  It is well known that a twice-differentiable real-valued function
$W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ on the group
$\operatorname{GL}^+(n)$ of invertible $n\times n-$matrices with positive
determinant is rank-one convex if and only if it is Legendre-Hadamard elliptic.
Many energy functions arising from interesting applications in isotropic
nonlinear elasticity, however, are not necessarily twice differentiable
everywhere on $\operatorname{GL}^+(n)$, especially at points with non-simple
singular values.
  Here, we show that if an isotropic function $W$ on $\operatorname{GL}^+(n)$
is twice differentiable at each $F\in\operatorname{GL}^+(n)$ with simple
singular values and Legendre-Hadamard elliptic at each such $F$, then $W$ is
already rank-one convex under strongly reduced regularity assumptions. In
particular, this generalization makes (local) ellipticity criteria accessible
as criteria for (global) rank-one convexity to a wider class of elastic energy
potentials expressed in terms of ordered singular values. Our results are also
directly applicable to so-called conformally invariant energy functions. We
also discuss a classical ellipticity criterion for the planar case by Knowles
and Sternberg which has often been used in the literature as a criterion for
global rank-one convexity and show that for this purpose, it is still
applicable under weakened regularity assumptions.
