Equip $C_0^\infty(D)$ with the <Dirichlet inner product>
$$
(f,g)_\nabla=\frac1{2\pi}\int_D\nabla f(x)\mathbin\cdot\nabla g(x)\,dx.
$$
The <zero-boundary Sobolev space> $H_0^1(D)$ is its <Hilbert space completion> in the norm $\|f\|_\nabla=(f,f)_\nabla^{1/2}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 4 / a / ii / Solution

{title2=$H_0^1(U)$}

The zero-boundary Sobolev space $H_0^1(U)$ is the closure of the <space of test functions> $C_c^\infty(U)$ in $H^1(U)$. Under standard regularity assumptions its elements have zero boundary trace.

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 Incoming links: Zero-boundary Sobolev space