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Let $M \subset \mathbb R^n$ be a submanifold of euclidean space whose boundary is locally a Lipschitz graph. Let $\omega \in L^2\Lambda^k(M)$ be a differential form with square-integrable coefficients.

The Poincaré constant for $k = 0$ can be estimated by $1/2 \cdot \operatorname{diam}(M)$ in case the domain is convex and bounded. I have not seen a formal statement in literature that provides a comparable statement for $k > 0$.

Could you provide me with a reference? I need this only for citation and an estimate in terms of the diameter of the domain would be completely sufficient. Thank you.

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  • $\begingroup$ Did you just go edit an accent into every question you could find about Poincare, John B? Thereby bumping a huge number of old questions to the front page (and an equal number of newer questions off the front page? Please don't do that sort of thing. If you want to edit 3 or 4 questions a day, fine, but not a dozen in a matter of minutes, please. $\endgroup$
    – Gerry Myerson
    Dec 31, 2015 at 4:02

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Check out Chapter 3 in the book by Agarwal et al.,Inequalities for differential forms

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