Fractional infinity Laplacian with obstacle

¹ Department of Mathematics and Statistics, College of Arts and Sciences, Qatar University, 2713, Doha, Qatar
² Department of Mathematics, Faculty of Arts and Sciences, American University of Beirut, Fisk Hall 305, PO Box 11-0236, Riad El Solh, Beirut 1107 2020 Lebanon

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 21 July 2025
Accepted: 1 February 2026

Abstract

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.