On nonlocal variational and quasi-variational inequalities with fractional gradient

Abstract

We extend classical results on variational inequalities with convex sets with gradient constraint to a newclass of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the σ-gradient (0 < σ < 1).We establish continuous dependence results with respect to the data, including the threshold of the fractional σ-gradient.Using these propertieswe give newresults on the existence to a class of quasi-variational variational inequalitieswith fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the gradient constrained problem, extending previous results for the classical gradient case, i.e., with σ = 1.