Sobolev's inequality for $u \in W^{1,p}(\Omega)$

Question

Let $\Omega \subset \mathbb{R}^n$, it is well known that we have the Sobolev's inequality $$ \|u\|_{L^q(\Omega)} \leq C\|\nabla u\|_{L^{p}(\Omega)}, \hspace{10pt} \text{ for all $u \in W_0^{1,p}(\Omega)$},$$ Here $p$, $q$ and $n$ are some suitable numbers.

However, I can not find the same result for $u \in W^{1,p}(\Omega)$. The best result I found is the Sobolev's embedding, that is $$ \|u\|_{L^q(\Omega)} \leq C\| u\|_{W^{1,p}(\Omega)}, \hspace{10pt} \text{ for all $u \in W^{1,p}(\Omega)$},$$ for some suitable $p$, $q$ and $n$ again. In this case, there is $\int|u|^p$ on the right-hand side. It can not be absorbed into left-hand side unless we use Holder's inequality and assume the domain $\Omega$ is small.

Can someone give the result for $u \in W^{1,p}(\Omega)$? May be some references? Thanks!

Answer 1

You may be looking at the Poincaré-Wirtinger inequality:

Assume that $1 \leq p \leq \infty$ and that $\Omega$ is a bounded connected open subset of the $n$ -dimensional Euclidean space $\mathbf{R}^{n}$ with a Lipschitz boundary (i.e., $\Omega$ is a Lipschitz domain). Then there exists a constant $C$, depending only on $\Omega$ and $p$, such that for every function $u$ in the Sobolev space $W^{1, p}(\Omega)$, $$ \left\|u-u_{\Omega}\right\|_{L^{p}(\Omega)} \leq C\|\nabla u\|_{L^{p}(\Omega)} $$ where $$ u_{\Omega}=\frac{1}{|\Omega|} \int_{\Omega} u(y) \mathrm{d} y $$ is the average value of $u$ over $\Omega$, with $|\Omega|$ standing for the Lebesgue measure of the domain $\Omega$.

Note that we must remove the constant $u_{\Omega}$ otherwise just take the constant function $u=1$, then $C\|\nabla u\|_{L^{p}(\Omega)}=0$ but $ \left\|u\right\|_{L^{p}(\Omega)} >0.$

Answer 2

Just to extend on @SonGohan's excellent answer:

1. By combining your second estimate and the estimate from @SonGohan's answer, you directly get $$\|u - u_\Omega\|_{L^q(\Omega)} \le C \|\nabla u\|_{L^p(\Omega)}.$$

2. Generally speaking, in order for such an estimate to hold in some space $X \subset W^{1, p}(\Omega)$, you need that $0$ is the only weak solution of $$-\Delta u = 0$$ in $X$. That is, if $u \in X$ satifies $$\int_\Omega \nabla u \cdot \nabla \varphi = 0 \qquad \text{for all $\varphi \in C^1(\overline \Omega)$},$$ then $u \equiv 0$. This works for instance for $X = \{u \in W^{1, p}(\Omega): u = 0 \text{ on } \partial \Omega\}$ or for $X = \{u \in W^{1, p}(\Omega): u_\Omega = 0 \}$.

The second remark remains true for more general operators (and is then probably also more interesting.)