Optimal control of the thermistor problem in three spatial dimensions, Part 2: Optimality conditions

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This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion ...