Relaxation theorems and necessary optimality conditions for semiconvex multidimensional control problems

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Up to now, the proof of Pontryagin's maximum principle for multidimensional control problems is closely related to the convexity or the convex relaxability of integrand and control domain. In order to overcome this limitation, the proposal aims to derive first-order optimality conditions for multidimensional control problems with first-order PDE's involving either polyconvex or quasiconvex data from the outset or requiring nonconvex relaxation. Proof techniques, which combine elements of polyconvex analysis with discretization methods and properties of gradient Young measures, will be applied. The connections between the proof of necessary optimality conditions and the semiconvex relaxation, which is mandatory for problems in the "vectorial" case, shall be systematically investigated.