The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

We consider conformally invariant energies W on the group GL (+) (2) of 2x2-matrices with positive determinant, i.e., W : GL(+) (2) -> R such that W(AFB) = W(F) for all A, B is an element of {a R is an element of GL(+) (2) vertical bar a is an element of (0, infinity), R is an element of SO(2)}, where SO(2) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation W(F) = h (lambda(1)/lambda(2)) of W in terms of the singular values lambda(1), lambda(2) of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion K := 1/2 parallel to F parallel to(2)/det F. Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grotzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755-4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the W-1,W- p-quasiconvex envelope on the domain GL(+) (n) which, in particular, ensure that a stricter version of Dacorogna's formula is applicable to conformally invariant energies on GL(+) (2).