CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Obstacle complementarity in distribution form

Statement

Assume Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $Ω \subseteq R^{n}$ be a bounded $C^{1}$ domain (Bounded C^k domains and boundary charts), let $L$ be a uniformly elliptic divergence-form operator with real $L^{\infty}$ coefficients, and let $a : H_{0}^{1} (Ω; R) \times H_{0}^{1} (Ω; R) \to R$ be the symmetric, bounded and coercive real restriction of its associated form (Uniformly elliptic divergence-form operators and their sesquilinear forms, Bounded C^k domains and boundary charts); let $f \in L^{2} (Ω; R)$ and let $F \in H^{- 1} (Ω; R)$ be its canonical functional $F (v) = \int_{Ω} f v$ on $H_{0}^{1} (Ω; R)$ , so in particular $F (φ) = \int_{Ω} f φ$ for test functions (The negative Sobolev space $H^{- 1} (Ω)$ , Locally integrable functions as regular distributions), let $ψ \in H^{1} (Ω; R)$ with $T ψ \leq 0$ (The notation $H^{k}$ and the reserved zero-boundary symbol), and let $u \in K$ be the obstacle solution of Existence and uniqueness for the obstacle problem. For real $φ \in C_{c}^{\infty} (Ω; R)$ put $Λ_{u} (φ) : = a (u, φ) - \int_{Ω} f φ .$ Extend $Λ_{u}$ complex linearly as in The closed convex obstacle set and the obstacle variational inequality. Then:

$Λ_{u}$ is a nonnegative distribution: $Λ_{u} (φ) \geq 0$ for every nonnegative test function $φ$ (Distribution, Test function space d of an open set).
If in addition $Λ_{u}$ is represented by a function $ζ \in L^{2} (Ω)$ , that is $Λ_{u} (φ) = \int_{Ω} ζ φ$ for every test function, and if $u$ and $ψ$ have continuous representatives on $Ω$ , then $ζ \geq 0$ a.e., $ζ = 0$ a.e. on the open set ${u > ψ}$ , and $(u - ψ) ζ = 0$ a.e. on $Ω$ .

Facts & Assumptions

Given: The obstacle setting above, with the obstacle solution $u \in K$ of Existence and uniqueness for the obstacle problem, the reaction $Λ_{u} (φ) = a (u, φ) - F (φ)$ on $C_{c}^{\infty} (Ω)$ , and, for the second assertion, a representation $Λ_{u} (φ) = \int_{Ω} ζ φ$ with $ζ \in L^{2} (Ω)$ together with continuous representatives of $u$ and $ψ$ on $Ω$ .

[F1]

The closed convex obstacle set and the obstacle variational inequality, Existence and uniqueness for the obstacle problem: $K = {v \in H_{0}^{1} (Ω) : v \geq ψ a.e.}$ is nonempty, and $u \in K$ satisfies $a (u, v - u) \geq F (v - u)$ for every $v \in K$ (Zero-boundary Sobolev space as a norm closure); the inequality and the membership are almost-everywhere statements about classes.

[F2]

A function with nonnegative test pairings is nonnegative a.e.: if $ζ \in L^{2} (O)$ and $\int_{O} ζ φ \geq 0$ for every nonnegative $φ \in C_{c}^{\infty} (O)$ , then $ζ \geq 0$ a.e. on $O$ .

[F3]

The fundamental lemma of the calculus of variations: if $g \in L_{l o c}^{1} (O)$ and $\int_{O} g φ = 0$ for every $φ \in C_{c}^{\infty} (O)$ , then $g = 0$ a.e. on $O$ ; in particular $L^{2} (O) \subseteq L_{l o c}^{1} (O)$ by the finite measure of the bounded domain.

[F4]

Uniformly elliptic divergence-form operators and their sesquilinear forms: the real Dirichlet form $a$ is used only on $H_{0}^{1} (Ω; R) \times H_{0}^{1} (Ω; R)$ ; under the stated hypotheses it is a bounded, coercive symmetric real bilinear form there, and the associated form on $H^{1} (Ω)$ restricts to it. In particular, both the obstacle solution and each test function lie in the form domain.

[F5]

The negative Sobolev space $H^{- 1} (Ω)$ : in the real convention, every $f \in L^{2} (Ω)$ defines $F (v) = \int_{Ω} f v$ in $H^{- 1} (Ω)$ , with $∣ F (v) ∣ \leq ∥ f ∥_{2} ∥ v ∥_{2} \leq ∥ f ∥_{2} ∥ v ∥_{H_{0}^{1}}$ and hence $∥ F ∥_{H^{- 1}} \leq ∥ f ∥_{2}$ .

