A useful subdifferential in the Calculus of Variations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-11-16 DOI:10.1016/j.na.2024.113697

Piernicola Bettiol , Giuseppe De Marco , Carlo Mariconda

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引用次数: 0

Abstract

Consider the basic problem in the Calculus of Variations of minimizing an energy functional depending on absolutely continuous functions Under suitable assumptions on the Lagrangian, a well-known result establishes that the minimizers satisfy the Du Bois-Reymond equation. Recent work (cf. Bettiol and Mariconda, 2020 [1], 2023; Mariconda, 2023 [2], 2021, 2024) highlights not only that a Du Bois-Reymond condition for minimizers can be broadened to cover the case of nonsmooth extended valued Lagrangians, but also that a particular subdifferential (associated with the generalized Du Bois-Reymond condition) plays an important role in the approximation of the energy via its values along Lispchitz functions, no matter minimizers exist. A crucial point is establishing boundedness properties of this subdifferential, based on weak local boundedness properties of the Lagrangian. This is the main objective of this paper. Our approach is based on a refined analysis of the metric that can be employed to evaluate the distance from the complementary of the effective domain of the reference Lagrangian. As an application of our findings we show how it is possible to deduce the non-occurrence of the Lavrentiev phenomenon, providing a new general result.

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变分法中一个有用的微分方程

考虑变分微积分的基本问题，即最小化绝对连续函数的能量函数在拉格朗日的适当假设下，一个著名的结果确定了最小化函数满足杜布瓦-雷蒙德方程。最近的工作（参见 Bettiol 和 Mariconda，2020 [1]，2023；Mariconda，2023 [2]，2021，2024）不仅强调了最小化子的 Du Bois-Reymond 条件可以扩展到涵盖非光滑扩展值拉格朗日的情况，而且还强调了一个特定的子微分（与广义 Du Bois-Reymond 条件相关）在通过沿 Lispchitz 函数的值逼近能量方面发挥着重要作用，无论最小化子是否存在。关键的一点是根据拉格朗日的弱局部有界性特性，建立该子微分的有界性特性。这是本文的主要目标。我们的方法基于对度量的精炼分析，该度量可用于评估与参考拉格朗日有效域互补的距离。作为我们研究成果的应用，我们展示了如何推导出拉夫连季耶夫现象的不发生，从而提供了一个新的一般结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Nonlinear Analysis-Theory Methods & Applications 数学-数学

CiteScore

3.30

自引率

0.00%

发文量

265

审稿时长

60 days

期刊介绍： Nonlinear Analysis focuses on papers that address significant problems in Nonlinear Analysis that have a sustainable and important impact on the development of new directions in the theory as well as potential applications. Review articles on important topics in Nonlinear Analysis are welcome as well. In particular, only papers within the areas of specialization of the Editorial Board Members will be considered. Authors are encouraged to check the areas of expertise of the Editorial Board in order to decide whether or not their papers are appropriate for this journal. The journal aims to apply very high standards in accepting papers for publication.