Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients. (January 2015)

• Record Type: Journal Article
• Authors: Ferreira, Rita; Kreisbeck, Carolin; Ribeiro, Ana Margarida
• Is Part Of: Nonlinear analysis. Volume 112(2015)
• Page Start: 199
• Page End: 214
• Publication Date: 2015-01
• Subjects: 46E35; Higher-order Sobolev spaces -- Singular functionals; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.09.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a k th-order Sobolev space. One of the main theorems is a new characterization of W k, p ( Ω ), k ∈ N and p ∈ ( 1, + ∞ ), for arbitrary open sets Ω ⊂ R n . In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis' overview article [Brézis (2002)] to the higher-order case, and extend the work [Borghol (2007)] to a more general setting.