变分问题中的偏微分方程公式

Uchechukwu Opara
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引用次数: 2

摘要

变分法在多变量问题中的应用可以产生一些经典的偏微分方程(PDE)。为此,人们承认,大量的经典偏微分方程最初是由变分问题形成的。本文从光滑黎曼流形上能量泛函优化的观点出发,拟合出这类方程。这些能量泛函以流形上定义的其他泛函的充分正则积分的形式给出。通过初步分析确定了包含最优泛函解的相关Banach域,然后在这些Banach空间中通过微分找到了必要的最优性条件。为了在简单设置中确定特定的最优函数,将较小的目标域作为Banach (Sobolev)空间的适当子集。针对上述简单设置以及更一般的案例场景,介绍了所提供的分析意义和方法。
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Partial Differential Equation Formulations from Variational Problems
The calculus of variations applied in multivariate problems can give rise to several classical Partial Differential Equations (PDE’s) of interest. To this end, it is acknowledged that a vast range of classical PDE’s were formulated initially from variational problems. In this paper, we aim to formulate such equations arising from the viewpoint of optimization of energy functionals on smooth Riemannian manifolds. These energy functionals are given as sufficiently regular integrals of other functionals defined on the manifolds. Relevant Banach domains which contain the optimal functional solutions are identified by preliminary analysis, and then necessary optimality conditions are discovered by differentiation in these Banach spaces. To determine specific optimal functionals in simple settings, smaller target domains are taken as appropriate subsets of the Banach (Sobolev) spaces. Briefings on analytical implications and approaches proffered are included for the aforementioned simple settings as well as more general case scenarios.
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来源期刊
CiteScore
0.60
自引率
0.00%
发文量
2
期刊介绍: The “Italian Journal of Pure and Applied Mathematics” publishes original research works containing significant results in the field of pure and applied mathematics.
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