Fractional Infinity Laplacian with Obstacle
Samer Dweik, Ahmad Sabra
TL;DR
The paper studies the fractional infinity Laplacian with a nonlinear right-hand side, L[u]=f(u), and Dirichlet boundary data on a bounded domain, where L is the sum of a nonlocal supremum and infimum with exponent α∈(0,1). Under f continuous and nondecreasing and boundary data g in C^{0,β}(∂Ω) with 0<β<α, it proves the existence of a viscosity solution u in Ω that attains g on ∂Ω and is β-Hölder continuous up to the boundary; it further develops a Perron-type construction and uniform Hölder estimates to guarantee regularity. The paper also treats an obstacle variant, showing that when f is nonnegative and nondecreasing and g≥0, there exists a nonnegative β-Hölder solution u to L[u]=f(u) in {u>0} with u=g on ∂Ω, obtained by approximating f and passing to the limit. Overall, the work extends the theory of fractional infinity Laplacians to nonlinear right-hand sides and obstacle problems, providing existence and Hölder regularity results and laying groundwork for further analysis of free boundaries in the nonlocal, nonlinear setting.
Abstract
This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, Ω\\ u=g &\qquad on\, \partial Ω\end{cases},$$ with $$L[u](x)=\sup_{y\in Ω,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^α}+\inf_{y\in Ω,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^α},\qquad 0<α<1.$$ 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.