Some Numerical Simulations Based on Dacorogna Example Functions in Favor of Morrey Conjecture

TL;DR

This paper uses numerical simulations with gradient descent on Dacorogna example functions to investigate Morrey Conjecture in the planar case, providing evidence supporting its validity.

Contribution

It introduces a numerical approach to test Morrey Conjecture in the planar case, an area previously explored mainly through analytical methods.

Findings

Numerical results support Morrey Conjecture in the planar case.

Gradient descent simulations indicate the conjecture holds true.

The approach offers a new tool for exploring complex properties of functions.

Abstract

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.