Some stability Estimates for the Identification of Conductivity in the one-dimensional Heat-Equation

The stability of position-dependent conductivity in one spatial dimension is considered. This inverse problem is assumed throughout to have at least one measurable, bounded and strictly positive solution. Since conductivity satisfies an ordinary differential equation (ODE), uniqueness conditions may result from information of local or non-local type. Local information corresponds to a Cauchy datum, which can be supplied either at a regular or at a critical point; at the latter, temperature is stationary (singular Cauchy problem). Non-local information is supplied as the domain average of thermal flux at a given instant of time. The main purpose of the paper is to provide a unified view over stability estimates pertaining to the unique solution. Some additional restrictions (regularisation) are imposed on the temperature data. If uniqueness is due to a regular Cauchy problem, then L^{infinity} estimates are obtained. Singular problems, on the other hand, yield L^r estimates, with 1 <= r < infinity. Non-local conditions are treated similarly. The unifying device is the defect equation, an ODE for conductivity differences in a space of distributions. Estimates are arrived at by suitably integrating said ODE. Some examples and counterexamples are provided.