= Dirichlet energy space
{c}
{title2=$H_0^1(D)$}
= Homogeneous Dirichlet space
{synonym}
For planar <Gaussian free field> theory, the <Dirichlet energy space> is the completion of real <test functions> $C_c^\infty(D)$ in the <norm> induced by the <Dirichlet inner product>,
$$
(f,g)_\nabla=(2\pi)^{-1}\int_D\nabla f\cdot\nabla g.
$$
On a bounded <domain>, the zero-boundary <Poincaré inequality> identifies this with $H_0^1(D)$ with an equivalent <Sobolev norm>. On a general unbounded <domain>, it must be distinguished from completion in the inhomogeneous $H^1$ norm.
A <conformal bijection> identifies the two corresponding energy completions by <conformal invariance of the planar Dirichlet inner product>. In particular, if $D$ is a nonempty simply connected proper <open subset> of the <complex plane>, the <Riemann mapping theorem> supplies a <conformal bijection> with the <unit disk>. This unit-disk identification is not asserted for an arbitrary multiply connected domain or the whole <complex plane>.
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