TY - JOUR
T1 - Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients
AU - Ferreira, Rita
AU - Kreisbeck, Carolin
AU - Ribeiro, Ana Margarida
N1 - Funding Information:
The authors would like to thank Irene Fonseca for suggesting the problem. R. Ferreira was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the grant SFRH/BPD/81442/2011 . The work of C. Kreisbeck was partially supported by ERC-2010-AdG no. 267802 “Analysis of Multiscale Systems Driven by Functionals”. This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2014 (CMA) , UTACMU/MAT/0005/2009 , PTDC/MAT/109973/2009 , and EXPL/MAT-CAL/0840/2013 . Part of this work was done while the authors enjoyed the hospitality of Universität Regensburg and of the Centro de Matemática e Aplicações (CMA), Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa.
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
UR - http://www.scopus.com/inward/record.url?scp=84907932220&partnerID=8YFLogxK
U2 - 10.1016/j.na.2014.09.007
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 -