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

Abstract

In this paper, we consider a transmission wave equation in N embedded domains with multiple interfaces of discontinuous coefficients in ℝ2. We study the global stability in determining the source term from a one-measurement data of wavefield velocity in a subboundary over a time interval. We prove the stability estimate for this inverse source problem by a combination of the local hyperbolic/elliptic Carleman estimates and the Fourier-Bros-Iagolniter transformation. Our method could be generalized to general dimensions since the weight functions and Carleman estimates are independent of the dimensions.

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 weight function we have defined and used in this work is the result of an adaptation of the well known function (39) to our particular transmission system. Indeed, the main features of ϕ in (12) used in this work are: it satisfies the transmission conditions (5), their properties allow us to deal with the traces on the interface and, finally, we can compute an explicit lower bound for the time in the stability of the inverse problem. These are the key properties which leaded us to the precise construction we are using, as we show in the following lines.…”

“…In fact, we can simply compute the value b M of the positive solution of the equation resulting when the second inequality in (45) is replaced by an identity, obtaining the following result. For instance, it is not difficult to see that a 2 < b M for all a ∈ (0, 1), (and equality is obtained for a = 1), which explain the choice of the time coefficient b = βa 2 with 0 < β < 1 in (12).…”