---
title: Fractional Infinity Laplacian
url: https://www.emergentmind.com/topics/fractional-infinity-laplacian
type: topic
---

# Fractional Infinity Laplacian

The fractional infinity Laplacian is a nonlocal, extremal, highly nonlinear operator characterized by its role as the limit case of the fractional $p$-Laplacian when $p \to \infty$. It lies at the intersection of classical Aronsson-type “$\infty$-Laplacian” equations and the framework of nonlocal (fractional) differential operators. The operator has several closely related definitions that arise in different settings—discrete dynamic programming, game theory (nonlocal tug-of-war), and as asymptotic limits of integral energies. It figures centrally in fully nonlinear nonlocal obstacle, boundary, and evolution problems, with a theory grounded in viscosity solutions, comparison principles, and explicit asymptotic regularity.

## 1. Operator Definitions and Extremal Structure

For a bounded open set $\Omega \subset \mathbb{R}^N$ and $0<\alpha<1$, the Dweik–Sabra definition gives the fractional infinity Laplacian:
\[
L[u](x) = \sup_{y \in \Omega, y \neq x} \frac{u(y) - u(x)}{|y-x|^\alpha} + \inf_{y \in \Omega, y \neq x} \frac{u(y) - u(x)}{|y-x|^\alpha}
\]
The operator thus combines the largest “incremental quotient” with the smallest, measuring the extremal spread of $u$ at $x$, and is degenerate elliptic in the viscosity sense [2507.04328].

For $s \in (\frac{1}{2},1)$, Bjorland–Caffarelli–Figalli introduce an integral form (the “infinity fractional Laplacian”):
\[
\Delta_\infty^s u(x) = \sup_{y \in S^{N-1}} \inf_{z \in S^{N-1}} \int_0^\infty \frac{u(x+\eta y) + u(x-\eta z) - 2u(x)}{\eta^{1+2s}}\,d\eta
\]
At points where $\nabla u(x) \neq 0$, the extremal directions align, and the expression reduces to a single-direction tail integral, reflecting nonlocal increments along $\nabla u$ [1011.1966, 2210.06414].

An alternate notation, the Hölder infinity Laplacian, becomes:
\[
(-\Delta^s_\infty) u(x) = {}_s^+u(x) + {}_s^-u(x), \quad\text{where}\quad 
{}_s^+u(x) = \sup_{y} \frac{u(x)-u(y)}{|x-y|^s},\ 
{}_s^-u(x) = \inf_{y} \frac{u(x)-u(y)}{|x-y|^s}
\]
This reflects the maximal and minimal pointwise directional increments of H\"older order $s$ [1807.01669].

## 2. Relationship to Classical and Fractional Laplacians

The operator interpolates between:

- The classical $\infty$-Laplacian (Aronsson’s equation): $\Delta_\infty u = D^2u\,\nabla u \cdot \nabla u$.
- The standard (linear) fractional Laplacian:
  \[
  (-\Delta)^s u(x) = C_{n,s} \,\mathrm{p.v.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\quad 0<s<1
  \]
- The “fractional $p$-Laplacian,” which reads
  \[
  L_p[u](x) = \int_\Omega \left| \frac{u(x)-u(y)}{|x-y|^\alpha} \right|^{p-1} \text{sgn}(u(x)-u(y))\,|x-y|^{-\alpha}\,dy
  \]
  and converges to the $\infty$-Laplacian as $p \to \infty$ [2507.04328, 1807.01669].

Unlike the linear fractional Laplacian, the fractional infinity Laplacian is defined via extremes rather than averages, representing a “max–min” principle over nonlocal increments. It is the $p\to\infty$ limit of the fractional $p$-Laplacian; analogously to the local case, this limit localizes the operator to the two extremal increments [2507.04328, 1807.01669].

## 3. Obstacle, Dirichlet, and Double Obstacle Problems

The fractional infinity Laplacian governs nonlocal versions of classical variational inequalities.

### Obstacle Problem
Given nondecreasing $f: [0,\infty) \to [0,\infty)$, boundary datum $g \in C^{0,\beta}(\partial\Omega)$ with $0<\beta<\alpha$, and $g \geq 0$, seek $u:\Omega \to [0,\infty)$ satisfying
\[
\begin{cases}
L[u](x) = f(u(x)) & x \in \{u > 0\} \\
u(x) \geq 0 & x \in \Omega \\
u(x) = g(x) & x \in \partial\Omega
\end{cases}
\]
in the sense of viscosity solutions. This extends the classical obstacle problem to the degenerate, nonlocal, and extremal context [2507.04328].

### Double Obstacle Problem
For two Lipschitz (and $C^{1,1}$ where active) obstacles $\Gamma^-, \Gamma^+$, the condition is:
- $u \geq \Gamma^-$, $u \leq \Gamma^+$ everywhere,
- $\Delta_\infty^s u \geq 0$ where $u > \Gamma^-$,
- $\Delta_\infty^s u \leq 0$ where $u < \Gamma^+$.

Under monotonicity hypotheses and suitable geometry, unique continuous viscosity solutions exist, with explicit regularity properties [1011.1966].

### Dirichlet Problem
For domains with prescribed boundary data and sufficient regularity, Perron’s method and comparison principles grant existence and (in certain geometries) uniqueness of viscosity solutions to $L[u]=f$ or $\Delta^s_\infty u = f$ [2507.04328, 1011.1966].

