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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<\/jats:italic>. We will demonstrate these methods using the planar isotropic rank-one convex function $$\\begin{aligned} W_\\textrm{magic}^+(F)=\\frac{\\lambda _\\textrm{max}}{\\lambda _\\textrm{min}}-\\log \\frac{\\lambda _\\textrm{max}}{\\lambda _\\textrm{min}}+\\log \\det F=\\frac{\\lambda _\\textrm{max}}{\\lambda _\\textrm{min}}+2\\log \\lambda _\\textrm{min}\\,, \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n W<\/mml:mi>\n magic<\/mml:mtext>\n +<\/mml:mo>\n <\/mml:msubsup>\n \n (<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \n \u03bb<\/mml:mi>\n max<\/mml:mtext>\n <\/mml:msub>\n \n \u03bb<\/mml:mi>\n min<\/mml:mtext>\n <\/mml:msub>\n <\/mml:mfrac>\n -<\/mml:mo>\n log<\/mml:mo>\n \n \n \u03bb<\/mml:mi>\n max<\/mml:mtext>\n <\/mml:msub>\n \n \u03bb<\/mml:mi>\n min<\/mml:mtext>\n <\/mml:msub>\n <\/mml:mfrac>\n +<\/mml:mo>\n log<\/mml:mo>\n det<\/mml:mo>\n F<\/mml:mi>\n =<\/mml:mo>\n \n \n \u03bb<\/mml:mi>\n max<\/mml:mtext>\n <\/mml:msub>\n \n \u03bb<\/mml:mi>\n min<\/mml:mtext>\n <\/mml:msub>\n <\/mml:mfrac>\n +<\/mml:mo>\n 2<\/mml:mn>\n log<\/mml:mo>\n \n \u03bb<\/mml:mi>\n min<\/mml:mtext>\n <\/mml:msub>\n \n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>where $$\\lambda _\\textrm{max}\\ge \\lambda _\\textrm{min}$$<\/jats:tex-math>\n \n \n \u03bb<\/mml:mi>\n max<\/mml:mtext>\n <\/mml:msub>\n \u2265<\/mml:mo>\n \n \u03bb<\/mml:mi>\n min<\/mml:mtext>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are the singular values of F<\/jats:italic>, 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 $$W:{\\text {GL}}^+(2)\\rightarrow {\\mathbb {R}}$$<\/jats:tex-math>\n \n W<\/mml:mi>\n :<\/mml:mo>\n \n \n GL<\/mml:mtext>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2192<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with an additive volumetric-isochoric split of the form $$\\begin{aligned} W(F)=W_\\textrm{iso}(F)+W_\\textrm{vol}(\\det F)={\\widetilde{W}}_\\textrm{iso}\\bigg (\\frac{F}{\\sqrt{\\det F}}\\bigg )+W_\\textrm{vol}(\\det F) \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n W<\/mml:mi>\n \n (<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n W<\/mml:mi>\n iso<\/mml:mtext>\n <\/mml:msub>\n \n (<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n \n W<\/mml:mi>\n vol<\/mml:mtext>\n <\/mml:msub>\n \n (<\/mml:mo>\n det<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \n W<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n iso<\/mml:mtext>\n <\/mml:msub>\n \n (<\/mml:mo>\n <\/mml:mrow>\n \n F<\/mml:mi>\n \n \n det<\/mml:mo>\n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msqrt>\n <\/mml:mfrac>\n \n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n \n W<\/mml:mi>\n vol<\/mml:mtext>\n <\/mml:msub>\n \n (<\/mml:mo>\n det<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>with a concave volumetric part. This example is therefore of particular interest with regard to Morrey\u2019s open question whether or not rank-one convexity implies quasiconvexity in the planar case.<\/jats:p>","DOI":"10.1007\/s00332-022-09820-x","type":"journal-article","created":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T13:03:57Z","timestamp":1662123837000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey\u2019s Conjecture"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8912-9625","authenticated-orcid":false,"given":"Jendrik","family":"Voss","sequence":"first","affiliation":[]},{"given":"Robert J.","family":"Martin","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1093-6374","authenticated-orcid":false,"given":"Oliver","family":"Sander","sequence":"additional","affiliation":[]},{"given":"Siddhant","family":"Kumar","sequence":"additional","affiliation":[]},{"given":"Dennis M.","family":"Kochmann","sequence":"additional","affiliation":[]},{"given":"Patrizio","family":"Neff","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,2]]},"reference":[{"key":"9820_CR1","doi-asserted-by":"crossref","unstructured":"Astala, K., Iwaniec, T., Prause, I., Saksman, E.: Burkholder integrals, Morrey\u2019s problem and quasiconformal mappings. 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