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Lemma 3.1: weighted Dirichlet Laplacian isomorphism

PDF p. 13: Lemma 3.1 and its proof

Statement

Work in ZFC. Let $(Σ, h)$ be a connected smooth three-dimensional exterior with compact smooth boundary $S$ , one fixed asymptotically Euclidean end chart, and $h - δ \in C_{- 1}^{3, α}$ . Let $0 < α < 1$ , $- 1 < β < 0$ , and $0 \leq V \in C_{- 4}^{2, α}$ . With the weighted Hölder norms defined using the fixed compact core and scaled end annuli, put $X_{β} = {w \in C_{β}^{4, α} : w ∣_{S} = 0}$ and $Y_{β} = C_{β - 2}^{2, α}$ . Then $L = Δ_{h} - V : X_{β} \to Y_{β}$ is a bounded linear bijection with bounded inverse. Here $Δ_{h} = {div}_{h} {grad}_{h}$ .

Statement references

Weighted exterior elliptic domains and admissible weights

Weighted Hölder norms on a three-dimensional exterior

Fredholm and Banach continuation interfaces

Facts & Assumptions

[F1]

The exterior interface supplies the end, boundary, weighted Dirichlet domain and Laplacian conventions; the weighted Hölder definition fixes the norm and its decay convention. Weighted exterior elliptic domains and admissible weights Weighted Hölder norms on a three-dimensional exterior

[F2]

Under the stated metric, weight and potential hypotheses, $L : X_{β} \to Y_{β}$ is Fredholm of index zero between Banach spaces. Exterior Dirichlet parametrix and compact decaying potentials

[F3]

A Fredholm operator has closed range and finite-dimensional kernel and cokernel, and its index is the kernel dimension minus the cokernel dimension. Fredholm and Banach continuation interfaces

[F4]

For a nonnegative potential of the stated regularity, the decaying Dirichlet kernel is zero on a connected exterior. Decaying Dirichlet kernel and the exact closed-component exception

[F5]

A bounded bijective linear map between Banach spaces has a bounded inverse under dependent choice. ZFC supplies AC, and AC implies dependent choice. Bounded inverse theorem The Axiom of Choice AC implies DC implies countable choice

Proof

**Given:**The hypotheses in the Statement.

Goal: Prove that $L : X_{β} \to Y_{β}$ is a bounded isomorphism.

1.1givenF1F2F4

The exterior and norm conventions match the scalar Dirichlet Fredholm result, including the compact inner boundary, $h - δ \in C_{- 1}^{3, α}$ and $- 1 < β < 0$ . Thus $L$ is bounded and Fredholm of index zero between the specified Banach spaces. The potential is nonnegative and the exterior is connected, so the decaying-kernel result gives $\ker L = {0}$ .

2.1F3F5step 1.1∎

The index formula now gives $\dim (Y_{β} / ran L) = 0$ . Hence the range equals $Y_{β}$ , so $L$ is bijective. By F5, ZFC supplies dependent choice, so the bounded inverse theorem applies to this bounded bijection between Banach spaces and gives a bounded linear inverse. This proves the assertion.

Source notes

The article's index, compactness and maximum-principle argument is attributed above. The proof here uses the separately supplied scalar exterior Fredholm and kernel results; the article is not a logical premise.

Depends on

Used by · 1

Openness via the weighted implicit-function branch and smooth boundary bootstrap

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

The uniqueness of smooth stationary black holes · PDF p. 13, Lemma 3.1 and its proof