Fractional infinity Laplacian with obstacle

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term f ( u ), where f : ℝ + ↦ ℝ + : { L [ u ] = f ( u ) in { u > 0} u ≥ 0 in Ω u = g on ∂ Ω with L [ u ]( x ) − sup y ∈Ω, y ≠ x u ( y ) − u ( x )/| y − x | α + inf y ∈ Ω , y ≠ x u ( y ) − u ( x )/| y − x | α , 0 < α < 1. Under the assumptions that f is a continuous and monotone function and that the boundary datum g is in C 0, β (∂Ω) for some 0 < ß < α, we prove existence of a solution u to this problem. Moreover, this solution u is β —Hölderian on ̅Ω. Our proof is based on an approximation of f by an appropriate sequence of functions f ɛ where we prove using Perron's method the existence of solutions u ɛ , for every ɛ > 0. Then, we show some uniform Hölder estimates on u ɛ that guarantee that u ɛ → u where this limit function u turns out to be a solution to our obstacle problem.