 Dirichlet energy space

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,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.