On conditions of validity of the Poincar\'e inequality

A. I. Nazarov ; S. V. Poborchi

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 43, Tome 410 (2013), pp. 104-109

Résumé

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.