Partial-trace displacement space on a polygon

Assume the Axiom of Choice. A bounded connected Lipschitz polygon $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is a bounded connected open set whose boundary is a simple closed polygonal curve and, near every boundary point after a rigid coordinate change, is the graph of a Lipschitz function with $\mathrm{\Omega }$ on one side. Assume $\mathrm{\Omega }$ is a $W^{1,2}$-extension domain in the sense of Sobolev extension domains and extension operators. Fix two disjoint closed straight segments ${\mathrm{\Gamma }}_D,{\mathrm{\Gamma }}_C\subset \mathrm{\partial }\mathrm{\Omega }$, each of positive length and each lying strictly inside a straight side of $\mathrm{\partial }\mathrm{\Omega }$.

As part of the geometry data, fix bounded $C^1$ subdomains $U_D,U_C\subset \mathrm{\Omega }$ (in the sense of Bounded C^k domains and boundary charts) whose boundaries have relatively open flat patches containing ${\mathrm{\Gamma }}_D,{\mathrm{\Gamma }}_C$, respectively. Such collars can be chosen as sufficiently thin stadium-shaped domains along slightly longer segments of the same polygon sides, with their rounded ends outside the prescribed segments. Let $T_D,T_C$ be the restrictions to these segments of the bounded trace operators on $U_D,U_C$ supplied by The $L^p$ trace operator on a bounded $C^1$ domain, applied componentwise to the restrictions of functions in $H^1(\mathrm{\Omega };{\mathbb{R}}^2)$. For $j\in \{D,C\}$, if $v\in H^1(\mathrm{\Omega };{\mathbb{R}}^2)$ has a representative on $U_j$ extending continuously to $\overline{U_j}$, then $T_{j}v$ agrees in $L^2({\mathrm{\Gamma }}_j;{\mathbb{R}}^2)$ with that extension's restriction to ${\mathrm{\Gamma }}_j$, by the trace theorem's agreement clause. In particular, this holds for $v\in H^1(\mathrm{\Omega };{\mathbb{R}}^2)\cap C(\bar{\mathrm{\Omega }};{\mathbb{R}}^2)$.

Define $V:=\{v\in H^1(\mathrm{\Omega };{\mathbb{R}}^2):T_{D}v=0\text{ in }{L}^2({\mathrm{\Gamma }}_D;{\mathbb{R}}^2)\},\mathrm{\parallel }v{\mathrm{\parallel }}_V:=\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1(\mathrm{\Omega })}\mathrm{.}$ The contact normal $n\in {\mathbb{R}}^2$ is a fixed unit vector on ${\mathrm{\Gamma }}_C$, oriented so that a positive value of $T_{C}v\cdot n$ opens the contact gap. No pointwise value of a general $H^1$ class is used in this definition.

The trace maps use fixed collars so their meaning is independent of pointwise representatives. The polygon and segment hypotheses here are the geometric branch used by the contact items; the collars are part of the fixed data.