Stability estimate in determination of a coefficient in transmission wave equation by boundary observation

Abstract

In this paper, we study the global stability in determination of a coefficient in the transmission wave equation from data of the solution in a subboundary over a time interval. Providing regular initial data, we prove a hölder stability estimate in the inverse problem with a single measurement. Moreover, the exponent in the stability estimate depends on the regularity of initial data.

An inverse problem for a transmission wave equation with a flat interface in Rn

“…Concerning global Carleman estimates for wave equations with discontinuous main coefficient, we mention [4], were the authors introduced a new weight function for interfaces which are a boundary of a strictly convex set, and the obtained Carleman estimates were applied to obtain stability of the inverse problem of recovering of the potential in a transmission system. In [12,16,19], it was used the weight function constructed in [4] to several related inverse problems. Also, in [5] Carleman estimates for hyperbolic equations with discontinuous main coefficient in dimension one are developed.…”

“…The case of variable main coefficients could be studied with the approach of this work together with estimates obtained, for instance, in [16]. Also, it would be interesting to find precise estimates for the minimal time in the case of the interface being the boundary of a convex inner domain (this is the geometric case studied in [4], where there are no explicit estimates for the minimal needed time for the stability of the inverse problem).…”

Carleman estimates for the wave equation in heterogeneous media with non-convex interface

“…The Carleman inequalities obtained are the main tool for the study of the Lipschitz stability of an inverse problem, the one of recovering the potential (a zero-order coefficient term) in the equation by means of an observation given by the trace of the normal derivative of the solution on some subset of the external boundary. Those Carleman estimates were also used in [24] in order to obtain Hölder stability of the related main-coefficient inverse problem. An analogous topic for the Schrödinger equation is investigated in [3], where the construction of the weight function is generalized for convex inner domains in R n .…”

Stability of inverse source problem for a transmission wave equation with multiple interfaces of discontinuity

“…In this article, we are interested in the case when there are more than one interface. Different from [4,25], it is difficult to find a weight function satisfying conditions (a)-(f) listed in Section 2 in the whole domain when N > 2, hence we cannot expect to establish a global Carleman estimate. We have to apply local Carleman estimates which will inevitably create some remainder terms by using cut-off functions.…”

“…In [4], Baudouin et al prove the Lipschitz stability for a one-measurement inverse problem of identifying the coefficient in the zero-order term for N = 2, by using a global Carleman estimate and the method of Bukhgeim-Klibanov (B-K) [13,14,21]. Later in [25], Riahi proves the Hölder stability for the inverse problem of determining the spatial varied coefficient a(x) in the principle term by two measurements, under an assumption that a(x) is a piecewise constant near the interface. For the one-dimension case, we refer to Bellassoued and Yamamoto [8], in which the authors establish the global Carleman estimate for multiple interfaces.…”