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

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 $u\in L^\infty(\Omega) \cap W_0^{1,p}(\Omega)$?

• I don't think that you can have an estimate like this one: $$\lVert u \rVert_\infty \le C \lVert \nabla u \rVert_p.$$ What you do have, as you already know very well, is Morrey's inequality (cfr. Brezis, Liguori ed., pag. 264): $$\frac{\lvert u(x)-u(y)\rvert}{\lvert x-y\rvert^\alpha}\le C \lVert \nabla u \rVert_p.$$ The reason why I am telling this is that in $\mathbb{R}^n$ the first inequality does not scale well while the second does. Sure, you are talking of bounded domains and so there is no scaling. But still I've got this feeling. Mar 26, 2013 at 0:14

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.

When $\Omega\subset% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}$, $n\geq2$ is a bounded domain, we have by the Sobolev embedding theorems that $W_{0}^{k,p}\left( \Omega\right) \subset L^{q}\left( \Omega\right) ,~1\leq q\leq\frac{np}{n-kp},~kp<n$ and that $W_{0}^{k,\frac{n}{k}}\left( \Omega\right) \subset L^{q}\left( \Omega\right) ,~1\leq q<\infty$. However, there are counterexamples to the embedding $W_{0}^{k,\frac{n}{k}}\left( \Omega\right) \subset L^{\infty}\left( \Omega\right)$. In this case, it was proposed independently by Yudovich, Pohozaev and Trudinger that $W_{0}^{1,n}\left( \Omega\right) \subset L_{\varphi_{n}}\left( \Omega\right)$ where $L_{\varphi_{n}}\left( \Omega\right)$ is the Orlicz space associated with the Young function $\varphi_{n}(t)=\exp\left( \beta_{n} \left\vert t\right\vert ^{n/(n-1)}% \right) -1$ for some positive $\beta_{n}$. Moreover, Moser explored more in this direction and further found the largest positive real number $\beta_{n}$. In fact, in his 1971 paper [A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71), 1077–1092.], J. Moser proved the following result: There exist sharp constant $\beta_{n}% =n\omega_{n-1}^{\frac{1}{n-1}}$, where $\omega_{n-1}$ is the area of the surface of the unit $n-$ball, such that $$\frac{1}{\left\vert \Omega\right\vert }\int_{\Omega}\exp\left( \beta \left\vert u\right\vert ^{\frac{n}{n-1}}\right) dx\leq c_{0}%$$ for any $\beta\leq\beta_{n}$, any $u\in W_{0}^{1,n}\left( \Omega\right)$ with $\int_{\Omega}\left\vert \nabla u\right\vert ^{n}dx\leq1$. This constant $\beta_{n}$ is sharp in the sense that if $\beta>\beta_{n}$, then the above inequality can no longer hold with some $c_{0}$ independent of $u$. The same result for $W_{0}^{k,p}\left( \Omega\right)$ was proved by D. Adams in his paper [A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. (2) 128 (1988), no. 2, 385–398.]

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.

• @Tomás you are right, let me correct this. Mar 25, 2013 at 23:29