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Fredholm solvability and residual stability for volume scattering

Statement

Assume AC and the hypotheses of The three-dimensional volume integral scattering model with $0 < ∣ D ∣ < \infty$ . If $I - K_{k}$ is injective, then for every $f \in L^{2} (D; C)$ there is a unique $u \in L^{2} (D; C)$ solving $(I - K_{k}) u = f$ , and $A_{k} : = I - K_{k}$ has a bounded inverse. Writing $R_{k} : = ∥ A_{k}^{- 1} ∥$ , every $v \in L^{2} (D; C)$ satisfies $∥ u - v ∥_{2} \leq R_{k} ∥ f - (I - K_{k}) v ∥_{2} .$ This is an operator-theoretic Fredholm branch; no physical uniqueness statement for the exterior PDE is asserted.

Facts & Assumptions

Given: AC, $D, q, k$ , compact $K_{k}$ , and injectivity of $I - K_{k}$ .

[A1]

AC supplies Countable Choice; complex $L^{2} (D)$ is complete under Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), Complex Lp completeness and almost-everywhere subsequences). Thus it is a Banach space (Banach space).

[F1]

The operator $K_{k}$ is compact (The volume integral scattering operator is compact).

[F2]

On a Banach space, if $K$ is compact and $I - K$ is injective, then $I - K$ is surjective with bounded inverse (Fredholm alternative for identity minus compact); this result explicitly assumes AC.

[F3]

A compact linear operator is bounded, so $A_{k} = I - K_{k}$ is bounded (Compact linear operator, A bounded linear operator between normed spaces).

Proof

Given: AC and the data in the Statement.

1.1givenA1F1F2

By [A1], $L^{2} (D; C)$ is Banach. By [F1] $K_{k}$ is compact, and the assumed injectivity permits [F2] to be applied to $A_{k} = I - K_{k}$ . Therefore $A_{k}$ is surjective and has a bounded inverse; for each $f$ the unique solution is $u = A_{k}^{- 1} f$ .

2.1givenF3step 1.1algebra∎

For arbitrary $v \in L^{2} (D; C)$ , linearity gives $A_{k} (u - v) = A_{k} u - A_{k} v = f - A_{k} v$ . Applying the bounded inverse from step 1.1 yields $u - v = A_{k}^{- 1} (f - A_{k} v)$ , so $∥ u - v ∥_{2} \leq ∥ A_{k}^{- 1} ∥ ∥ f - A_{k} v ∥_{2} = R_{k} ∥ f - (I - K_{k}) v ∥_{2}$ .

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Sources

Andreas Neubauer and Ronny Ramlau, Lecture Notes: Integral Equations and Boundary Value Problems, Chapter 3, §3.3 (standard reference, not scraped)
Habib Ammari, Brian Fitzpatrick, and Sanghyeon Yu, Mathematical and Computational Methods in Photonics: Tutorial Notes, §1.3 (standard reference, not scraped)