  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.
