Homogenization problems in the Calculus of Variations: an overview

Stochastic homogenization of deterministic control problems

In this paper we study homogenization of a class of control problems in a stationary and ergodic random environment. This problem has been mostly studied in the calculus of variations setting in connection to the homogenization of the Hamilton-Jacobi equations. We extend the result to the control problems with fairly general state dynamics and macroscopically inhomogeneous Lagrangians. Moreover, our approach proves homogenization under weaker growth assumptions on the Lagrangian even in the well-studied calculus of variations setting.

分数阶梯度下降法及其应用综述

An overview of the fractional-order gradient descent method and its applications

本文综述了分数阶梯度下降法及其应用，探讨了分数阶运算在优化问题中的收敛性与稳定性，通过实验验证了其在复杂问题中的有效性。

The Variable-Order Fractional Calculus of Variations

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivativ…

涉及有限时域分数阶梯度的积分泛函的变分理论

A variational theory for integral functionals involving finite-horizon fractional gradients

本文通过构造平移算子建立经典、分数阶与非局部梯度之间的联系，证明拟凸性等价于非局部积分泛函的弱下半连续性，并利用 Gamma 收敛得到松弛、均匀化及分数阶参数趋于经典局部模型的严格局部化结果。

Structured Deformations of Continua: Theory and Applications

The scope of this contribution is to present an overview of the theory of structured deformations of continua, together with some applications. Structured deformations aim at being a unified theory in which elastic and plastic behaviours, as well as fractures and defects can be described in a single setting. Since its introduction in the scientific community of rational mechanicists (Del Piero-Owen, ARMA 1993), the theory has been put in the framework of variational calculus (Choksi-Fonseca, ARMA 1997), thus allowing for solution of problems via energy minimization. Some background, three prob…