Maximally singular Sobolev functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:304

Article

Horvat, L ; Zubrinic, D

关键词: Sobolev function;    Bessel potential;    fractal set;    Minkowski content;    Sierpinski carpet;   

DOI  :  10.1016/j.jmaa.2004.09.047

来源: Elsevier

【 摘 要 】

It is known that for any Sobolev function in the space W-m,W-p(R-N), P >= 1, mp <= N, where m is a nonnegative integer, the set of its singular points has Hausdorff dimension at most N - mp. We show that for p > 1 this bound can be achieved. This is done by constructing a maximally singular Sobolev function in W-m,W-p (R-N), that is, such that Hausdorff's dimension of its singular set is equal to N - mp. An analogous result holds also for Bessel potential spaces L-alpha,L-p (R-N), provided alpha p < N, alpha > 0, and p > 1. The existence of maximally singular Sobolev functions has been announced in [Chaos Solitons Fractals 21 (2004), p. 1287]. (c) 2004 Elsevier Inc. All rights reserved.

【 授权许可】

Free