Fractional Infinity Laplacian with Obstacle

TL;DR

This paper establishes the existence and regularity of solutions for an obstacle problem involving the fractional infinity Laplacian with a nonhomogeneous term, using approximation and Perron's method.

Contribution

It introduces a novel approach to solve the obstacle problem for the fractional infinity Laplacian with nonhomogeneous terms, proving existence and Hölder regularity of solutions.

Findings

Existence of solutions under specified conditions.

Solutions are Hölder continuous up to the boundary.

Method involves approximation of the nonlinearity and Perron's method.

Abstract

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, \Omega\\ u=g &\qquad on\, \partial \Omega\end{cases},$$ with $$L[u](x)=\sup_{y\in \Omega,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^{\alpha}}+\inf_{y\in \Omega,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^\alpha},\qquad 0<\alpha<1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,\beta}(\partial\Omega)$ for some $0<\beta<\alpha$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $\beta-$H\"olderian on $\overline{\Omega}$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform H\"older estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem.