Fluorescent optical tomography (FOT) is a new biomedical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses nonionizing red and infrared light. Mathematically, FOT can be modeled as an inverse parameter identification problem, associated with a coupled elliptic system with Robin boundary conditions. Herein we utilize novel methods of calculus of variations in L-infinity to lay the mathematical foundations of FOT which we pose as a PDE-constrained minimization problem in L-p and L-infinity.