Stability of Coupled-Physics Inverse Problems with internal measurements

In this paper, we develop a general approach to prove stability for the non linear second step of hybrid inverse problems. We work with general functionals of the form $\sigma \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p$, $0<p\le 1$, where $u$ is the solution of the elliptic partial differential equation $\mathrm{\nabla }\cdot \sigma \mathrm{\nabla }u=0$ on a bounded domain $\mathrm{\Omega }$ with boundary conditions $u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=f$. We prove stability of the linearization and H"older conditional stability for the non-linear problem of recovering $\sigma$ from the internal measurement.