  We develop a theory of existence of minimizers of energy functionals in
vectorial problems based on a nonlocal gradient under Dirichlet boundary
conditions. The model shares many features with the peridynamics model and is
also applicable to nonlocal solid mechanics, especially nonlinear elasticity.
This nonlocal gradient was introduced in an earlier work, inspired by Riesz'
fractional gradient, but suitable for bounded domains. The main assumption on
the integrand of the energy is polyconvexity. Thus, we adapt the corresponding
results of the classical case to this nonlocal context, notably, Piola's
identity, the integration by parts of the determinant and the weak continuity
of the determinant. The proof exploits the fact that every nonlocal gradient is
a classical gradient. Contrary to classical elasticity, this existence result
is compatible with cavitation and fracture.
