Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

TL;DR

This paper explores numerical methods to test quasiconvexity of functions, focusing on a specific isotropic function related to Morrey's conjecture, aiming to find counterexamples and deepen understanding of convexity implications.

Contribution

It introduces and demonstrates numerical approaches for investigating quasiconvexity, particularly applied to a key example linked to Morrey's conjecture in the planar case.

Findings

Numerical methods can effectively search for counterexamples to quasiconvexity.

The example function relates to Morrey's conjecture and rank-one convexity.

Results inform the relationship between rank-one convexity and quasiconvexity.

Abstract

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function \[ W_{\rm magic}^+(F)=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-\log\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+\log\det F=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+2\log\lambda_{\rm min}\,, \] where $\lambda_{\rm max}\geq\lambda_{\rm min}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.