inequality of the form $\lVert\phi\rVert_q\leq C\lVert\nabla\phi\rVert_p$ for $\phi\in C^{\infty}_c(\mathbb{R}^n)$ must satisfy $q^{-1}=p^{-1}-n^{-1}$)