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For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies $$W{:}{\\text {GL}}^+(n)\\rightarrow \\mathbb {R}$$<\/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 n<\/mml:mi>\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, i.e. $$\\begin{aligned} W(F)=W_{\\mathrm{iso}}(F)+W_{\\mathrm{vol}}(\\det F)={\\widetilde{W}}_{\\mathrm{iso}}\\bigg (\\frac{F}{\\sqrt{\\det F}}\\bigg )+W_{\\mathrm{vol}}(\\det F)\\,, \\end{aligned}$$<\/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:mi>\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:mi>\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:mi>\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:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n det<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\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>which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of \u201cleast\u201d rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function $$\\begin{aligned} W_{\\mathrm{magic}}^{+}(F)=\\frac{\\lambda _{\\mathrm{max}}}{\\lambda _{\\mathrm{min}}}-\\log \\frac{\\lambda _{\\mathrm{max}}}{\\lambda _{\\mathrm{min}}}+\\log \\det F=\\frac{\\lambda _{\\mathrm{max}}}{\\lambda _{\\mathrm{min}}}+2\\log \\lambda _{\\mathrm{min}} \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n W<\/mml:mi>\n \n magic<\/mml:mi>\n <\/mml:mrow>\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:mi>\n <\/mml:msub>\n \n \u03bb<\/mml:mi>\n min<\/mml:mi>\n <\/mml:msub>\n <\/mml:mfrac>\n -<\/mml:mo>\n log<\/mml:mo>\n \n \n \u03bb<\/mml:mi>\n max<\/mml:mi>\n <\/mml:msub>\n \n \u03bb<\/mml:mi>\n min<\/mml:mi>\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:mi>\n <\/mml:msub>\n \n \u03bb<\/mml:mi>\n min<\/mml:mi>\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:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>is quasiconvex. In addition, we demonstrate that under affine boundary conditions, $$W_{\\mathrm{magic}}^+(F)$$<\/jats:tex-math>\n \n \n W<\/mml:mi>\n \n magic<\/mml:mi>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msubsup>\n \n (<\/mml:mo>\n F<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis.<\/jats:p>","DOI":"10.1007\/s00332-022-09827-4","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T18:04:10Z","timestamp":1661364250000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Morrey\u2019s Conjecture for the Planar Volumetric-Isochoric Split: Least Rank-One Convex Energy Functions"],"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":[]},{"given":"Ionel-Dumitrel","family":"Ghiba","sequence":"additional","affiliation":[]},{"given":"Patrizio","family":"Neff","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"9827_CR1","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830114","volume-title":"Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane","author":"K Astala","year":"2008","unstructured":"Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. 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