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.

Sound-soft exterior Helmholtz problem for a spherical obstacle

Definition

Fix $a > 0$ , $k > 0$ , and $f \in C^{\infty} (S^{2}; C)$ . Write $S_{a} = {x \in R^{3} : ∣ x ∣ = a}$ . The sound-soft exterior Helmholtz problem is to find a complex-valued $u \in C^{2} ({x \in R^{3} : ∣ x ∣ > a})$ that extends continuously to ${x \in R^{3} : ∣ x ∣ \geq a}$ and satisfies $(Δ + k^{2}) u = 0$ for $∣ x ∣ > a$ , with boundary restriction $u ∣_{S_{a}} (a ω) = f (ω)$ , and $\lim_{r \to \infty} \sup_{ω \in S^{2}} ∣ r (\partial_{r} u (r ω) - i k u (r ω)) ∣ = 0.$ The time factor is $e^{- i ω t}$ , so this is the outgoing convention. The obstacle is the closed ball ${∣ x ∣ \leq a}$ ; its boundary is smooth and its exterior is connected. This definition makes no claim for nonspherical obstacles or nonsmooth data.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Plamen Stefanov, Scattering and Inverse Scattering in R^n, graduate lecture notes (standard reference, not scraped)
Pierre Capel, Introduction to Scattering Theory, TALENT lecture notes (standard reference, not scraped)