On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient
Tóm tắt
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
$$\sigma =1$$
.
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