Singular sets of Sobolev functions
type: Item
instance of: scholarly article
title: Singular sets of Sobolev functions
zbMATH Open document ID: 1007.46033
DOI: 10.1016/S1631-073X(02)02316-6
author: Darko Žubrinić
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-03-14T00:00:00Z
review text: Let \(u:\mathbb R^N\rightarrow \overline{\mathbb R}\) be a measurable function. If \(x_0\in \mathbb R^N\) and \(\alpha>0\), say that \(u\) has a singularity at least of order \(\alpha\) at \(x_0\) if there exist \(R>0\) and \(C>0\) such that \(u(x)\geq C\left|x-x_0\right|^{-\alpha}\) for all \(x\in B(x_0,R)\). If \(A\) is a nonempty subset of \(\mathbb R^N\), say that \(u\) has a singularity of order at least \(\alpha\) on \(A\) if there exist \(R>0\) and \(C>0\) such that \(u(x)\geq Cd(x,A)^{-\alpha}\) for all \(x\in A_R\), the \(R\)-neighborhood of \(A\). The singular set of \(u\) is the set of all points in \(\mathbb R^N\) for which there exists \(\alpha>0\) such that \(u\) has a singularity at \(x_0\) at least of order \(\alpha\). It will be denoted by Sing \(u\). If \(X\) is a Banach space of measurable functions \(u\), the lower singular dimension of \(X\) is \[ s-\underline{\text{dim}}X=\sup\left\{\text{dim}_H(\text{Sing }u);\;u\in X\right\} \] where \(\text{dim}_H\) denotes the Hausdorff dimension. The upper singular dimension of \(X\) is \[ s-\overline{\text{dim}}X=\sup\left\{\text{dim}_H(e-\text{Sing }u);\;u\in X\right\} \] where \[ e-\text{Sing }u=\left\{x_0\in \mathbb R^N;\;\overline{\lim}_{r\rightarrow 0} \frac 1{r^N}\int_{B(x_0,r)}u(x)dx=+\infty\right\}. \] For \(A\subset \mathbb R^N\), denote also by \(\overline{\text{dim}}_BA\) the upper box dimension of \(A\) [see \textit{K. Falconer}, ``Fractal geometry: Mathematical foundations and applications'', New-York (1990; Zbl 0689.28003)]. The main result of the present paper is as follows: if \(1<p<+\infty\), \(k\in \mathbb N\), \(kp<N\) and \(A\) is a compact subset of \(\mathbb R^N\) such that \(\overline{\text{dim}}_BA<N-kp\), then there exists a function \(u\in W^{k,p}(\mathbb R^N)\) which is singular exactly on \(A\). Moreover, the lower and upper singular dimensions of \(W^{k,p}(\mathbb R^N)\) coincide and are equal to \(N-kp\). If \(kp=N\), the upper singular dimension of \(W^{k,p}(\mathbb R^N)\) i…
reviewed by: Emmanuel Russ
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 1754814
zbMATH Keywords: singular set
zbMATH Keywords: Bessel potential
zbMATH Keywords: singular dimension
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1754814