Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

Rita Ferreira; Carolin Kreisbeck; Ana Margarida Ribeiro

doi:10.1016/j.na.2014.09.007

Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

Rita Ferreira^*, Carolin Kreisbeck, Ana Margarida Ribeiro

^*Corresponding author for this work

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

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 kth-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⊂ℝⁿ. 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.

Original language	English (US)
Pages (from-to)	199-214
Number of pages	16
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	112
DOIs	https://doi.org/10.1016/j.na.2014.09.007
State	Published - Jan 2015

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Ltd. All rights reserved.

Keywords

Higher-order Sobolev spaces
Singular functionals

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2014.09.007

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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 kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂ℝn. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Br{\'e}zis' overview article [Br{\'e}zis (2002)] to the higher-order case, and extend the work [Borghol (2007)] to a more general setting.",

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AU - Ferreira, Rita

AU - Kreisbeck, Carolin

AU - Ribeiro, Ana Margarida

PY - 2015/1

Y1 - 2015/1

N2 - 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 kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂ℝn. 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.

AB - 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 kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂ℝn. 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.

KW - Higher-order Sobolev spaces

KW - Singular functionals

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DO - 10.1016/j.na.2014.09.007

M3 - Article

AN - SCOPUS:84907932220

SN - 0362-546X

VL - 112

SP - 199

EP - 214

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

ER -

Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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