For $u,g\in L^1_{\mathrm{loc}}(U)$, the function $g$ is the $i$th <weak derivative> $D_i u$ when
$$
\int_UuD_i\varphi=-\int_Ug\varphi
$$
for every <test function> $\varphi\in C_c^\infty(U)$. This is the <integration by parts> identity with no boundary term and agrees with the ordinary derivative whenever $u$ is classically differentiable.

For $1\leq p\leq\infty$, the <first-order Sobolev space> is
$$
W^{1,p}(U)=\{u\in L^p(U):D_i u\in L^p(U),\ 1\leq i\leq n\},
$$
with norm, for example,
$$
\|u\|_{W^{1,p}(U)}=\|u\|_{L^p(U)}+\sum_{i=1}^n\|D_i u\|_{L^p(U)}.
$$
Functions equal <almost everywhere> represent the same Sobolev element.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 105 / 2 / a / Solution

{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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