Gagliardo-Nirenberg inequality for bounded domain

Question

For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that $$ \|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}. $$ This is also true if $u\in W^{1,2}_0(\Omega)$ for a bounded domain $\Omega$ in $\Bbb R^2$.

Is it still true if we modify this in the same way as Poincare ineqaulity, i.e. $$ \|u-\overline u\|_4\le C \|u-\overline u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}, $$ where $\overline u=\frac 1{|\Omega|}\int u dx$, for any $u\in W^{1,2}(\Omega)$?

I have seen the version for $u\in W^{1,2}(\Omega)$ where we need to add another term $C'\|u\|_s$ but that is not what I want.

Edit: The usual proof of this interpolation inequality use the fact that $$ \|u\|_{L^{n/(n-1)}(\Bbb R^n)} \le \|\nabla u\|_{L^1(\Bbb R^n)} $$ for compactly supported $u$. In bounded domain $\Omega$, the $u$ on the LHS cannot be easily replaced with $u-\overline u$ since Rellich-Kondrachov theorem doesn't give compactness for the embedding $W^{1,1}$ into $L^{1^*}$. Thus it seems to me that we'd need to modify the proof somewhere else if the statement is true.

Answer 1

If you assume that $\Omega$ is a bounded uniform extension domain, then your desired inequality holds true. By uniform extension domain, I mean that there exists a linear extension operator $E$ which maps $L^p(\Omega)$ to $L^p(\mathbb{R}^n)$, where $1<p<\infty$, and $W^{1,2}(\Omega)$ to $W^{1,2}(\mathbb{R}^n)$ at the same time, and both continuously.

First, for all functions $u \in W^{1,2}(\Omega)$ you have from the extension property for $\Omega$ and the standard Poincaré inequality a constant $C_P$ such that $$\|w-\bar w\|_{W^{1,2}(\Omega)} \leq C_P \|\nabla w\|_{L^2(\Omega)}$$ for all $w \in W^{1,2}(\Omega)$, where $\bar w$ is the average of $w$ as you defined it.

Now, using the extension operator properties and the classical Gagliardo-Nirenberg inequality, we have \begin{align*}\|u\|_{L^4(\Omega)} \leq \|Eu\|_{L^4(\mathbb{R}^n)} & \leq C_{GN} \|Eu\|_{L^2(\mathbb{R}^n)}^{\frac12}\|\nabla Eu\|_{L^2(\mathbb{R}^n)}^{\frac12} \\ & \leq C_{GN} \|Eu\|_{L^2(\mathbb{R}^n)}^{\frac12}\|Eu\|_{W^{1,2}(\mathbb{R}^n)}^{\frac12} \leq C_{GN} C_E \|u\|_{L^2(\Omega)}^{\frac12}\|u\|_{W^{1,2}(\Omega)}^{\frac12},\end{align*} again for all $u \in W^{1,2}(\Omega)$. But then inserting $w-\bar w$ for $w \in W^{1,2}(\Omega)$ and using the above consequence of the Poincaré inequality gives you the result $$\|w-\bar w\|_{L^4(\Omega)} \leq C_{GN}C_EC_P^\frac12 \|w-\bar w\|_{L^2(\Omega)}^{\frac12}\|\nabla w\|_{L^{2}(\Omega)}^{\frac12}$$ as desired.

This procedure of course always works when there are suitable extension operators available (and in principle, we only needed $E$ to simultaneously act on $L^2, L^4$ and $W^{1,2}$). Such operators were constructed even for extremely irregular domains, see e.g. the paper of Rogers cited below, or Thm. 7.25 in Gilbarg/Trudinger for the a bit less irregular but still quite general Lipschitz domains.

Rogers, Luke G., Degree-independent Sobolev extension on locally uniform domains, J. Funct. Anal. 235, No. 2, 619-665 (2006). ZBL1158.46025.