• According to (0x67420435) we can find smooth functions which approximate in \(W^{k,p}_{\text{loc}}(U)\) with \(p\in [1,\infty)\) for arbitrary \(U\).

• If \(U\) is bounded there exist functions in \(\mathcal{C}^\infty(U)\cap W^{k,p}(U)\) which approximate in \(W^{k,p}(U)\). [1, 5.3 Theorem 2]

• Furthermore, by definition elements in \(W^{k,p}_0(U)\) can be approximated by test functions .

References Link to heading

1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.