## 4. Existence, Regularity, and Viscosity Solutions

Existence and regularity properties for the fractional infinity Laplacian rely on viscosity solution theory, barrier constructions, and compactness.

- **Existence (Obstacle Problem):** The main result [2507.04328] asserts that if $f$ is continuous, nondecreasing, and $g \in C^{0,\beta}(\partial\Omega)$, there exists a viscosity solution $u \in C^{0,\beta}(\overline\Omega)$.
- **Approximation Techniques:** The proof proceeds by mollifying $f$ to get $f_\varepsilon$, constructing sub- and super-solutions with explicit barriers of the form $\pm C|x-x_0|^\beta$, applying Perron’s method, and passing to the limit while deriving uniform Hölder estimates.
- **Regularity:** Any viscosity solution $u$ or $U$ to $L[u]=0$ or $\Delta^s_\infty u=0$ is $C^{0,\beta}$ locally and globally for every $0<\beta<\alpha$ (or $0<\beta<2s-1$ in the nonlocal tug-of-war setting), with sharp a priori seminorms. In the homogeneous case, interior $C^{0,\alpha}$ regularity follows by comparison with explicit power barriers [2507.04328, 1011.1966].
- **Comparison Principle:** Strong comparison principles hold in suitable domains, providing uniqueness (in strip-like geometries for the tug-of-war operator) [1011.1966].

## 5. Asymptotic Expansions and Mean Value Characterization

The operator admits a probabilistic and analytic characterization through asymptotic expansions of nonlocal averages.

Let $A_\varepsilon^+\varphi(x) = \sup_{|y|=1} \int_{t=\varepsilon}^\infty \varphi(x+ty)\,d\mu_s(t)$, $A_\varepsilon^-\varphi(x) = \inf_{|y|=1} \int_{t=\varepsilon}^\infty \varphi(x+ty)\,d\mu_s(t)$, with $\mu_s(t) = \alpha_s t^{-1-2s}dt$. For $\varphi$ smooth at $x$,
\[
A_\varepsilon^{\pm}\varphi(x) = \varphi(x) + s \epsilon^{2s} \Delta_\infty^s\varphi(x) + o(\epsilon^{2s}) \quad \text{as } \varepsilon \to 0^+
\]
Thus, $\Delta_\infty^s\varphi(x)$ is the coefficient of the $\epsilon^{2s}$ term in the expansion, and
\[
\Delta_\infty^s\varphi(x) = \lim_{\epsilon \to 0} \frac{A_\varepsilon^{\pm}\varphi(x) - \varphi(x)}{s \epsilon^{2s}}
\]
This frames the operator as a nonlinear, nonlocal “mean-value” limit, analogous to classical results for the Laplacian and the local $\infty$-Laplacian [2007.15765].

## 6. Connections to Nonlocal Games and Parabolic Flow

The “infinity fractional Laplacian” arises as the dynamic-programming operator for a nonlocal tug-of-war game:

- At each (discrete) step, two players select directions; the actual move is determined by sampling an $s$-stable Lévy process, leading to heavy-tailed jump distributions.
- Passing to the $\epsilon\to 0$ limit in the dynamic programming yields the operator definition [1011.1966].
- In the parabolic context, the evolution equation $u_t = I^s_\infty [u]$ is studied via viscosity solutions. Existence results are obtained via semi-discrete time schemes, and uniqueness is shown within the class of classical solutions. For radial and monotone profiles, the operator reduces exactly to the one-dimensional fractional Laplacian (after even reflection), allowing explicit smoothing kernel formulas for the evolution [2210.06414].

A global Harnack inequality holds: Solutions with nonnegative, decaying initial data satisfy sharp two-sided pointwise bounds in terms of the 1D fractional heat kernel at all positive times. Long-time asymptotics are governed by self-similar fractional diffusion profiles, with convergence up to constants [2210.06414].

## 7. Limit Equations and Asymptotic Analysis

The fractional infinity Laplacian serves as the unique viscosity limit of fractional $p$-Laplacians and Orlicz-type nonlocal equations as the growth parameter diverges. In the general Orlicz case, the asymptotic equation for $u_\infty$ is
\[
\begin{aligned}
{}_s^+ u_\infty &= 1 &\text{in } \{x: f(x)>0\} \\
{}_s^- u_\infty &= -1 &\text{in } \{x: f(x)<0\} \\
{}_s^+ u_\infty + {}_s^- u_\infty &= 0 &\text{in } \Omega \setminus \mathrm{supp}\,f
\end{aligned}
\]
with precise regularity $u_\infty\in C^{0,s}$, strong comparison, uniqueness, and explicit identification as distance-to-boundary power in the pure obstacle case [1807.01669].

This convergence provides the variational underpinning for the operator, linking the extremal structure to more classical nonlocal energies as growth parameters are sent to infinity.

---

**References:**
- [2507.04328] Dweik, Sabra: "Fractional Infinity Laplacian with Obstacle"
- [1011.1966] Bjorland, Caffarelli, Figalli: "Non-Local Tug-of-War and the Infinity Fractional Laplacian"
- [2007.15765] del Teso, Endal, Lewicka: "On asymptotic expansions for the fractional infinity Laplacian"
- [2210.06414] del Teso, Endal, Jakobsen, Vázquez: "Evolution Driven by the Infinity Fractional Laplacian"
- [1807.01669] Demengel, Sire, Valdinoci: "A Hölder Infinity Laplacian obtained as limit of Orlicz Fractional Laplacians"

Source: https://www.emergentmind.com/topics/fractional-infinity-laplacian