×

Sobolev functions whose inner trace at the boundary is zero. (English) Zbl 1021.46027

Let \(k\) be a positive integer, let \(1<p<\infty\) and let \(W^{k,p}(\Omega)\) be the Sobolev space on an arbitrary open set \(\Omega\subset \mathbb{R}^n\). Let \(W^{k,p}_0(\Omega)\) be the closure in \(W^{k,p}(\Omega)\) of the family of \(C^{\infty}\) functions in \(\Omega\) with compact support.
The main result of the paper looks as follows. A function \(f\) is in \(W_0^{k,p}(\Omega)\) if, and only if, \[ \lim\limits _{r\to 0}\frac{1}{r^n} \int _{B(x,r)\cap \Omega}|D^{\beta}f(y)|dy =0 \] for all \(x\in \mathbb{R}^n\setminus\Omega\) with the exception of a set of zero \(C_{k-|\beta |,p}\)-capacity and for all multi-indices \(\beta\) with \(0\leq |\beta |\leq k-1\). The paper is self-contained and very clearly written.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A45 Functions of bounded variation, generalizations
28A78 Hausdorff and packing measures
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [AH]Adams, D. R., andHedberg, L. I.,Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg, 1996.
[2] [B]Bagby, T., Quasi topologies and rational approximation,J. Funct. Anal. 10 (1972), 259–268. · Zbl 0266.30024
[3] [GZ]Gariepy, R. F. andZiemer, W. P.,Modern Real Analysis, PWS Publishing Co., Boston, Mass., 1994.
[4] [F]Federer, H.,Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1969. · Zbl 0176.00801
[5] [FR]Fleming, W. H. andRishel, R., An integral formula for total variation,Arch. Math. (Basel) 11 (1960), 218–222. · Zbl 0094.26301
[6] [H]Havin, V. P., Approximation in the mean by analytic functions,Dokl. Akad. Nauk SSSR 178 (1968), 1025–1028 (Russian). English transl.:Soviet Math. Dokl. 9 (1968), 245–248.
[7] [Z]Ziemer, W. P.,Weakly Differentiable Functions, Springer-Verlag, New York, 1989. · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.