Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey’s Conjecture

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 Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλmin,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W_\ extrm{magic}^+(F)=\\frac{\\lambda _\ extrm{max}}{\\lambda _\ extrm{min}}-\\log \\frac{\\lambda _\ extrm{max}}{\\lambda _\ extrm{min}}+\\log \\det F=\\frac{\\lambda _\ extrm{max}}{\\lambda _\ extrm{min}}+2\\log \\lambda _\ extrm{min}\\,, \\end{aligned}$$\\end{document}where lambda _textrm{max}ge lambda _textrm{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:{text {GL}}^+(2)rightarrow {mathbb {R}} with an additive volumetric-isochoric split of the form W(F)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W(F)=W_\ extrm{iso}(F)+W_\ extrm{vol}(\\det F)={\\widetilde{W}}_\ extrm{iso}\\bigg (\\frac{F}{\\sqrt{\\det F}}\\bigg )+W_\ extrm{vol}(\\det F) \\end{aligned}$$\\end{document}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.