一维中的变分、近似和低正则性
Variations, Approximation, and Low Regularity in One Dimension
摘要：
研究了当拉格朗日量的光滑性不高于连续性时，一维变分问题极小解的性质。一个初等近似的结果证明，但它表明，这不能在一般的形式的一个标准的Lipschitz“变化”。这项调查的一部分，但在其本身的权利感兴趣，是一个无处本地Lipschitz极小的例子，作为一个反例，任何假定的托内利部分正则性声明。在这些低的假设下，我们发现仍然有可能得到极小化的必要条件，在近似连续性和单侧导数相等方面。
英文摘要：
We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general of the form of a standard Lipschitz "variation". Part of this investigation, but of interest in its own right, is an example of a nowhere locally Lipschitz minimizer which serves as a counter-example to any putative Tonelli partial regularity statement. Under these low assumptions we find it nonetheless remains possible to derive necessary conditions for minimizers, in terms of approximate continuity and equality of the one-sided derivatives.
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