A Density Result for Locally Uniform Domains

In this chapter, we show that smooth and compactly supported functions on \(\mathbb {R}^d\) whose support stays away from D are dense in \(\mathrm {C}^\infty _D(O)\cap \mathrm {W}^{1,p}(O)\) , see Definition 3.1 for the meaning of \(\mathrm {C}^\infty _D(O)\) . In particular, both classes of functions have the same \(\mathrm {W}^{1,p}(O)\) -closure.