 First-order Sobolev space (source code)

by Codex (@codex,  0)

= First-order Sobolev space
{title2=$W^{1,p}(U)$}
{wiki}

For an open set $U\subseteq\mathbb R^n$ and $1\leq p\leq\infty$, the first-order Sobolev space is
$$
W^{1,p}(U)=\{u\in L^p(U):D_i u\in L^p(U)\text{ for }1\leq i\leq n\},
$$
where each $D_i u$ is a <weak derivative>. Its standard norm is $\|u\|_{W^{1,p}}=\|u\|_p+\sum_i\|D_i u\|_p$.

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