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

Swanson, David; Ziemer, William P.

Ark. Mat. 37, No. 2, 373-380 (1999).

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.

Reviewer: Walter Farkas (München)

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

Keywords:

Sobolev functions; BV functions; \(p\)-capacity; Hausdorff measure

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