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Let $U$ be a bounded Lipschitz domain and ${\Gamma }_D$ a measurable portion of its boundary. The spaceis a closed vector subspace of the H1 space. The Sobolev trace operator maps $H^1(U)$ continuously into $L^2(\partial U)$, and restriction to ${\Gamma }_D$ is bounded, so $V$ is the kernel of a continuous linear map. Boundary regularity is what makes this ordinary trace formulation available.

No positive-measure assumption is needed for closedness. If $U$ is also connected and ${\Gamma }_D$ has positive surface measure, the Poincare inequality with a partial Dirichlet boundary controls the full Sobolev norm by the gradient norm on $V$.

Let $\Omega$ be a nonempty bounded Lipschitz domain, and define $A:H^1(\Omega )\to H^1(\Omega {)}^{\ast }$ byThis operator is strictly monotone: the difference pairing is the sum of $\Vert \nabla (v-w){\Vert }_2^2$ and the nonnegative integral $(\tanh v-\tanh w)(v-w)$, which is strictly positive wherever $v\ne w$. The bounded-domain hypothesis also makes nonzero constant functions admissible in $H^1$.

For constant $v=t$,Thus $A$ is not a nonlinear coercive operator in the large-norm sense. On $H_0^1(\Omega )$, the Poincaré inequality instead giveswhich proves coercivity on that zero-boundary space. The example separates being a strictly monotone operator from coercivity and identifies the uncontrolled constant direction.

On a bounded Lipschitz domain, the Sobolev trace operator is the continuous linear extension to $H^1(\Omega )$ of the boundary restriction of smooth functions. Continuity and density make this extension unique; its existence is the Sobolev trace theorem. The kernel prescribes a homogeneous Dirichlet boundary condition. Vanishing Sobolev trace on only part of the boundary instead gives the form domain for a mixed boundary condition, with the natural Neumann boundary condition on the remainder. On an assembly of congruent tiles, taking a Sobolev trace commutes with a constant matrix acting on the tile restrictions. This lets a transplantation theorem be proved on quadratic form domains even when corner eigenfunctions have singular classical derivatives.

On a bounded Lipschitz domain, the Poisson equation $-\Delta u=f$ with Robin boundary condition ${\partial }_{n}u+\beta u=0$ has the displayed weak formulation in $H^1(\Omega )$ when $\beta$ is a bounded boundary coefficient. Boundary integrals use Sobolev traces. It follows from Green's first identity, requiring only first weak derivatives of $u$. If $\beta$ is uniformly positive, testing the difference of two solutions against itself proves uniqueness: its gradient and its boundary Sobolev trace vanish. The Poincare inequality with a boundary trace also makes this a coercive bilinear form.

Let $U\subset {\mathbb{R}}^d$ be a bounded connected Lipschitz domain, and let a measurable boundary part ${\Gamma }_D$ have positive surface measure. Impose a homogeneous Dirichlet boundary condition there and a homogeneous Neumann boundary condition on the remainder. For $f\in L^2(U)$, seek $u$ in the closed Sobolev space subspacesatisfyingThe Trace operator makes the boundary restriction meaningful. The Poincare inequality with a partial Dirichlet boundary bounds the full $H^1$ norm by the gradient norm on $V$. Thus the left side is a bounded coercive bilinear form, and the Lax-Milgram theorem gives a unique weak solution. The same proof allows any bounded linear functional on $V$ as the right side.

The Neumann condition is natural: integration by parts leaves the boundary term on the non-Dirichlet portion, and requiring the identity for all $v\in V$ makes that term vanish. If ${\Gamma }_D$ is empty, constants remain in the kernel of a linear map; for homogeneous Neumann data, compatibility requires ${\int }_{U}f=0$, and uniqueness requires a normalization such as zero mean. Positive Dirichlet surface measure is what removes this ambiguity.

The zero-mean bounded-variation space isOn a bounded connected Lipschitz domain, removing the constant nullspace makes total variation coercive on this subspace by the Poincaré inequality for total variation.