On a class of nonlocal problems with fractional gradient constraint

TL;DR

This paper investigates nonlocal fractional gradient constraint problems, extending existence results and analyzing their convergence to local gradient constrained problems as the fractional order approaches one.

Contribution

It extends previous work by establishing existence and Lagrange multipliers for fractional gradient constraints and proves their convergence to local problems as the fractional order nears one.

Findings

Existence of solutions for fractional gradient constraint problems.

Existence of Lagrange multipliers in the fractional setting.

Convergence of fractional solutions to local solutions as the fractional order approaches one.

Abstract

We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type $|D^\sigma u|\leq g$, 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 $\sigma\nearrow1$ towards their local counterparts with the gradient constraint $|D u|\leq g$.