Dirichlet inner product ($(f,g{)}_{\nabla }$)

On a domain $U\subseteq {\mathbb{R}}^n$, a common normalization of the Dirichlet inner product isThe Poincaré inequality makes its induced seminorm a norm on $H_0^1(U)$.

For a conformal bijection $f:D\to \tilde{D}$ and smooth $u,v$ of compact support on $\tilde{D}$, $\boxed{(u\circ f,v\circ f{)}_{\nabla ,D}=(u,v{)}_{\nabla ,\tilde{D}}}$. The Jacobian matrix of $f$ is a rotation times $|f^{\prime }|$. Consequently the chain rule introduces $|f^{\prime }{|}^2$ into the gradient pairing, while the change of variables formula introduces precisely the same factor into area. They cancel. The identity extends by completion to the Dirichlet energy spaces; it does not preserve the $L^2$ term of an inhomogeneous Sobolev norm.

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.

Let $U\subseteq D$ be open. Regard $H_{\mathrm{supp}}=H_0^1(U)$ as functions on $D$ by zero extension of H01, and let $H_{\mathrm{harm}}$ consist of the weakly harmonic Sobolev functions on $U$ belonging to $H_0^1(D)$. For $h\in H_{\mathrm{harm}}$, the Dirichlet inner product $(h,\varphi {)}_{\nabla }$ vanishes for each test function supported in $U$. Approximate any $u\in H_{\mathrm{supp}}$ by those test functions in the gradient norm and use the Cauchy-Schwarz inequality. This gives $\boxed{(h,u{)}_{\nabla }=0}$. The proof also works with the inhomogeneous zero-boundary Sobolev space convention, and with a homogeneous completion realized as weak functions.