Inverse boundary value problem for the dynamical heterogeneous Maxwell's system

Abstract

We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of R 3 , from a finite number of boundary measurements. Our main result is a Hölder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell's system with boundary conditions. for some λ 0 > 0 and µ 0 > 0. Next, in view of deriving existence and uniqueness results for (1.1), introduce the following functional spaceand denote by γ τ the unique linear continuous application from H(curl ,we see that the operator iA, whereFurther, in light of the last line of (1.1), set H(div 0, Ω) := {u ∈ L 2 (Ω) 3 , div u = 0} and H 0 (div 0, Ω) := {u ∈ H(div 0, Ω), γ n u = 0}, where γ n is the unique linear continuous mapping from H(div , Ω) := {u ∈ L 2 (Ω) 3 , div u ∈ L 2 (Ω)} onto H −1/2 (Γ), such that γ n u = u · ν when u ∈ C ∞ 0 (Ω) (see [16][Chap. IX A, Theorem 1]). Since H 0 := H(div 0; Ω) × H 0 (div 0; Ω) is a closed subspace of H and that H ⊥ 0 ⊂ ker A, the restriction A 0 Φ = A H 0 Φ := AΦ, Φ ∈ Dom(A 0 ) = Dom(A) ∩ H 0 := V, is, by Stone's Theorem [17][Chap. XVII A, §4, Theorem 3], the infinitesimal generator of a unitary group of class C 0 in H 0 . Thus, by rewriting (1.1)-(1.2) into the equivalent form Φ ′ = A 0 Φ Φ(0) = Φ 0 , with Φ = (D, B) T and Φ 0 = (D 0 , B 0 ) T , we get that: Lemma 1.1 Given (D 0 , B 0 ) ∈ V there exists a unique strong solution (D, B) to (1.1) starting from (D 0 , B 0 ) within the following class (D, B) ∈ C 0 (R; V) ∩ C 1 (R; H). (1.4) Moreover it holds true from [16][Chap. IX A, Remark 1] that V = H τ,0 (curl , div 0; Ω) × H n,0 (curl , div 0; Ω),where H * ,0 (curl , div 0; Ω) = {u ∈ H 1 (Ω) 3 , div u = 0 and γ * u = 0}, * = τ, n.

An inverse problem for an electroseismic model describing the coupling phenomenon of electromagnetic and seismic waves

“…Since V is separable, there exists a sequence of linearly independent functions {v (n) } n≥1 which form a basis of V . Let us define S m = span v (1) , v (2) , . .…”

“…For the Maxwell equations with σ = 0, Carleman estimates can be found, for example, in [12,2]. The arguments in these references easily generalize to the case σ = 0.…”

“…Assume that all the parameters in the electroseismic system satisfy the hypotheses of Lemma 3.2 and satisfy(3.3). Then, there exists a constant C 0 such that Q e 2τϕ τ 3 |D| 2 + |B| 2 + |u| 2 + | div u| 2 + | div w| 2 + | curl u|2 …”

Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain

“…Here, we used the fact that x n+1 = t ∈ [0, T]. Due to (3) to (5), the right-hand side of (27) is lower bounded, up to the multiplicative constant 4 2 n+1 , by the left-hand side of (21). Since n+1 is nonzero by (4) and (25), then we obtain…”

“…We first give an upper bound u (2) (·, 0) in the e s (·,0) -weighted H 1 (Ω L )-norm topology, by the corresponding norms of u (2) and u (3) in Q L . For the proof, see Lemma 3.2 in p 13 of Bellassoued et al 21 We apply Lemma 2.6 with z = e s i u (2) for i ∈ N * n and j = 0, 1, getting…”