Existence for Nonlocal Variational Problems in Peridynamics

We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in solid mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincaré inequality. We cover Dirichlet, Neumann, and mixed boundary conditions. The existence theory is set in the Lebesgue Lp spaces and in the fractional Sobolev W s,p spaces, for 0 < s < 1 and 1 < p < ∞.