| Abstract: |
| Motivated by peridynamic models capturing discontinuities and singularities in material behavior, and building on recent advances in nonlocal hyperelasticity, this talk studies a class of variational problems involving integral functionals with nonlocal gradients. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the 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. We will discuss properties of the associated Sobolev spaces, focusing in particular on the analysis of a trace operator and the proof of a Poincar\`e inequality. A central ingredient of our approach is to exploit connections with pseudo-differential operator theory. As an application, we show 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. |
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