LemmaStatement: Literature-sourcedProof: 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.

Lemma 3.11: extension preserving the obstacle Dirichlet condition

PDF p. 16: Lemma 3.11

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Θ \subset R^{n}$ be nonempty and compact with smooth boundary, and suppose its exterior $Ω = R^{n} ∖ Θ$ is connected. Choose $R_{0} > 0$ with $Θ \subset B_{R_{0}} (0)$ and put $Ω_{0} = B_{R_{0}} (0) ∖ Θ$ and $Γ_{2} = \partial Θ$ . For $1 \leq s \leq 2$ , let $X^{s} (Ω_{0}) = {u \in H^{s} (Ω_{0}) : γ_{Γ_{2}} u = 0}$ with the inherited $H^{s}$ norm, where $H^{s}$ is the Bessel-potential restriction space and $γ_{Γ_{2}}$ is the value trace. Let $H_{D}^{s} (Ω) = D ((I - Δ_{D})^{s / 2})$ with its spectral norm, where $Δ_{D}$ is the Dirichlet Laplacian. Then there is a bounded linear operator $E : X^{s} (Ω_{0}) \to H_{D}^{s} (Ω)$ such that $(E u) ∣_{Ω_{0}} = u$ for every $u \in X^{s} (Ω_{0})$ .

Statement references

Exterior and truncated obstacle domains

States compatible with the uncontrolled obstacle boundary

Dirichlet spectral Sobolev scale and duality

The Axiom of Choice

A bounded linear operator between normed spaces

Facts & Assumptions

Given: The hypotheses and conventions in the Statement.

[F1]

The geometry and spaces are those of Exterior and truncated obstacle domains, States compatible with the uncontrolled obstacle boundary and Dirichlet spectral Sobolev scale and duality. The Choice assumption is The Axiom of Choice. Bounded linear operators between the resulting complex normed spaces are understood as in A bounded linear operator between normed spaces.

[F2]

The fixed annular construction Annular extension and cutoff gluing construction, with the partial-extension and gluing result Partial extension across an artificial sphere and annular cutoff gluing, gives a bounded linear map $G : H^{s} (Ω_{0}) \to H^{s} (Ω)$ which restricts to the identity on $Ω_{0}$ and maps $X^{s} (Ω_{0})$ into $H^{s} (Ω) \cap H_{0}^{1} (Ω)$ . Here $H_{0}^{1} (Ω)$ is the $H^{1} (Ω)$ closure of $C_{c}^{\infty} (Ω)$ .

[F3]

Under the stated geometry and Choice assumption, Low-order identifications of the Dirichlet scale identifies $H_{D}^{s} (Ω)$ with $H^{s} (Ω) \cap H_{0}^{1} (Ω)$ for $1 \leq s \leq 2$ , with the spectral and inherited $H^{s}$ norms equivalent.

Proof

Given: The hypotheses and conventions in the Statement.

[F1] The geometry and spaces are those of Exterior and truncated obstacle domains, States compatible with the uncontrolled obstacle boundary and Dirichlet spectral Sobolev scale and duality. The Choice assumption is The Axiom of Choice. Bounded linear operators between the resulting complex normed spaces are understood as in A bounded linear operator between normed spaces.

[F2] The fixed annular construction Annular extension and cutoff gluing construction, with the partial-extension and gluing result Partial extension across an artificial sphere and annular cutoff gluing, gives a bounded linear map $G : H^{s} (Ω_{0}) \to H^{s} (Ω)$ which restricts to the identity on $Ω_{0}$ and maps $X^{s} (Ω_{0})$ into $H^{s} (Ω) \cap H_{0}^{1} (Ω)$ . Here $H_{0}^{1} (Ω)$ is the $H^{1} (Ω)$ closure of $C_{c}^{\infty} (Ω)$ .

[F3] Under the stated geometry and Choice assumption, Low-order identifications of the Dirichlet scale identifies $H_{D}^{s} (Ω)$ with $H^{s} (Ω) \cap H_{0}^{1} (Ω)$ for $1 \leq s \leq 2$ , with the spectral and inherited $H^{s}$ norms equivalent.

Given: The hypotheses and data in the Statement.

Goal: Construct a bounded linear right inverse to restriction with values in $H_{D}^{s} (Ω)$ .

1.1F2given

Fix the cutoffs and annular extension in the construction of $G$ . Restrict its domain to $X^{s} (Ω_{0})$ and call the resulting map $E$ . The gluing result gives $E u \in H^{s} (Ω) \cap H_{0}^{1} (Ω)$ and $(E u) ∣_{Ω_{0}} = u$ . It also gives a constant $C_{s}$ such that $∥ E u ∥_{H^{s} (Ω)} \leq C_{s} ∥ u ∥_{H^{s} (Ω_{0})}$ . The fixed construction is linear.

2.1F1F3step 1.1∎

The scale identification gives a constant $C_{s}^{'}$ such that $∥ E u ∥_{H_{D}^{s} (Ω)} \leq C_{s}^{'} ∥ E u ∥_{H^{s} (Ω)}$ on $H^{s} (Ω) \cap H_{0}^{1} (Ω)$ . Combining this with the previous bound and the inherited norm of $X^{s} (Ω_{0})$ proves that $E$ is bounded into $H_{D}^{s} (Ω)$ . Its restriction identity and linearity were established above.

Source notes

The original article is cited for attribution of Lemma 3.11 and its annular construction. The proof here applies the separately supplied gluing and spectral-identification results; the article's assertion is not used as a premise.

Depends on

Used by · 4

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.

Sources

Nonlinear Schrödinger equation in an exterior domain, v2