论文标题
几何变异问题的适应性特性：存在，规律性和唯一性结果
Well-posedness properties of geometric variational problems: existence, regularity and uniqueness results
论文作者
论文摘要
本论文致力于研究某些几何变异问题的适应性特性：解决方案的存在，规律性和独特性。我们研究了在变化的几何计算和共享强大类比的背景下引起的两个特定问题：高原的问题和最佳的分支运输问题。论文的第一部分讨论了存在理论。这两个问题都是用费德勒和弗莱明的电流理论的语言提出的。在阐述了主要结果之后，我们将介绍（内部）规律性理论的核心思想，用于最小水电流和最佳传输路径。论文的最后一部分包含两个原始结果：解决高原问题的解决方案的通用唯一性（在任何维度和编辑中）以及最佳的分支传输问题。
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.
