Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schr\"odinger type equation via local Cauchy data

Abstract

We consider the inverse problem of the simultaneous identification of the coefficients $\sigma$ and $q$ of the equation div$(\sigma\nabla u) + qu=0$ from the knowledge of the complete Cauchy data pairs. We assume that $\sigma=\gamma A$ where $A$ is a given matrix function and $\gamma, q$ are unknown piecewise affine scalar functions. No sign, nor spectrum condition on $q$ is assumed. We derive a result of global Lipschitz stability in dimension $n\geq 3$. The proof relies on the method of singular solutions and on the quantitative estimates of unique continuation.