变分法中的泛奇异集
Universal singular sets in the calculus of variations
摘要：
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。
英文摘要：
For regular one-dimensional variational problems, Ball and Nadirashvilli introduced the notion of the universal singular set of a Lagrangian L and established its topological negligibility. This set is defined to be the set of all points in the plane through which the graph of some absolutely continuous L-minimizer passes with infinite derivative. Motivated by Tonelli's partial regularity results, the question of the size of the universal singular set in measure naturally arises. Here we show that universal singular sets are characterized by being essentially purely unrectifiable -- that is, they intersect most Lipschitz curves in sets of zero length and that any compact purely unrectifiable set is contained within the universal singular set of some smooth Lagrangian with given superlinear growth. This gives examples of universal singular sets of Hausdorff dimension two, filling the gap between previously known one-dimensional examples and Sychev's result that universal singular sets are Lebesgue null. We show that some smoothness of the Lagrangian is necessary for the topological size estimate, and investigate the relationship between growth of the Lagrangian and the existence of (pathological) rectifiable pieces in the universal singular set. We also show that Tonelli's partial regularity result is stable in that the energy of a `near' minimizer u over the set where it has large derivative is controlled by the how far u is from being a minimizer.
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