Carleman estimate for a 1D linear elastic problem involving interfaces: Application to an inverse problem

The aim of this work is to study the stability of the reconstruction of some mechanical parameters that characterize interfaces within linear elastic bodies. The model considered consists of two bonded elastic solids. The adhesive elastic layer between them is a thin interphase that is approximated, by asymptotic analysis, as an interface. The resulting interface model is in turn characterized by a typical spring-type linear elastic behavior. In this context and on a 1D configuration, we investigate a Carleman-type estimate that relates to a unique continuation principle. The proof is based on the construction of suitable weight functions: their gradients are non-zero, the jumps of the derivatives are positive across the interface, and the averages of the derivatives vanish. The design of such weight functions enables the control of the interface terms in the estimates. With this at hand, we establish a Lipschitz stability estimate for the inverse problem of identifying an interface stiffness parameter from measurements that are available on both sides of the external boundary.