Variational methods for hyperbolic obstacle-type problems, k-harmonic maps with defects and optimal Steiner-type networks

In this thesis, we present existence results for a class of hyperbolic obstacle-type problems by using a variational scheme in the spirit of minimizing movements. We consider both linear and nonlinear cases, as well as non-local (fractional) operators. We discuss some applications to singular limits of nonlinear wave equations and to nonlinear waves in adhesive phenomena. Then, we move to discuss the relation between energy minimizing maps with prescribed singularities and (Gilbert-)Steiner optimal transport networks. More precisely, we show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb.