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A Poincare-Wirtinger inequality holds over a domain $\Omega \subseteq \mathbb R^n$ with exponentnt $1 \leq p \leq \infty$ if there exists $C(p,\Omega) > 0$ such that

$\| u - \operatorname{avg}(u) \|_{L^p(\Omega)} \leq C \| \nabla u \|_{L^p(\Omega)}$

for every $u \in L^p(\Omega)$. We have written $\operatorname{avg}(u)$ for the average of $u$ over the domain $\Omega$.

For which domains does such an inequality hold, with possible dependence on $p$? I expect the answer to be dependent on that integrability exponent.

The statement is true for bounded Lipschitz domains, as one may easily find in the literature. I am wondering what are necessary and sufficient conditions.

EDIT: If you think no such conditions are known, I am interested in domains which do not support such an inequality.

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    $\begingroup$ I don't know exactly what necessary or sufficient conditions are, but they'll need to account for domains that have infinitely many "nearly disconnected" components, at least for $p=2$. Such domains have a Neuman Laplacian with essential spectrum containing $0$. I believe this is called the "regions and passages" construction, for example see Hempel-Seco-Simon 1990 "The Essential Spectrum of Neumann Laplacians on Some Bounded Singular Domains" $\endgroup$
    – Neal
    Commented Nov 9, 2020 at 21:49
  • $\begingroup$ I recently learned that there is the notion/field of research "Poincaré domain" which is concerned exactly with your question. See e.g. jstor.org/stable/2001337. (I am not an expert on this myself, just wanted to point to the term.) $\endgroup$
    – Hannes
    Commented Nov 10, 2020 at 9:11

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