= Sobolev approximation by mollifiers
{c}
{title2=$\rho_\varepsilon*u\to u\text{ in }W^{k,p}(\mathbb R^n)$}
For $1\leq p<\infty$, an integer $k\geq0$, and a normalized nonnegative <mollifier> $\rho\in C_c^\infty(\mathbb R^n)$, put $\rho_\varepsilon(x)=\varepsilon^{-n}\rho(x/\varepsilon)$. If $u\in W^{k,p}(\mathbb R^n)$, its <mollification> is smooth and converges to $u$ in the <Sobolev norm>. The definition of a <weak derivative> and <Fubini's theorem> give $\partial^\alpha(\rho_\varepsilon*u)=\rho_\varepsilon*\partial^\alpha u$ for every <multi-index> with $|\alpha|\leq k$. The <approximate identity> theorem in the <Lp space> $L^p$ applied to these finitely many <weak derivatives> proves the claim. On a domain with boundary this direct <convolution> argument applies only away from the boundary unless an extension is first supplied.
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