DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

The finite-energy double-layer transmission space

Definition

Use the exterior energy space $E^{+} (Ω^{+})$ and weak conormal conventions of Weak Laplace single-layer potentials and energy spaces. For $ϕ \in H^{1 / 2} (Γ)$ let $A_{ϕ} = {(u^{-}, u^{+}) \in H^{1} (Ω^{-}) \times E^{+} (Ω^{+}) : γ^{-} u^{-} - γ^{+} u^{+} = - ϕ} .$ When it exists uniquely, denote by $D ϕ$ the pair in $A_{ϕ}$ minimizing $E (u^{-}, u^{+}) = \int_{Ω^{-}} ∣ \nabla u^{-} ∣^{2} d x + \int_{Ω^{+}} ∣ \nabla u^{+} ∣^{2} d x .$ Existence, uniqueness, harmonicity, and equality of the weak conormals are proved in The double-layer potential has the stated jumps and bounds ↗.

For $ϕ \in H^{1 / 2} (Γ)$ and $x \notin Γ$ , the corresponding classical potential, whose agreement with the minimizer is proved in The double-layer potential has the stated jumps and bounds ↗, is $D ϕ (x) = \int_{Γ} \partial_{n_{y}} G_{3} (x, y) ϕ (y) d S_{y}$ . The sign in the imposed jump, $γ^{-} D ϕ - γ^{+} D ϕ = - ϕ$ , is the convention for this kernel.

Depends on

Used by

Calderon boundary operators and trace conventions Definition
The double-layer potential has the stated jumps and bounds Theorem

Dependency tree · two levels

23 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

Martin Costabel, Boundary Integral Operators on Lipschitz Domains: Elementary Results (1988) (standard reference, not scraped)
Francisco-Javier Sayas, First Steps in the Boundary Element Method (2015 lecture notes) (standard reference, not scraped)