publications

2022

1. arXiv

   Some Numerical Simulations in Favor of the Morrey’s Conjecture

   Xinghao Dong, and Koffi Enakoutsa

   Abstract Bib PDF

   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.

   @article{dong2022some,
     title = {Some Numerical Simulations in Favor of the Morrey’s Conjecture},
     author = {Dong, Xinghao and Enakoutsa, Koffi},
     year = {2022},
   }