Fractional Infinity Laplacian with Obstacle

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in {u>0}\ u \geq 0 &\qquad in, Ω\ u=g &\qquad on, \partial Ω\end{cases},$$ with $$Lu=\sup_{y\in Ω,,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^α}+\inf_{y\in Ω,,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^α},\qquad 0<α<1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,β}(\partialΩ)$ for some $0<β<α$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $β-$Hölderian on $\overlineΩ$. 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 Hölder 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.

💡 Research Summary

The paper studies a Dirichlet–obstacle problem for the fractional infinity Laplacian
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