论文标题
限制了变化的计算的非平滑问题
Constrained Nonsmooth Problems of the Calculus of Variations
论文作者
论文摘要
该论文致力于分析具有不同类型约束的变体的非平滑多维多维问题的最佳条件，例如边界上的其他约束和等值限制。为了获得最佳条件，我们研究了称为Codiverentible和Ampriverentible的非平滑函数可不同函数的广义概念。在一些自然且易于验证的假设下，我们证明在Sobolev空间上定义的非平滑积分功能是不断编纂的，并计算其编码和准差。然后，我们将一般的最佳条件应用于Banach空间中非平滑优化问题的一般最佳条件，以获取变化计算的非平滑问题问题的最佳条件。通过一系列简单的示例，我们证明了我们的最佳条件有时比现有的条件更好，从某种意义上说，当我们的最佳条件可以检测到给定点的非偏见性时，当基于亚分化的最佳条件不足以剥夺这一点时，这一点是非优势。
The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.
