Sobolev functions whose inner trace at the boundary is zero (Q1810333)

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.