Let \(u\in W^{k,p}_{\text{loc}}(\Omega)\) with \(p\in [1,\infty)\). Then the mollification \(u_\varepsilon\) converges to \(u\) in \(W^{k,p}_{\text{loc}(\Omega)}\), as \(\varepsilon\to 0\). To be more precise, for every compact subset \(U\subset \Omega\) the mollification \(u_\varepsilon\) converges in \(W^{k,p}(U)\). [1, 5.3 Theorem 1]

Proof idea Link to heading

Using the relation \(D^\alpha u^\varepsilon=\eta_\varepsilon * D^\alpha u\) this is an immediate consequence of (0x6741bab6) .

References Link to heading

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