变化微积分经典问题中强最小值的二阶必要条件

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-13 DOI:10.1007/s00526-024-02795-5

A. D. Ioffe

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引用次数: 0

摘要

本文提出了变分微积分标准问题中强最小值的二阶必要条件。经典理论中似乎从未出现过这样的结果。但文中给出的一个简单例子表明，当所有已知条件都失效时，这个条件也能起作用。同时，该命题的证明也相当简单。论文中还解释说，该条件只对积分不凸的问题有效。

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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Second order necessary condition for a strong minimum in the classical problem of calculus of variations

The paper offers a second order necessary condition for a strong minimum in the standard problem of calculus of variations. No idea of such a result seems to have appeared in the classical theory. But a simple example given in the paper shows that the condition can work when all known conditions fail. At the same time, the proof of the proposition is fairly simple. It is also explained in the paper that the condition effectively works only for problems with integrands not convex with respect to the last (derivative) argument.

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.