Stability of Discontinuous Diffusion Coefficients and Initial Conditions in an Inverse Problem for the Heat Equation

Abstract

We consider the heat equation with a discontinuous diffusion coefficient and give uniqueness and stability results for both the diffusion coefficient and the initial condition from a measurement of the solution on an arbitrary part of the boundary and at some arbitrary positive time. The key ingredient is the derivation of a Carleman-type estimate. The diffusion coefficient is assumed to be discontinuous across an interface with a monotonicity condition.

Inverse problem for a parabolic system with two components by measurements of one component

“…(i) the uniqueness and the stability in determining coefficients: Especially for parabolic equations, see Benabdallah, Dermenjian and Le Rousseau [5], Benabdallah, Gaitan and Le Rousseau [6], Imanuvilov and Yamamoto [15], [17], Imanuvilov, Puel and Yamamoto [19], Isakov [21], Klibanov [23], [24] Klibanov and Timonov [26], Yuan and Yamamoto [32] and the references therein. For hyperbolic problems, among many works, we restrict ourselves to a few works such as Imanuvilov and Yamamoto [16], Isakov [20], [21], Klibanov [23], Klibanov and Timonov [26] and see the references also in Isakov [21] and Klibanov and Timonov [26].…”

On the determination of the principal coefficient from boundary measurements in a KdV equation

“…Other related papers on this type of inverse problem for pde's can be listed: [1,26] (one dimensional fourth order parabolic equation), [10,4,3] (discontinuous coefficients), [2] (network of one dimensional waves) [9,8] (logarithmic stability estimates), [21] (parabolic system), [41] (unknown coefficient in the nonlinearity), [46,15,21] (two unknown coefficients). We also mention some important books which can be the starting point to study inverse problems [30], Carleman estimates [22], and control theory for partial differential equations [18].…”

Four variations in global Carleman weights applied to inverse and controllability problems

“…As an application of these inequalities, we study one measurement inverse problems for the heat and wave equations using the general BukhgeimKlibanov approach [11]. The results explained here have been collected from the articles [13], [6], [7] and the preprint [2].…”