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The Signorini contact set is nonempty, closed, and convex

Statement

Under The vector Signorini contact set and variational inequality, $K$ is nonempty, closed, and convex in the Hilbert space $V$ .

Facts & Assumptions

Given: The polygon and partial-trace Hilbert space $V$ , a continuous obstacle $ψ \leq 0$ , a unit normal $n$ , and $K = {v \in V : T_{C} v \cdot n \geq ψ a.e. on Γ_{C}}$ .

[A1]

The Axiom of Choice is inherited through the trace map and the Hilbert-space interface for $V$ . (The Axiom of Choice, The partial-trace displacement space is a Hilbert space)

[F1]

The contact trace $T_{C} : V \to L^{2} (Γ_{C}; R^{2})$ is bounded and linear. (The partial-trace displacement space is a Hilbert space)

[F2]

The contact set is the preimage of the nonnegative cone under $v \mapsto T_{C} v \cdot n - ψ$ . (The vector Signorini contact set and variational inequality)

[F3]

The contact segment has finite length and its continuous obstacle belongs to $L^{2} (Γ_{C})$ . (Partial-trace displacement space on a polygon, The vector Signorini contact set and variational inequality)

Proof

technique · direct

Given: $V$ , $T_{C}$ , $n$ , and $ψ \leq 0$ as above.

1.1A1F1F2givenalgebra

The zero field belongs to $V$ , and $T_{C} 0 \cdot n = 0 \geq ψ$ almost everywhere, so $0 \in K$ . If $v, w \in K$ and $0 \leq t \leq 1$ , linearity of $T_{C}$ gives $T_{C} (t v + (1 - t) w) \cdot n - ψ = t (T_{C} v \cdot n - ψ) + (1 - t) (T_{C} w \cdot n - ψ) \geq 0$ almost everywhere, so $t v + (1 - t) w \in K$ .

2.1A1F1F2F3algebrastep 1.1∎

If $g_{m} \in L_{+}^{2} (Γ_{C})$ and $g_{m} \to g$ in $L^{2}$ , then $g^{-} : = \max {- g, 0} \leq ∣ g - g_{m} ∣$ almost everywhere for each $m$ , hence $∥ g^{-} ∥_{2} \leq ∥ g - g_{m} ∥_{2} \to 0$ and $g \geq 0$ almost everywhere. Thus the cone is closed; it is convex by pointwise convexity. By [F1] and [F3], the affine map $v \mapsto T_{C} v \cdot n - ψ$ is continuous into $L^{2} (Γ_{C})$ , so its preimage of that closed convex cone is closed in $V$ . This proves the claim.

Depends on

Used by

Cited to discharge well-definedness by The vector Signorini contact set and variational inequality.

Dependency tree · two levels

12 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, complete 52-page KTH lecture notes (standard reference, not scraped)
Richard S. Falk, Error Estimates for the Approximation of a Class of Variational Inequalities, complete 10-page paper (standard reference, not scraped)