Characterization of Lower Semicontinuity and Relaxation of Fractional Integral and Supremal Functionals

Abstract

In light of recent developments regarding fractional function spaces, new avenues have opened up for studying fractional variational problems and partial differential equations. We take the viewpoint of the calculus of variations by investigating the minimization of nonlocal integral and supremal functionals depending on the Riesz fractional gradient. With the aim of establishing the existence of minimizers, we give full characterizations of the lower semicontinuity of these fractional functionals. Interestingly, the characterizations are in terms of notions intrinsic to variational problems involving classical gradients, that is, quasiconvexity and level-quasiconvexity. The key ingredient in the proofs is an inherent connection between classical and fractional gradients, which we extend to Sobolev functions, enabling us to transition between the two settings.

In the absence of lower semicontinuity, we determine representation formulas for the relaxations, i.e. lower semicontinuous envelopes, of the fractional integral and supremal functionals. They are obtained by taking the relevant convex hulls of the integrand and supremand, but only inside a prescribed region. As such, we observe that, unlike in the classical case, the integrand and supremand change structure through the relaxation process, going from homogeneous to inhomogeneous. Finally, to draw the connection between the integral and supremal case, we present an Lp-approximation result showing the Gamma-convergence of the nonlocal integral functionals to their supremal counterpart.