Lipschitz Stability in Inverse Source Problems for Singular Parabolic Equations

Abstract

We address the question of Lipschitz stability results in inverse source problems for the heat equation perturbed by a singular inverse-square potential − μ/|x|2. Since the works of Baras and Goldstein [2], the existence of a critical parameter μ☆(N) = (N − 2)2/4 (the optimal constant appearing in the so-called Hardy inequality) is by now well-known. Recently specific Carleman inequalities (involving suitable weight functions to take into account the singularity) have been developed by Vancostenoble and Zuazua [42] and Ervedoza [20] in the sub-critical and critical cases. They have been used to deduce observability and null controllability results. In this paper, we are interested in inverse source problems with a locally distributed observation, such as those studied by Immanuvilov and Yamamoto [25]: using the classical Carleman estimates by Fursikov and Immanuvilov [21], they obtain Lipschitz stability results in the case of the classical heat equation. Here we study the case of the heat equation with inverse-square potentials. In this purpose, the specific Carleman inequalities by Vancostenoble and Zuazua and Ervedoza are not sufficient to conclude. Hence we first complete them by deriving sharper Carleman estimates. Major steps in the proof rely on several improved forms of Hardy-Poincaré inequalities. Next we prove Lipschitz stability results in the case of the heat equation with sub-critical or critical inverse-square potentials that extend the results by Immanuvilov-Yamamoto.