{"ID":22938761,"CreatedAt":"2026-09-17T01:37:05.451791504Z","UpdatedAt":"2026-09-17T01:37:05.451791504Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18660","arxiv_id":"2609.18660","title":"Polyconvexity for incompressible inversion-symmetric energies of Valanis-Landel type","abstract":"{Let $λ_i = ν_i(F)$ denote the three singular values of the deformation gradient $F \\in {\\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_ψ(F)=\\sum_i ψ\\left(|\\!\\logλ_i|\\right)$ with $ψ:[0,\\infty)\\mapsto\\mathbb{R}$. Set $g(s)=ψ\\left({\\rm arcosh}\\frac{s}{2}\\right)$ for all $s\\geq2$. If $g$ has a convex and non-decreasing extension $\\bar g$ to $[0,\\infty)$, then $W_ψ$ is the restriction to ${\\rm SL}(3)$ of the explicit polyconvex function \\ {$ F\\mapsto\\sum_i \\bar g\\left(ν_i(F+{\\rm Cof} F)\\right). $} The proof uses the identity $ν_i(F+{\\rm Cof} F)=λ_i+λ_i^{-1}$, up to permutation, on ${\\rm SL}(3)$ and Ball's convexity theorem for functions of the singular values. We also give a direct proof along rank-one lines contained in ${\\rm SL}(3)$ and derive a convenient one-dimensional differential sufficient condition. In particular, $ F\\mapsto\\sum_i e^{\\log^2\\!λ_i} $ is rank-one convex on ${\\rm SL}(3)$ and possesses the stated polyconvex extension. A simple-shear computation shows that scalar convexity in $\\log λ_i$ alone is insufficient; the quadratic Hencky energy $\\sum_i \\log^2λ_i$ is not rank-one convex on ${\\rm SL}(3)$.","short_abstract":"{Let $λ_i = ν_i(F)$ denote the three singular values of the deformation gradient $F \\in {\\rm GL}^+(3)$.} We consider the family of incompressible isotropic energies $ W_ψ(F)=\\sum_i ψ\\left(|\\!\\logλ_i|\\right)$ with $ψ:[0,\\infty)\\mapsto\\mathbb{R}$. Set $g(s)=ψ\\left({\\rm arcosh}\\frac{s}{2}\\right)$ for all $s\\geq2$. If $g$...","url_abs":"https://arxiv.org/abs/2609.18660","url_pdf":"https://arxiv.org/pdf/2609.18660v1","authors":"[\"Ionel-Dumitrel Ghiba\",\"Maximilian P. Wollner\",\"Patrizio Neff\"]","published":"2026-09-16T13:37:28Z","proceeding":"math.AP","tasks":"[\"math.AP\"]","methods":"[]","has_code":false}