具有连续拉格朗日量和奇异极小值的一维变分问题

一类不连续 Lagrangian 极小元的部分正则性

Partial regularity for minimizers of a class of discontinuous Lagrangians

本文研究含离散 Monge-Ampère 引力模型变分形式的一维不连续 Lagrangian 问题，借助 Voronoi 分解的几何结构证明局部极小元在有限冲击时刻外具有 C^{1,1} 正则性。

Causal Variational Principles in the $σ$-Locally Compact Setting: Existence of Minimizers

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous…

Causal Variational Principles in the Infinite-Dimensional Setting: Existence of Minimizers

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler-Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under …

控制仿射问题广义极小元的奇异性度量

Measuring Singularity of Generalized Minimizers for Control-Affine Problems

一个开放的问题由Yu提供。Orlov最近出版的一卷“未解决的问题在数学系统和控制理论“，V.D. Blondel，A. Megretski（eds），Princeton Univ. Press，2004，关注最优控制-仿射问题的正则化。这些非强制性的问题一般承认“廉价（广义）控制”作为最小化，它一直受到质疑是否和在什么条件下的正规化问题的下确界收敛到原问题的下确界。从这个问题的研究，我们表明，简单的功能理论推理，它承认，在一般情况下，积极的答案。这个答案不依赖于受控向量场的交换性/非交换性。相反，它取决于拉夫连季耶夫间隙的存在与否。提出了一个用奇异度来度量控制仿射最优控制问题极小化序列奇异性的替代问题。结果表明，在奇异线性二次型问题的特殊情况下，这个程度是密切相关的“奇异性的顺序”的问题。我们提出了一个类似的非线性控制问题-仿射问题，并建立了部分结果。提出了一些建议和有待解决的问题。

变分法中的泛奇异集

Universal singular sets in the calculus of variations

对于正则一维变分问题，Ball和Nadirashvilli引入了拉格朗日L的泛奇异集的概念，并建立了它的拓扑可忽略性。这个集合被定义为某个绝对连续的L-极小元的图以无穷导数通过的平面上所有点的集合。受Tonelli的部分正则性结果的启发，普遍奇异集在测度上的大小问题自然产生了。在这里，我们表明，普遍奇异集的特点是基本上是纯粹的unrectifiable -也就是说，他们相交于大多数Lipschitz曲线在零长度的集合，任何紧凑的纯unrectifiable集包含在一些光滑拉格朗日的普遍奇异集给定的超线性增长。这给出了豪斯多夫维数2的通用奇异集的例子，填补了以前已知的一维例子和Sychev的结果之间的空白，通用奇异集是勒贝格零。我们表明，一些光滑的拉格朗日是必要的拓扑尺寸估计，并调查之间的关系增长的拉格朗日和存在（病态）可求长件的普遍奇异集。我们还表明，Tonelli的部分正则性结果是稳定的，在一个“近”极小U的能量超过它有大的衍生物的集合是由有多远U是一个极小控制。

Singular Support of Minimizers of the Causal Variational Principle on the Sphere

The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $τ>\sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $τ>\sqrt{6}$, the support is proven to have Hausdorff dimension at most $6/7$.