= Solution
For $I=(-a,a)$, the <first-order Sobolev space> is
$$
\boxed{H^1(I)=\{u\in L^2(I):\exists v\in L^2(I),\ \int_Iu\varphi'=-\int_Iv\varphi\text{ for every }\varphi\in C_c^\infty(I)\}.}
$$
The <weak derivative> $v=u'$ is unique as an <equivalence class> modulo equality <almost everywhere>. We use the standard <Sobolev norm> $\|u\|_{H^1(I)}^2=\|u\|_{L^2(I)}^2+\|u'\|_{L^2(I)}^2$. Pointwise values of an arbitrary representative have no intrinsic meaning; the <one-dimensional Sobolev representative> supplies the continuous representative used later.
Back to article page
