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

TL;DR

This thesis investigates the well-posedness of geometric variational problems, focusing on existence, regularity, and uniqueness of solutions for Plateau's problem and optimal branched transport, using currents theory.

Contribution

It provides new results on the existence, regularity, and generic uniqueness of solutions for these classical geometric problems.

Findings

Existence of solutions established for both problems.

Regularity theory developed for area-minimizing currents and transport paths.

Proved generic uniqueness of solutions in any dimension and codimension.

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.