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⁺: cases L[u]=f(u) & in \,\,\,\,\, >0\\\ u ≥ 0 & in \,\,\,\,\, Ω\\ u=g & on\,\,\,\,\, ∂ Ωcases, with [[EQUATION]]\\ Under the assumptions that f is a continuous and monotone function and that the boundary datum g is in C0,β(∂Ω) for some 0<β<α, we prove existence of a solution u to this problem. Moreover, this solution u is β-H\"olderian on Ω̄. Our proof is based on an approximation of f by an appropriate sequence of functions f_ε where we prove using Perron's method the existence of solutions u_ε, for every ε>0. Then, we show some uniform H\"older estimates on u_ε that guarantee that u_ε → u where this limit function u turns out to be a solution to our obstacle problem.