On how Poincaré inequalities imply weighted ones

Martin Rathmair

doi:10.1007/s00605-019-01266-w

Titel

On how Poincaré inequalities imply weighted ones

Autor*in

Martin Rathmair

Fakultät für Mathematik, Universität Wien

Abstract

We consider a domain Ω⊂Rd equipped with a nonnegative weight w and are concerned with the question whether a Poincaré inequality holds on Ω, i.e., if there exists a finite constant C independent of f such that media.springernature.com/original/springer-static/image/art:10.1007/s00605-019-01266-w/MediaObjects/605_2019_1266_Figa_HTML.png It turns out that it is essentially sufficient that on all superlevel sets of w there hold Poincaré inequalities w.r.t. the constant weight 1 and that the corresponding Poincaré constants satisfy an integrability condition. Furthermore we provide an explicit bound of the constant C in the weighted inequality (1) in terms of the Poincaré constants of the superlevel sets. A similar statement holds true in the more general asymmetric case where we allow for certain weights ρ different from w on the right hand side of (1).

Stichwort

Weighted Poincaré inequalityPoincaré constantSobolev inequalitySuperlevel sets

Objekt-Typ

journal article

Sprache

Englisch [eng]

Persistent identifier

phaidra.univie.ac.at/o:1063666

DOI

10.1007/s00605-019-01266-w

Erschienen in

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