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 new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the $$\sigma $$ -gradient ( $$0<\sigma <1$$ ). We establish continuous dependence results with respect to the data, including the threshold of the fractional $$\sigma $$ -gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with 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 $$\sigma $$ -gradient constrained problem, extending previous results for the classical gradient case, i.e., with $$\sigma =1$$ .