I want to check the validity of my approach to finding a tigehter bound on the Poincare constant in $W^{1,p}(\Omega)$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set. This will be somewhat long, so I will divide this post into sections.
Problem Statement: Let $u\in W_0^{1,p}(\Omega)$, with $2<p<\infty$. There is a constant $C$, depending only on $\Omega$ and $p$ such that $\forall u\in W^{1,p}(\Omega)$ , we have $||u||_{L^p(\Omega)}\leq C||\nabla u||_{L^p(\Omega)}$
Results: Optimal $L^1(\Omega)$ and $L^2(\Omega)$ estimates have been found already for convex domains.
If $u\in W_0^{1,p}(\Omega)$, then $u\in L^p(\Omega)$. I will add the assumption $u\in W_0^{1,q}(\Omega)$, Now consider the Poincare inequality.
$||u||_{L^p(\Omega)}\leq C||\nabla u||_{L^p(\Omega)}$
Note that the Lebesgue measure $\mu(\Omega)$ is finite, as $\Omega$ is a bounded set. This allows us to use $L^p$ "nesting theorem".
The statement of this theorem is:
Let $1\leq p<q\leq \infty$, $\mu(\Omega)<\infty$, then $||u||_{L^p(\Omega)}\leq \mu(\Omega)^{(1/p)-(1/q)}||u||_{L^q(\Omega)}$, which shows $L^q(\Omega)\subset L^p(\Omega)$. The proof of this statement can be found here:
How to show that $L^p$ spaces are nested?
This essentially tells us if we have $u\in L^q(\Omega)$, we also have $u\in L^p(\Omega)$. That said, depending on the constant $L=\mu(\Omega)^{(1/p)-(1/q)}$, $||u||_{L^p}$ is not smaller than the $L^q$ norm if $0<L<1$. We can partition the result into two cases, starting with $L\geq 1$.
My plan here is to play around with the $L^p$ and $L^q$ norms of $u$ and $\nabla u$, and the constant $L$, to get a decent, if not optimal bound on the Poincare constant $C$. I am optimistic about when the geometry of $\Omega$ is nice enough, where $\mu(\Omega)=1$.
Addendum: I also would like to reconcile this with the optimal $L^1$ and $L^2$ estimates already known, and I am curious as to what happens if $\Omega$ is disconnected. My hunch is that the value of the norms would change, but my approach wouldn't, as it does not utilize connectedness of the domain $\Omega$