Branched transport and fractal structures
Branched transport and fractal structures
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分支传输和分形结构
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发表时间：
2017
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通讯作者：
Paul Pegon
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作者：
Paul Pegon
本论文致力于研究分支输运、相关的变分问题以及可能出现的分形结构。分支运输问题包括通过一个网络连接两个相同质量的测量，以最小化一定的成本，在我们的研究中，为了将质量m移动距离L，该成本将与mLα成比例。已经提出了几个连续的模型来制定这个问题，我们专注于两个主要的静态模型：拉格朗日和欧拉的，重点放在第一个。在适当地设置这些模型的基础，我们建立严格的等价性使用Smirnov分解的向量测度的分歧是一个措施。其次，我们研究了形状优化问题的分支运输，其中包括在寻找单位体积的分支运输的意义上最接近的原点。我们证明存在一个解决方案，描述为一个子水平集的景观功能，现在标准的分支运输。景观功能的保持器的规律性，在这里获得没有先验假设的考虑解决方案，使我们能够获得一个上限的闵可夫斯基维数的边界，这是非整数，我们推测是它的确切尺寸。数值模拟，基于变分近似一拉Modica Mortola的分支运输功能，已支持这一猜想。论文的最后一部分集中在景观功能，这是必不可少的变分问题的研究，涉及分支运输，因为它出现在灌溉成本的第一个变化。我们的目标是将其定义和基本属性扩展到扩展源的情况下，我们在具有有限根系的网络的情况下实现，例如，如果措施有不相交的支持。在这种情况下，我们给出了一个令人满意的定义的景观功能，它满足第一变分性质，我们证明了其保持器正则性的措施，我们要连接在合理的假设。
This thesis is devoted to the study of branched transport, related variational problems and fractal structures that are likely to arise. The branched transport problem consists in connecting two measures of same mass through a network minimizing a certain cost, which in our study will be proportional to mLα in order to move a mass m over a distance L. Several continuous models have been proposed to formulate this problem, and we focus on the two main static models : the Lagrangian and the Eulerian ones, with an emphasis on the first one. After setting properly the bases for these models, we establish rigorously their equivalence using a Smirnov decomposition of vector measures whose divergence is a measure. Secondly, we study a shape optimization problem related to branched transport which consists in finding the sets of unit volume which are closest to the origin in the sense of branched transport. We prove existence of a solution, described as a sublevel set of the landscape function, now standard in branched transport. The Holder regularity of the landscape function, obtained here without a priori hypotheses on the considered solution, allows us to obtain an upper bound on the Minkowski dimension of its boundary, which is non-integer and which we conjecture to be its exact dimension. Numerical simulations, based on a variational approximation a la Modica-Mortola of the branched transport functional, have been made to support this conjecture. The last part of the thesis focuses on the landscape function, which is essential to the study of variational problems involving branched transport as it appears as a first variation of the irrigation cost. The goal is to extend its definition and fundamental properties to the case of an extended source, which we achieve in the case of networks with finite root systems, for instance if the measures have disjoint supports. We give a satisfying definition of the landscape function in that case, which satisfies the first variation property and we prove its Holder regularity under reasonable assumptions on the measures we want to connect.