Let $\Omega$ be a bounded connected open subset of $\mathbb{R}^n$ with Lipschitz boundary and $1\leq q \leq \infty$. By the Poincaré inequality, there is $c>0$ such that
$$\|u\|_{L^q(\Omega)} \leq c \|\nabla u \|_{L^q(\Omega)}$$
for any $u \in C^1(\overline{\Omega})$ satisfying $\int_\Omega u=0$. I would like to find a demonstration for it that uses induction on $n$. The reason is that I am trying to prove it in a different class of spaces than $L^q$ (in which I have already checked that the case $n=1$ holds), and I think the proof in $L^q$ spaces would serve as a kind of guidance.
The classical proof for it uses compact embeddings, which I am avoiding since I do not know if they hold in my class of functional spaces. As for a proof by induction, until now, I have found it only under the hypothesis that $u=0$ on $\partial \Omega$ instead of $\int_\Omega u=0$.
Do you guys know some references?