I have an exam in a couple of days so am going through some practice questions, and am stumped by the following:
i) Does $u\in W^{1,3}(\mathbb{R}^2)\implies u\in L^\infty(\mathbb{R}^2)$?
ii) Does $u\in W_0^{1,2}(\mathbb{R}^3)\implies u\in L^\infty(\mathbb{R}^3)$?
I'm not really sure which embeddings theorems can be used to tackle these; I have a Poincare-type inequality for an $L^\infty$ bound but that doesn't apply to these cases. These are just two questions of a bunch of similar examples so if someone could help me with these I could hopefully crack the rest. Thanks!
The answer to the first is yes, using the general Sobolev inequality in Section 5.6. Evans uses bounded domains, but you can subdivide $\mathbb R^m$ into nonoverlapping cubes (or balls with controlled overlap) and get $L^\infty$ estimates on each cube individually, where the estimate is controlled in a uniform way by $\|u\|_{W^{1,3}(\mathbb R^2)}$.
For the second, the answer is no. Look at a function of the form $f(x) = |x|^{-\alpha} - 1$ if $|x| \le 1$ and $f(x) = 0$ if $|x| > 1$. As long as $\alpha$ is chosen appropriately (I think $\alpha = \frac 14$ will work) both $f$ and $Df$ are sufficiently integrable but $f$ is not bounded.