Geodesic
On conditions of validity of the Poincaré inequality
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 104-109 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $l=1,2,\dots$, $p,q\ge1$, let $G$ be a domain in $\mathbb R^n$, and let $\mathcal P_l$ be the space of polynomials in $\mathbb R^n$ of degree less than $l$. We show that inclusion $\mathcal P_l\subset L_q(G)$ (and hence $\mathrm{mes}_n (G)<\infty$) is necessary for validity of the generalized Poincaré inequality $$ \inf\{\|u-P\|_{L_q(G)}\colon P\in\mathcal P_l\}\le\mathrm{const}\,\|\nabla_l u\|_{L_p(G)},\quad u\in L_p^l(G). $$ Thus, this inequality is equivalent to continuity of the embedding $L_p^l(G)\to L_q(G)$. In the case of critical Sobolev exponent $q=np/(n-lp)$ for $lp this fact is not true. We give some sufficient conditions for validity of the Poincaré inequality in domains of infinite volume. 
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     author = {A. I. Nazarov and S. V. Poborchi},
     title = {On conditions of validity of the {Poincar\'e} inequality},
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     pages = {104--109},
     year = {2013},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_410_a3/}
}
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A. I. Nazarov; S. V. Poborchi. On conditions of validity of the Poincaré inequality. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 104-109. http://geodesic.mathdoc.fr/item/ZNSL_2013_410_a3/

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