Band 19, Heft 2

A dynamical Maxwell system is e t = curl h, h t = – curl e in Ω × (0, T ), e | t =0 = 0, h | t =0 = 0 in Ω, e θ = ƒ in ∂ Ω × [0, T ], where Ω is a smooth compact oriented 3-dimensional Riemannian manifold with boundary, (·) θ is a tangent component of a vector at the boundary, e = e ƒ ( x, t ) and h = h ƒ ( x, t ) are the electric and magnetic components of the solution. One associates with this system a response operator R T : ƒ ↦ ν ∧ h ƒ | ∂ Ω×(0, T ) , where ν is an outward normal to ∂ Ω. The time-optimal setup of the inverse problem, which is relevant to the finiteness of the wave speed propagation, is as follows: given R 2 T to recover the part Ω T ≔ { x ∈ Ω| dist( x, ∂ Ω) < T } of the manifold. As was shown by Belishev, Isakov, Pestov, Sharafutdinov (Doklady Mathematics 61: 353–356, 2000), for T small enough the operator R 2 T determines Ω T uniquely up to isometry. Here we prove that uniqueness holds for arbitrary T > 0 and provide a procedure that recovers Ω T from R 2 T . Our approach is a version of the boundary control method (Belishev, 1986).

Inversion methods are very important components of nuclear-geophysical technologies. Their main aim is to evaluate the set of parameters of a formation, using measurements data. The parameters describe the properties and the elemental content of it, and they enter into coefficients of the transport equation. In this paper we develop and formalize an approach to problems on mathematical interpretation of the data, we state two inverse problems to the equation and construct successive approximations of the “prime” iteration type to solve a corresponding system of parameters evaluation. The approach is based on concepts defined of “characteristic” interactions, elements, and matrix, it uses the superposition principle for transport processes and linear a priori constraints for the unknowns. Earlier a few concrete problems were examined within this approach.

We consider the inverse problem of determining simultaneously two unknown electric potential coefficients for a system of two general strongly coupled Schrödinger equations, with magnetic potential terms, and with Neumann boundary conditions, from single Dirichlet measurements on a portion Γ 1 of the boundary. Under suitable geometrical assumptions on the complementary unobserved portion Γ 0 of the boundary, we show that one can uniquely determine the two unknown potential coefficients in one shot, from respective Dirichlet boundary measurements on Γ 1 over an arbitrarily short time interval. The proof is based on a recent Carleman estimate in [Lasiecka, Triggiani and Zhang, J. Inv. Ill-Posed Problems 12: 43–123, 2004] for single Schrödinger equations. It also takes advantage of a convenient route “post-Carleman estimates” suggested by [Isakov, Inverse Problems for Partial Differential Equations, Springer, 2006, Theorem 8.2.2, p. 231].

We recover unknown nonlinear terms in quasilinear first-order partial differential equations and give some applications to conservation laws.

We consider the problem of identifying the constant α ∈ ℝ in the Cauchy problem , when , from an additional measurement on the solution u .

We consider the estimation of the accuracy of finite dimensional approximations of regularized solutions for inverse problems in a separable Hilbert space. A connection of the regularization parameter with the estimates of error for these approximations and with the estimates of initial date is obtained. For the establishment method the estimation of error of the finite-dimensional approximation is obtained. This estimate is exact with respect to the order.