One-dimensional Sobolev spaces on intervals

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix complex scalars. Let $I\subseteq \mathbb{R}$ be a nonempty open interval, bounded or unbounded, and write $L^2(I;\mathbb{C})$ for the complex Hilbert space of Lebesgue (equivalence classes of) square-integrable functions with the inner product $⟨u,v⟩={\int }_{I}u\bar{v}$ linear in the first argument (Complex Lp classes and Euclidean test-function conventions, The space $L^p(\mu )$ as the quotient by null functions, $L^2$ with the integral pairing is a Hilbert space).

First-order space. Define $H^1(I):=\{u\in L^2(I;\mathbb{C}):\text{ the weak derivative }Du\text{ of the class }u\text{ exists and }Du\in L^2(I;\mathbb{C})\},$ where $Du$ is the weak derivative of Weak derivative of a locally integrable function: a class $Du\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(I;\mathbb{C})$ with ${\int }_{I}u{\phi }^{\mathrm{\prime }}=(-1){\int }_I(Du)\phi \text{for every }\phi \in C_c^{\mathrm{\infty }}(I),$ an identity that is choice-free (Locally integrable functions as regular distributions). Equip $H^1(I)$ with the sesquilinear form $⟨u,v{⟩}_{H^1}:={\int }_{I}u\bar{v}+{\int }_I(Du)\overline{Dv},$ whose associated norm is $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}=(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}^2{)}^{1\mathrm{/}2}$; the symbol $u^{\mathrm{\prime }}$ denotes the same weak derivative $Du$ throughout the page.

Second-order space. Define $H^2(I):=\{u\in H^1(I):Du\in H^1(I)\},$ and for $u\in H^2(I)$ write $D^{2}u:=D(Du)\in L^2(I;\mathbb{C})$; the norm is $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^2}=(\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}^2+\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^2}^2{)}^{1\mathrm{/}2}$. Thus $H^2(I)$ consists exactly of the classes in $L^2(I)$ whose first and second weak derivatives exist and lie in $L^2(I)$, and $H^2(I)\subseteq H^1(I)$.

Endpoint-vanishing first-order space and the Dirichlet form. For the bounded interval $(0,L)$, $L>0$, define $H_0^1(0,L):={\overline{C_c^{\mathrm{\infty }}(0,L)}}^{\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^1}},$ the closure of the test functions in $H^1(0,L)$, and define the Dirichlet form on it by $q_D[u]:=\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2(0,L)}^2={\int }_0^L\mathrm{\mid }Du{\mathrm{\mid }}^2$.

Elementary consequences, recorded for reuse. Weak derivatives are unique as locally integrable classes under Countable Choice, are additive and $\mathbb{C}$-homogeneous, and classical derivatives of continuously differentiable functions are weak derivatives. The cited published items supply these facts, so $H^1(I)$ and $H^2(I)$ are well-defined linear subspaces of $L^2(I;\mathbb{C})$. No claim is made here that $H_0^1(0,L)$ equals the set of classes with vanishing endpoint values; that identification is proved on this page from the trace estimates.