论文标题
部分可观测时空混沌系统的无模型预测
Some Numerical Simulations Based on Dacorogna Example Functions in Favor of Morrey Conjecture
论文作者
论文摘要
Morrey的猜想涉及两种函数的属性，称为准跨性别性和等级为单位。众所周知，满足准分子属性属性的每个函数也可以满足排名一的凸度。莫里（Morrey）（1952）猜想，反向含义并非总是如此。 1992年，弗拉基米尔·斯维拉克（Vladimir Sverak）发现了一个反例，证明莫雷（Morrey）的猜想在三维情况下是正确的。然而，由于其与复杂分析，谐波分析，几何函数理论，概率，martingales，差异包含和平面非线性弹性的联系，因此平面案例仍然保持开放和有趣。通过分析检查这些概念是一项非常困难的任务，因为准分子标准标准是非本地类型的，尤其是对于矢量值函数。这就是为什么我们使用dacorogna和Marcellini示例函数基于梯度下降算法进行一些数值模拟的原因。我们的数值结果表明莫雷的猜想是正确的。
Morrey Conjecture deals with two properties of functions which are known as quasi-convexity and rank-one convexity. It is well established that every function satisfying the quasi-convexity property also satisfies rank-one convexity. Morrey (1952) conjectured that the reversed implication will not always hold. In 1992, Vladimir Sverak found a counterexample to prove that Morrey Conjecture is true in three dimensional case. The planar case remains, however, open and interesting because of its connections to complex analysis, harmonic analysis, geometric function theory, probability, martingales, differential inclusions and planar non-linear elasticity. Checking analytically these notions is a very difficult task as the quasi-convexity criterion is of non-local type, especially for vector-valued functions. That's why we perform some numerical simulations based on a gradient descent algorithm using Dacorogna and Marcellini example functions. Our numerical results indicate that Morrey Conjecture holds true.
