Nonlocal variational models are used in various science and engineering applications, like continuum mechanics and image processing models. Yet, their analysis is far from straightforward, since classical methods in the calculus of variations, which tend to rely intrinsically on localization arguments, break down.
Motivated by new nonlocal models in hyperelasticity, we discuss a class of variational problems with integral functionals depending on nonlocal gradients, with focus on finite-horizon versions of the Riesz fractional gradient. We address several aspects regarding the existence theory of these problems and their asymptotic behavior. Our analysis relies on suitable translation operators that allow us to switch between nonlocal and classical gradients and are thus helpful technical tools for transferring results from one setting to the other. Based on this approach, we identify the appropriate notion of convexity for these nonlocal problems and derive new relaxation and homogenization results. Finally, we analyze localization limits, which provide a rigorous bridge between nonlocal and classical variational theories. This is based on joint work with Javier Cueto (Universidad Autónoma de Madrid) and Hidde Schönberger (UCLouvain).