  In this paper we address nonlocal vector variational principles obtained by
substitution of the classical gradient by the Riesz fractional gradient. We
show the existence of minimizers in Bessel fractional spaces under the main
assumption of polyconvexity of the energy density, and, as a consequence, the
existence of solutions to the associated Euler--Lagrange system of nonlinear
fractional PDE. The main ingredient is the fractional Piola identity, which
establishes that the fractional divergence of the cofactor matrix of the
fractional gradient vanishes. This identity implies the weak convergence of the
determinant of the fractional gradient, and, in turn, the existence of
minimizers of the nonlocal energy. Contrary to local problems in nonlinear
elasticity, this existence result is compatible with solutions presenting
discontinuities at points and along hypersurfaces.
