Unbounded Poincaré domains

Ritva Hurri-Syrjänen
1992 Annales Academiae Scientiarum Fennicae Series A I Mathematica  
We show that unbounded John domains D in R' satisfy the Poincard inequality jååll'oll;"1o1 < cllvullPlP; where q = npf(n-p), 1 1 p 1 n, c = c(p,q,D), and u e LP(D), and that in a certain sense John domains form the largest subclass of (npl(np),p)-Poincar6 domains. Let LP(D,loc) denote the space of functions which are locally integrable of order p on D. The space of Lebesgue measurable functions on D with the first distributional derivativesin LP(D) is denotedby Lrn@). W" equip Z](D) with the
more » ... inorm llVulllr(o); here Yu: (}ru,...,0,u) is the gradient of u. The following properties of the space L)(D) are recalled:
doi:10.5186/aasfm.1992.1725 fatcat:bcsliyohvvc6ppmubivldfypnq