On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient

Applied Mathematics & Optimization - Tập 80 - Trang 835-852 - 2019
José Francisco Rodrigues1, Lisa Santos2
1CMAFcIO – Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa, Portugal
2CMAT and Departamento de Matemática, Escola de Ciências, Universidade do Minho, Braga, Portugal

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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