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 $|D^{\ddot{I}}ƒu|\le g$, involving the distributional fractional Riesz gradient $D^{\ddot{I}}ƒ$, $0<\ddot{I}ƒ<1$, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems. We also prove their convergence as $\ddot{I}ƒ\nearrow 1$ towards their local counterparts with the gradient constraint $|Du|\le g$.