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.

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

Exterior Dirichlet realization, H² domain and form domain

Statement

Assume the Axiom of Choice (AC). Let $Ω = R^{n} ∖ Θ$ , $n \geq 2$ , where $Θ$ is compact with smooth boundary and $Ω$ is open. Work in $L^{2} (Ω; C)$ with inner product $⟨ u, v ⟩ = \int_{Ω} u \overline{v}$ , linear in the first variable. For $k = 1, 2$ , write $H^{k} (Ω) = W^{k, 2} (Ω; C)$ with its weak-derivative sum-of-squares norm, and let $H_{0}^{1} (Ω) = W_{0}^{1, 2} (Ω; C)$ be the $H^{1}$ norm closure of $C_{c}^{\infty} (Ω; C)$ . Self-adjointness means equality of a densely defined operator with its Hilbert-space adjoint, including equality of domains. The form $q (u, v) = \int_{Ω} (u \overline{v} + \nabla u \cdot \overline{\nabla v})$ on $H_{0}^{1} (Ω)$ represents a self-adjoint $A \geq I$ . Its domain is $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ , $A u = u - Δ u$ , with $∥ A u ∥_{2}$ equivalent to $∥ u ∥_{H^{2}}$ . Moreover $D (A^{1 / 2}) = H_{0}^{1} (Ω)$ with $∥ A^{1 / 2} u ∥_{2}^{2} = ∥ u ∥_{2}^{2} + ∥ \nabla u ∥_{2}^{2}$ . Thus $Δ_{D} = I - A$ is self-adjoint and nonpositive.

Facts & Assumptions

Given: The hypotheses and conventions of the Statement.

[F1]

Assume the Axiom of Choice The Axiom of Choice. It is the choice hypothesis of [F2] and [F3] and supplies the Countable Choice hypothesis of [F4] and [F5].

[F2]

Weak shifted Dirichlet form and energy bound on an open set supplies the self-adjoint shifted weak realization, its distributional domain, the energy bound and the exact square-root form norm on any nonempty open set.

[F3]

Collar and tail assembly of the exterior graph domain proves the exterior $H^{2}$ domain and graph norm by bounded-collar regularity and the separately justified unbounded Fourier tail.

[F4]

The weak-derivative spaces and norms are supplied by Integer-order Sobolev spaces and their norms with $p = 2$ , complex scalars and $k = 1, 2$ . The closure convention for $H_{0}^{1} = W_{0}^{1, 2}$ is supplied by Zero-boundary Sobolev space as a norm closure. The open-set and dimensional hypotheses hold here; their Countable Choice hypothesis follows from [F1].

[F5]

Self-adjointness on the ambient complex Hilbert space means equality with the densely defined Hilbert-space adjoint, including the domain, as in Symmetric, self-adjoint and essentially self-adjoint operators. Its Countable Choice hypothesis follows from [F1]; the ambient $L^{2}$ Hilbert space and pairing are those in [F2].

Proof

Given: The hypotheses and data in the Statement.

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

1.1F1F2F4F5given

The exterior of a compact subset of $R^{n}$ is nonempty. Apply the weak-form lemma with $U = Ω$ and the stated Dirichlet form. It gives a self-adjoint $A \geq I$ , $D (A) = {u \in H_{0}^{1} : Δ u \in L^{2}}$ , $A u = u - Δ u$ , and $D (A^{1 / 2}) = H_{0}^{1}$ with the exact form identity.

2.1F1F2F3F4F5step 1.1algebra∎

The graph-domain lemma applies to this same weak domain and gives $D (A) = H^{2} \cap H_{0}^{1}$ and $∥ A u ∥_{2} ≍ ∥ u ∥_{H^{2}}$ . Since $A$ is self-adjoint, $I - A$ is self-adjoint on $D (A)$ , and $⟨ (I - A) u, u ⟩ = - ∥ \nabla u ∥_{2}^{2} \leq 0$ . This identifies the nonpositive Dirichlet Laplacian and proves every assertion.

Source notes

This is a local collar/free-tail composition of the named library results. The article under study is not a prerequisite source. The five intermediate items supply the substantial local argument; no independent proof-review credit is claimed.

Depends on

Used by · 12

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.