Solution
= Solution
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 $\|f\|_\nabla=(f,f)_\nabla^{1/2}$.