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Morrey Conjecture deals with two properties of functions which are known asquasi-convexity and rank-one convexity. It is well established that everyfunction satisfying the quasi-convexity property also satisfies rank-oneconvexity. Morrey (1952) conjectured that the reversed implication will notalways hold. In 1992, Vladimir Sverak found a counterexample to prove thatMorrey 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 analyticallythese notions is a very difficult task as the quasi-convexity criterion is ofnon-local type, especially for vector-valued functions. That's why we performsome numerical simulations based on a gradient descent algorithm usingDacorogna and Marcellini example functions. Our numerical results indicate thatMorrey Conjecture holds true.

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