## 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
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### References:

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