Alphabeta MathDefinitionDefinition: 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 three-dimensional Laplace Newton kernel
Definition
For in , the Newton kernel is . Its distributional fundamental-solution property is proved in The three-dimensional Newton kernel is a fundamental solution ↗. The kernel fixes the sign and normalization for every three-dimensional layer operator on this page.
Used by
- Weak Laplace single-layer potentials and energy spaces Definition
- A constant density on the sphere gives a constant trace Example
- A unit sphere checks both layer jump signs Example
- A positive far-panel quadrature residual and normalized cluster combination Lemma
- Certified touching-panel and disjoint-panel quadrature for the Laplace P0 matrix Lemma
- Finite-energy transmission pairs have the double-layer kernel representation across polyhedral edges Lemma
- Lipschitz finite-energy transmission pairs have the double-layer kernel representation Lemma
- The three-dimensional Newton kernel is a fundamental solution Lemma
- The double-layer potential has the stated jumps and bounds Theorem
- The Helmholtz single-layer and double-layer traces have the Laplace jump signs Theorem
- The three-dimensional single layer is bounded and coercive Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
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)