Characterization of polynomials and higher-order Sobolev spaces in terms of nonlocal functionals involving difference quotients
- Ferreira, Rita
- Kreisbeck, Carolin
- Ribeiro, Ana Margarida
2010 Mathematics Subject Classification
- Higher-order Sobolev spaces, nonlocal functionals
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 [Russ. Math. Surv. 57 (2002), pp. 693-708] to the higher-order case, and extend the work by Borghol [Asymptotic Anal. 51 (2007), pp. 303-318] to a more general setting.
- Nonlinear Anal. Real World Appl., 112 (2015), pp. 199-214, DOI 10.1016/j.na.2014.09.007 .