Sobolev inequality in $W_0^{1,p}$

Question

If $\Omega \subseteq \mathbb{R}^N$ is an open bounded domain and $1<p<N$, then the classical Sobolev Inequality: $$\| u\|_{p^*,\Omega} \leq C\ \| \nabla u\|_{p,\Omega}$$ holds with $C=C(p,N,\Omega)>0$ and $p^*:= Np/(N-p)$ for any $u\in W_0^{1,p}(\Omega)$.

What about the case $p\geq N$? May I take the $L^\infty$-norm in the LHside?

If I remember correctly, in general I cannot get the inequality with $\| \cdot \|_\infty$, for there are counterexemples of unbounded Sobolev functions... But, what if I know "a priori" that $u\in L^\infty(\Omega) \cap W_0^{1,p}(\Omega)$?

Any reference? (Adams-Fournier? Brezis?)

Thanks in advance.

Answer 1

Sobolev inequality gives you that $W_0^{m,p}$ is embedded $L^{p^*}$, and by an interpolation argument, you can embed $W_0^{m,p}$ in $L^q$ for $p\leq q\leq p^*$ for $mp<N$, in your case, $m = 1$, $p<N$.

If you want $mp=N$, $W_0^{m,p}$ is embedded in $L^q$ with $1<p\leq q <\infty$. In your case $m=1$, $p=N$, but I am not sure what happens if $p>N$.

For a proof of these facts, see page 20 of http://people.bath.ac.uk/masgrb/Sobolev/notes.pdf

I am not sure if this answers your question. I am also only a beginner in this area.

Answer 2

For the case $p=N$ take a look here. There you will find all you want. Note that the function $f$ defined by me is a counter example for what you want, also, even if you ask $u\in L^\infty$, you dont get what you want. Take a look in the answer and you will have all the explanations you need.

When $p>N$ and you have some regularity in the boundary, then your functions are continuous, even Holder continuous. I suggest you to take a look in any good book about Sobolev Spaces. For example the book ok Leoni is a good one, there you will find all you need.