On a class of nonlocal problems with fractional gradient constraint

Abstract

We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type |D σ u| ≤ g, involving the distributional fractional Riesz gradient D σ , 0 < σ < 1, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems.We also prove their convergence as σ 1 towards their local counterparts with the gradient constraint |Du| ≤ g.