  Based on recent developments in the theory of fractional Sobolev spaces, an
interesting new class of nonlocal variational problems has emerged in the
literature. These problems, which are the focus of this work, involve integral
functionals that depend on Riesz fractional gradients instead of ordinary
gradients and are considered subject to a complementary-value condition. With
the goal of establishing a comprehensive existence theory, we provide a full
characterization for the weak lower semicontinuity of these functionals under
suitable growth assumptions on the integrands. In doing so, we surprisingly
identify quasiconvexity, which is intrinsic to the standard vectorial calculus
of variations, as the natural notion also in the fractional setting. In the
absence of quasiconvexity, we determine a representation formula for the
corresponding relaxed functionals, obtained via partial quasiconvexification
outside the region where complementary values are prescribed. Thus, in contrast
to classical results, the relaxation process induces a structural change in the
functional, turning the integrand from a homogeneous into an inhomogeneous one.
Our proofs rely crucially on an inherent relation between classical and
fractional gradients, which we extend to Sobolev spaces, enabling us to
transition between the two settings.
