Fractional Infinity Laplacian with Obstacle

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:{\mathbb{R}}^{+}\mapsto {\mathbb{R}}^{+}$: $$\{\begin{array}{ll}L[u]=f(u)& in\{u>0\}\\ u\ge 0& in\mathrm{\Omega }\\ u=g& on\mathrm{\partial }\mathrm{\Omega }\end{array},$$ with $$L[u](x)=\underset{y\in \mathrm{\Omega },y\mathrm{\ne }x}{\sup }{\displaystyle \frac{u(y)-u(x)}{\mathrm{\mid }y-x{\mathrm{\mid }}^{\alpha }}}+\underset{y\in \mathrm{\Omega },y\mathrm{\ne }x}{\inf }{\displaystyle \frac{u(y)-u(x)}{\mathrm{\mid }y-x{\mathrm{\mid }}^{\alpha }}},0<\alpha <1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,\beta }(\mathrm{\partial }\mathrm{\Omega })$ for some $0<\beta <\alpha$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $\beta -$Hölderian on $\bar{\mathrm{\Omega }}$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_{\epsilon }$ where we prove using Perron’s method the existence of solutions $u_{\epsilon }$, for every $\epsilon >0$. Then, we show some uniform Hölder estimates on $u_{\epsilon }$ that guarantee that $u_{\epsilon }\to u$ where this limit function $u$ turns out to be a solution to our obstacle problem.