LemmaStatement: AI-generatedProof: AI-generated

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.

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Zero extension of a free exterior tail by distribution tests

Statement

Assume AC and let $Ω, η$ be as in Bounded collar regularity for an exterior weak Dirichlet function. For $u \in H^{1} (Ω)$ with $Δ u = g \in L^{2} (Ω)$ , define $v = (1 - η) u$ on $Ω$ and $v = 0$ on $Θ$ . Then $v \in H^{1} (R^{n})$ , and its distributional Laplacian is the zero extension of $h = (1 - η) g - 2 \nabla η \cdot \nabla u - (Δ η) u .$ In particular $∥ v ∥_{H^{1} (R^{n})} + ∥ Δ v ∥_{2} \leq C_{η} (∥ u ∥_{H^{1} (Ω)} + ∥ g ∥_{2})$ . The support of $v$ need not be bounded.

Facts & Assumptions

Given: The hypotheses and conventions of the Statement.

[F1]

Assume the Axiom of Choice The Axiom of Choice; in particular its countable and dependent instances are available.

[F2]

Bounded collar regularity for an exterior weak Dirichlet function fixes a smooth $η$ equal to $1$ on a neighborhood of $Θ$ ; all terms multiplied by $1 - η$ or derivatives of $η$ vanish on a smaller such neighborhood.

[F3]

Weak derivatives and Sobolev classes use Integer-order Sobolev spaces and their norms, with compactly supported tests even though the function itself may have an unbounded tail.

[F4]

Weak Leibniz rule with a smooth factor supplies cutoff first derivatives, while Test function cutoffs and euclidean localization supplies compact test cutoffs separating the obstacle from a given test support.

[F5]

For an open set $U$ and $f \in L_{l o c}^{1} (U)$ , Locally integrable functions as regular distributions supplies the regular distribution $T_{f} (ϕ) = \int_{U} f ϕ$ , with a bilinear pairing and no conjugation. Using Distributional derivative, write $Δ f = \sum_{j = 1}^{n} \partial_{j}^{2} T_{f}$ . Thus $Δ f = q$ for $q \in L_{l o c}^{1} (U)$ means $\sum_{j = 1}^{n} \partial_{j}^{2} T_{f} = T_{q}$ , equivalently $\int_{U} f Δ ϕ = \int_{U} q ϕ$ for every $ϕ \in C_{c}^{\infty} (U)$ . This convention applies on both $Ω$ and $R^{n}$ . The $L^{2}$ classes used here have locally integrable representatives by the Sobolev conventions in [F3]; the assumed AC includes the Countable Choice required by the regular-distribution supplier.

Proof

Given: The hypotheses and data in the Statement.

Goal: Prove the conclusions in the statement under its stated hypotheses.

1.1F2F3F4given

The asserted fields $v$ , the zero extensions of $(1 - η) \partial_{j} u - (\partial_{j} η) u$ , and the zero extension of $h$ have the claimed $L^{2}$ bounds by boundedness of the cutoff and its derivatives. All vanish near $Θ$ , so assigning zero on $Θ$ gives well-defined almost-everywhere classes; no values of a representative on a boundary are used.

2.1F2F3F4step 1.1

Fix $ϕ \in C_{c}^{\infty} (R^{n})$ . The product $(1 - η) ϕ$ belongs to $C_{c}^{\infty} (Ω)$ , so the weak first-derivative identity on $Ω$ gives $\int_{R^{n}} v \partial_{j} ϕ = - \int_{Ω} ((1 - η) \partial_{j} u - (\partial_{j} η) u) ϕ$ . Thus the fields from the previous step really are the global weak derivatives, proving $v \in H^{1} (R^{n})$ .

3.1F2F3F4F5step 1.1algebra∎

Testing $Δ u = g$ against $(1 - η) ϕ$ and expanding its Laplacian gives $\int v Δ ϕ = \int_{Ω} ((1 - η) g + (Δ η) u) ϕ + 2 \int_{Ω} u \nabla η \cdot \nabla ϕ$ . Integrate the last term once in distributions, with $u \nabla η$ supported away from $Θ$ , to obtain $\int v Δ ϕ = \int_{Ω} h ϕ$ . This is the global Laplacian identity and proves its $L^{2}$ bound. Every test is compactly supported; no integration by parts at infinity or compact-support extension theorem is invoked.

Depends on

Used by · 1

Collar and tail assembly of the exterior graph domain

Dependency tree · 2 levels

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