Local boundary conditions in nonlocal hyperelasticity via heterogeneous horizons

Abstract: In this paper, we consider a class of variational problems with integral functionals involving nonlocal gradients. These models have been recently proposed as refinements of classical hyperelasticity, aiming for an effective framework to capture also discontinuous and singular material effects. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the nonlocal operator depends only on values within the domain and localizes to the classical gradient at the boundary, which allows for a seamless integration of nonlocal modeling with local boundary values. The main contribution of this work is a comprehensive theory for the newly introduced associated Sobolev spaces, including the rigorous treatment of a trace operator and Poincar\'e inequalities. A central aspect of our technical approach lies in exploiting connections with pseudo-differential operator theory. As an application, we establish the existence of minimizers for functionals with quasiconvex or polyconvex integrands depending on heterogeneous nonlocal gradients, subject to local Dirichlet, Neumann or mixed-type boundary conditions.