Gâteaux differentiability of hyperelastic energies with blow-up at minimizers

Determine whether the hyperelastic variational integral E[y] = ∫_Ω W(x, ∇y(x) G(x)^{-1}) det(G(x)) dx is Gâteaux differentiable at its minimizers y when the stored energy density W(x,F) satisfies the physically standard assumptions of frame-indifference and blow-up as det(F)→0, where G is a given growth tensor and the elastic part of the deformation gradient is F_el = ∇y G^{-1}.

Background

The analysis of the quasi-static equilibrium equation in nonlinear elasticity typically proceeds via a variational formulation. However, for stored energies W that blow up as det(F)→0 (to model the impossibility of interpenetration/volume collapse), the smoothness of the variational integral at minimizers is not guaranteed.

In the morphoelastic setting considered, the elastic part of the deformation gradient is F_el = ∇y G{-1}. Establishing Gâteaux differentiability at minimizers would justify the Euler–Lagrange equation and facilitate analysis without additional regularization assumptions.

References

Under these circumstances, it is largely unknown, whether the energy is Gateaux differentiable at a given minimizer.

— Morphoelastic Growth in the Presence of Nutrients at Small Strains  (2604.01812 - Abels et al., 2 Apr 2026) in Introduction (paragraph following the quasi-static equilibrium equation)