Zero-boundary Sobolev space ($H_0^1(U)$)

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.

On a nonempty bounded domain where the Poincaré inequality holds, no $a\in H_0^1(\Omega )$ can satisfy $a\ge a_{-}>0$ almost everywhere. For $g(s)=\min (\max (s,0),a_{-})$, Lipschitz truncation preserves zero-boundary Sobolev spaces, so $g(a)\in H_0^1(\Omega )$. But $g(a)$ would be the nonzero constant $a_{-}$, whose zero gradient contradicts the Poincaré inequality. In a uniformly elliptic divergence-form elliptic operator, the homogeneous boundary condition belongs to the unknown and test functions, not to a coefficient bounded below by a positive constant.

If $v\in H_0^1(\Omega )$ and $g:\mathbb{R}\to \mathbb{R}$ is Lipschitz continuous with $g(0)=0$, then $g(v)\in H_0^1(\Omega )$. The Lipschitz version of the Sobolev chain rule controls its weak derivatives, and its boundary trace is $g(0)=0$. Equivalently, approximate $v$ by compactly supported smooth functions, apply the derivative bound to their compositions, and pass weakly in $H^1$ and strongly in $L^2$. The closed linear space $H_0^1$ is weakly closed. This allows positive-part and bounded truncations in weak formulations without losing the Dirichlet boundary condition.

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, this 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 general unbounded domain this convention must be distinguished from completion in the inhomogeneous $H^1$ norm. A conformal map identifies its energy norm with that on the unit disc by conformal invariance.

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.