  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 $\sigma|\nabla u|^p$, $0 < p \leq 1$, where $u$ is the solution of
the elliptic partial differential equation $\nabla\cdot \sigma \nabla u =0$ on
a bounded domain $\Omega$ with boundary conditions $u|_{\partial \Omega} = f$.
We prove stability of the linearization and H\"older conditional stability for
the non-linear problem of recovering $\sigma$ from the internal measurement.