Proof

technique · direct

Given: The setting above, in particular the obstacle solution $u \in K$ and the reaction $Λ_{u}$ .

1.1givenF1F4F5

(Nonnegativity) Let $φ \in C_{c}^{\infty} (Ω)$ with $φ \geq 0$ , and put $v : = u + φ$ . Then $v \in H_{0}^{1} (Ω; R)$ because $u \in H_{0}^{1} (Ω; R)$ and $φ \in C_{c}^{\infty} (Ω; R) \subseteq H_{0}^{1} (Ω; R)$ , and $v \geq u \geq ψ$ a.e. on $Ω$ , so $v \in K$ [F1]. By [F5], $F$ is the continuous $H^{- 1}$ functional used by the variational inequality; testing that inequality at $v$ gives $a (u, φ) \geq F (φ)$ , that is $Λ_{u} (φ) \geq 0$ ; hence $Λ_{u}$ is a nonnegative distribution.

1.2givenF1F4

(Vanishing on the noncontact set) Assume now that $Λ_{u} (φ) = \int_{Ω} ζ φ$ for all test functions and that $u, ψ$ have continuous representatives, and put $O : = {u > ψ}$ , an open subset of $Ω$ because the difference of the continuous representatives is continuous and positive exactly on $O$ . Let $φ \in C_{c}^{\infty} (O)$ be arbitrary. If $φ = 0$ , then $Λ_{u} (φ) = 0$ . Otherwise its support $K_{φ}$ is a nonempty compact subset of $O$ ; as $u - ψ$ is continuous and positive there, there is $m > 0$ with $u - ψ \geq m$ on $K_{φ}$ . Choose $ε > 0$ with $ε ∥ φ ∥_{\infty} < m$ . Then $u \pm ε φ \geq u - ε ∥ φ ∥_{\infty} \geq ψ$ on $K_{φ}$ , while on $Ω ∖ K_{φ}$ one has $u \pm ε φ = u \geq ψ$ a.e.; also $u \pm ε φ \in H_{0}^{1} (Ω; R)$ . Hence both competitors belong to $K$ [F1]. Testing the variational inequality at them gives $ε Λ_{u} (φ) \geq 0$ and $- ε Λ_{u} (φ) \geq 0$ , so $Λ_{u} (φ) = 0$ ; that is, $\int_{Ω} ζ φ = 0$ for every $φ \in C_{c}^{\infty} (O)$ .

2.1step 1.1F2

For the second assertion, the hypothesis of the representation gives $\int_{Ω} ζ φ = Λ_{u} (φ) \geq 0$ for every nonnegative test function by step 1.1, so [F2] applied with $O = Ω$ yields $ζ \geq 0$ a.e. on $Ω$ .

2.2step 1.2F3

Step 1.2 gives $\int_{Ω} ζ φ = 0$ for every $φ \in C_{c}^{\infty} (O)$ with $ζ \in L^{2} (Ω) \subseteq L_{l o c}^{1} (Ω)$ ; consequently [F3] yields $ζ = 0$ a.e. on $O = {u > ψ}$ .

3.1step 2.2F1

Finally $(u - ψ) ζ = 0$ a.e. on $Ω$ : on $O$ this follows from $ζ = 0$ a.e. by step 2.2, and on $Ω ∖ O$ the class $u - ψ$ vanishes a.e. — indeed $u \geq ψ$ a.e. by [F1] while on $Ω ∖ {u > ψ}$ one has $u \leq ψ$ pointwise for the continuous representatives, so the set where $u < ψ$ is contained in the complement of $O$ and is null.

4.1step 1.1step 2.1step 2.2step 3.1∎

Step 1.1 proves the nonnegativity of the reaction, step 2.1 the a.e. nonnegativity of its $L^{2}$ representative, step 2.2 its vanishing on the noncontact set and step 3.1 the complementarity product; all three conclusions of the second assertion use exactly the stated $L^{2}$ -representation and continuity hypotheses, and no product of a distribution with a Sobolev class is formed.

Depends on

Used by

Dependency tree · two levels

63 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Andersson, The Obstacle Problem, KTH lecture notes, 16 December 2015 (complete 52-page notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)