The closed convex obstacle set and the obstacle variational inequality

Assume the Axiom of Choice (The Axiom of Choice). Let $n\ge 1$, and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a bounded open set of one of the following two kinds (Bounded C^k domains and boundary charts).

Trace conventions. If $n=1$, then $\mathrm{\Omega }=(a,b)$ is a bounded interval and $T$ is the endpoint-pair trace of Endpoint trace commutes with Sobolev truncation on an interval, ordered componentwise. If $n\ge 2$, then $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded $C^1$ domain and $T$ is the trace operator of The $L^p$ trace operator on a bounded $C^1$ domain. An obstacle is a real function $\psi$ with $\psi \in H^1(\mathrm{\Omega };\mathbb{R})$ (The notation $H^k$ and the reserved zero-boundary symbol, Integer-order Sobolev spaces and their norms) such that $T\psi \le 0$ — componentwise at the two endpoints when $n=1$, and almost everywhere on $\mathrm{\partial }\mathrm{\Omega }$ when $n\ge 2$. By Endpoint trace commutes with Sobolev truncation on an interval for $n=1$ and A function whose trace is at most a level has positive part in the zero-boundary space for $n\ge 2$, this boundary order condition is equivalent to ${\psi }^{+}\in H_0^1(\mathrm{\Omega })$.

The obstacle admissible set is $K:=\{v\in H_0^1(\mathrm{\Omega };\mathbb{R}):v\ge \psi \text{ a.e. on }\mathrm{\Omega }\},$ where the inequality $v\ge \psi$ is imposed on almost-everywhere classes and is therefore independent of the chosen representatives (The space $L^p(\mu )$ as the quotient by null functions, Zero-boundary Sobolev space as a norm closure).

Data. Let $a:H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })\to \mathbb{R}$ be a bounded coercive bilinear form with constants $M,\alpha >0$ (Bounded, coercive and symmetric sesquilinear forms) and let $F\in H^{-1}(\mathrm{\Omega })$ (The negative Sobolev space $H^{-1}(\mathrm{\Omega })$).

The obstacle variational inequality is the problem: find $u\in K$ with $a(u,v-u)\ge F(v-u)\text{for every }v\in K\mathrm{.}$

Energy and reaction. In the symmetric case the associated energy is $J(v):=\frac{1}{2}a(v,v)-F(v)$ for $v\in H_0^1(\mathrm{\Omega })$; the reaction distribution of a solution $u$ is the distribution on $\mathrm{\Omega }$ defined by ${\mathrm{\Lambda }}_u(\phi ):=a(u,\phi )-F(\phi )(\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })),$ the right-hand side being well defined on real test functions because $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})\subseteq H_0^1(\mathrm{\Omega };\mathbb{R})$ (Test function space d of an open set). It is linear and continuous: boundedness gives $\mathrm{\mid }{\mathrm{\Lambda }}_u(\phi )\mathrm{\mid }\le (M\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}+\mathrm{\parallel }F{\mathrm{\parallel }}_{H^{-1}})\mathrm{\parallel }\phi {\mathrm{\parallel }}_{H^1}$, and for tests supported in a compact $S\subseteq \mathrm{\Omega }$, $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{H^1}\le \sqrt{(n+1)\mathrm{\mid }S\mathrm{\mid }}{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 1}\mathrm{\parallel }{D}^{\alpha }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$. Thus $\mathrm{\mid }{\mathrm{\Lambda }}_u\mathrm{\mid }$ is a seminorm continuous on every fixed-support test space and hence continuous for Test function topology. Its complex-linear extension is ${\mathrm{\Lambda }}_u({\phi }_1+i{\phi }_2)={\mathrm{\Lambda }}_u({\phi }_1)+i{\mathrm{\Lambda }}_u({\phi }_2)$ for real tests ${\phi }_1,{\phi }_2$, giving a distribution in the convention of Distribution; nonnegativity is tested on real nonnegative tests.