Speaker
Description
In this talk, 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 discontinuous and singular material effects, such as fracture. Specific to our set-up is that the nonlocal gradient has a space-dependent interaction range that vanishes at the boundary of the reference domain. In particular, the nonlocal operator only depends on values inside the domain and localizes to the classical gradient on the boundary. Our main contribution consists of a comprehensive treatment of the associated Sobolev spaces, including the analysis of a trace operator and the validity of a Poincaré inequality. As an application, we obtain the existence of minimizers for quasiconvex or polyconvex integral functionals involving these heterogeneous nonlocal gradients subject to local boundary conditions of Dirichlet-, Neumann- or mixed-type.
