  The center of interest in this work are variational problems with integral
functionals depending on special nonlocal gradients. The latter correspond to
truncated versions of the Riesz fractional gradient, as introduced in [Bellido,
Cueto & Mora-Corral 2022] along with the underlying function spaces. We
contribute several new aspects to both the existence theory of these problems
and the study of their asymptotic behavior. Our overall proof strategy builds
on finding suitable translation operators that allow to switch between the
three types of gradients: classical, fractional, and nonlocal. These provide
useful technical tools for transferring results from one setting to the other.
Based on this approach, we show that quasiconvexity, which is the natural
convexity notion in the classical -- and as shown in [Kreisbeck & Sch\"onberger
2022] also in the fractional -- calculus of variations, gives a necessary and
sufficient condition for the weak lower semicontinuity of the nonlocal
functionals as well. As a consequence of a general Gamma-convergence statement,
we obtain relaxation and homogenization results. The analysis of the limiting
behavior for varying fractional parameters yields, in particular, a rigorous
localization with a classical local limit model.
