Inverse Hyperbolic Problems with Time-Dependent Coefficients

Abstract

We consider the inverse problem for the second order self-adjoint hyperbolic equation in a bounded domain in R n with lower order terms depending analytically on the time variable. We prove that, assuming the BLR condition, the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the hyperbolic equation up to a diffeomorphism and a gauge transformation. As a by-product we prove a similar result for the nonself-adjoint hyperbolic operator with time-independent coefficients.−V (x, t)u = 0,

Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field

“…The uniqueness by a local DN map is well solved (e.g., Belishev [4], Eskin [17], [19], Katchlov, Kurylev and Lassas [26], Kurylev and Lassas [28]). The stability estimates in the case where the DN map is considered on the whole lateral boundary were established in, Stefanov and Uhlmann [39], Sun [42], Bellassoued ans Dos Santos Ferriera [10].…”

Inverse problems for general second order hyperbolic equations with time-dependent coefficients

“…Proof The proof of Lemma 3.1 is a simplification of the proof of Lemmas 2.2 and 3.2 in [10]. We shall prove Lemma 3.1 for the case s 0 = T 1 .…”

“…The method is the extension of the approach in [8,9,11] to the case of time-dependent metrics. We adapt some lemmas of [8][9][10][11] to the time-dependent situation and simultaneously give sharper and simpler proofs.…”

“…In [8,9] the author proposed a new approach to hyperbolic inverse problems that uses substantially the idea of BC method. This approach was extended in [10] to some class of timeindependent metrics with time-dependent vector potentials and in [11] to the case of hyperbolic equations of general form with time-independent coefficients without vector potentials. The generalization to the case of Yang-Mills potentials was considered in [14].…”

The Calderón problem in transversally anisotropic geometries

“…The boundary control method for the wave equation also requires the coefficients to be independent of the time variable, although a variant of this method due to Eskin [14] allows lower order coefficients that are real analytic in time. The Gel'fand problem for time-dependent coefficients is interesting in its own right; see [43], [45], [48] for some results when the background metric is Euclidean.…”