论文标题
平均距离问题与曲率惩罚进行数据参数化：最小化器的规律性
Average-distance problem with curvature penalization for data parameterization: regularity of minimizers
论文作者
论文摘要
我们提出了一个用于在给定度量中找到一维结构的模型。我们的方法是基于最小化目标函数，该目标功能结合了平均距离功能，以测量近似值的质量并惩罚曲率，类似于弹性功能。引入曲率惩罚克服了平均距离功能的某些缺点，特别是缺乏最小化器的规律性。我们确定了所提出功能的最小化器的存在，独特性和规律性。特别是我们在最小化器上建立了$ c^{1,1} $估计。
We propose a model for finding one-dimensional structure in a given measure. Our approach is based on minimizing an objective functional which combines the average-distance functional to measure the quality of the approximation and penalizes the curvature, similarly to the elastica functional. Introducing the curvature penalization overcomes some of the shortcomings of the average-distance functional, in particular the lack of regularity of minimizers. We establish existence, uniqueness and regularity of minimizers of the proposed functional. In particular we establish $C^{1,1}$ estimates on the minimizers.
