Inverse scattering for the Laplace operator with boundary conditions on Lipschitz surfaces

Abstract

We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of selfadjoint operators ( ∆, ∆), where ∆ is the free Laplacian in L 2 (R 3 ) and ∆ is one of its singular perturbations, i.e., such that the set {u ∈ H 2 (R 3 ) ∩ dom( ∆) : ∆u = ∆u} is dense. Typically ∆ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular our results apply to standard, either separating or semi-transparent, boundary conditions at Γ = ∂Ω, where Ω ⊂ R 3 is a bounded Lipschitz domain. Similar results hold in the case the boundary conditions are assigned only on Σ ⊂ Γ, a relatively open subset with a Lipschitz boundary. We show that either the obstacle Ω or the screen Σ are determined by the knowledge of the Scattering Matrix, equivalently of the Far Field Operator, at a single frequency.

Inverse wave scattering in the Laplace domain: A factorization method approach

“…Since R Σ : X s ♯ → X s ♯ is an orthogonal projector (see [16,Lemma 5.1], the coercivity of M λ implies the coercivity of R * Σ M λ R Σ ; likewise if M λ is sign-definite then R * Σ M λ R Σ is signdefinite as well. Then (4.10) is consequence, by the inf-criterion in [12,Theorem 1.16], of Lemma 5.2 in the Appendix.…”

“…, the self-adjoint operator ∆ Λ α represents a bounded from above Laplace operator with the semi-transparent boundary conditions [16, [16,Lemma 5.1]), where…”

“…In our recent paper [16], we used Kreȋn-type resolvent formulae, combined with the limiting absorption principle and the factorization method, to provide inverse scattering (reconstrucion) results for Lipschitz obstacles and screens. In what follows analogous results are obtained for the time-dependent obstacle scattering problem; our approach exploits the Laplace transform analysis of the time-propagator leading to a factorized representation of the data operator in the Laplace transform domain.…”

“…(♯ = N) As in the previous case, if Σ • ⊂ Σ then g Σ• λ = SL λ 1 Σ• with 1 Σ• ∈ H 1/2−ǫ Σ (Γ), ǫ > 0. Since γ 1 DL λ ∈ B(H 1/2 (Γ), H −1/2 (Γ)) is coercive (see, e.g., [17, Lemma 3.2]), R Σ γ 1 DL λ R * Σ ∈ B(H 1/(Γ), H −1/2 (Σ)) is coercive as well (see [16, Remark 5.2]) and hence (R Σ γ 1 DL λ R * Σ ) −1 ∈ B(H −1/2 (Σ), H 1/(Γ)) (see[16, Remark 4.6]); then, by the mapping properties of γ 1 DL λ (see[7, Theorem 3]), one gets (R Σ γ 1 DL λ R * Σ ) −1 ∈ B(H s−1/2 (Σ), H s+1/2 Σ (Γ)), s ∈ [0, 1/2]. Since both g Σ• λ and DL λ φ solve (−∆ + λ)u = 0 in R n \Σ • with boundary condition R Σ γ 1 u = R Σ γ 1 g Σ• λ whenever φ = (R Σ γ 1 DL λ R * Σ ) −1 R Σ γ 1 g Σ• λand such a problem has a unique radiating solution (see[1, Theorem 3.3]), one gets g Σ• λ = DL λ φ = G N λ φ, φ ∈ H s+1/2 Σ (Γ).…”

Inverse wave scattering in the time domain for point scatterers

“…In our previous work [17], we considered inverse wave scattering in the time domain for a wide class of self-adjoint Laplacians, including those with hard, soft and semi-transparent bounded obstacles with Lipschitz boundaries. By applying to ∆ Y r the results there provided (which build on our previous works [18], [19], [15], [16]), one gets the following: denoting by u Y r f and u ∅ f the solutions of the wave equations corresponding to ∆ Y r and to the free Laplacian ∆ respectively, with a source term f concentrated at time t = 0 (a pulse) one has that for any fixed λ ≥ λ • > 0 and any fixed open B ⊂⊂ R n \Y r , the obstacle Y r can be reconstructed by the knowledge of the data operator F Y r ,B λ : L 2 (B) → L 2 (B),…”