On a class of nonlocal problems with fractional gradient constraint

We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type $\mathrm{\mid }{D}^{\sigma }u\mathrm{\mid }\le g$, involving the distributional fractional Riesz gradient $D^{\sigma }$, $0<\sigma <1$, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems. We also prove their convergence as $\sigma \nearrow 1$ towards their local counterparts with the gradient constraint $\mathrm{\mid }Du\mathrm{\mid }\le g$.