Hybrid inverse problems and redundant systems of partial differential equations

Hybrid inverse problems are mathematical descriptions of coupled-physics (also called multi-waves) imaging modalities that aim to combine high resolution with high contrast. The solution of a high-resolution inverse problem, a first step that is not considered in this paper, provides internal information combining unknown parameters and solutions of differential equations. In several settings, the internal information and the differential equations may be described as a redundant system of nonlinear partial differential equations. We propose a framework to analyze the uniqueness and stability properties of such systems. We consider the case when the linearization of the redundant system is elliptic and with boundary conditions satisfying the Lopatinskii conditions. General theories of elliptic systems then allow us to construct a parametrix for such systems and derive optimal stability estimates. The injectivity of the nonlinear problem or its linearization is not guaranteed by the ellipticity condition. We revisit unique continuation principles, such as the Holmgren theorem and the uniqueness theorem of Calder\'on, in the context of redundant elliptic systems of equations. The theory is applied to the case of power density measurements, which are internal functionals of the form $\gamma|\nabla u|^2$ where $\gamma$ is an unknown parameter and $u$ is the solution to the elliptic equation $\nabla\cdot\gamma\nabla u=0$ on a bounded domain with appropriate boundary conditions.