Well-posedness properties of geometric variational problems: existence, regularity and uniqueness results

Abstract: This thesis is devoted to the study of well-posedness properties of some geometric variational problems: existence, regularity and uniqueness of solutions. We study two specific problems arising in the context of geometric calculus of variations and sharing strong analogies: the Plateau's problem and the optimal branched transport problem. The first part of the thesis discusses the existence theory. Both problems are formulated in the language of Federer and Fleming's theory of currents. After an exposition of the main results, we will present the core ideas of the (interior) regularity theory for area-minimizing currents and for optimal transport paths. The last part of the thesis contains two original results: the generic uniqueness of solutions both for the Plateau's problem (in any dimension and codimension) and for the optimal branched transport problem.