Lorina Varvaruca University of Bath mapel@maths.bath.ac.uk

On the Quasi-convexity of a functional in Nonlinear Elasticity

Abstract: Let Ω ⊂ ℝ^{3} denote the reference configuration of a nonlinear elastic material, and consider deformationsu : Ω → ℝ^{3}. We determine a sufficient condition for the stored energy density W (F) = |F|^{p} +h(det F), where h is a smooth and convex function, and 2 < p < 3, to be W^{1,p}-quasiconvex at λI on the restricted class of deformations u that satisfy condition (INV), det ∇u > 0 a.e., and which open a single hole anywhere in the material. The condition takes the form λ^{3−p}h′(λ^{3}) ≤ Υ_{p}, where Υ_{p} > 0 is explicitly determined, and is in a certain sense optimal for the class of energy densities under consideration. The proof makes use of isoperimetric inequalities and of estimates on radial p-harmonic functions.