Fractional Infinity Laplacian

• Fractional infinity Laplacian is a nonlocal, extremal operator defined as the limit of the fractional p-Laplacian as p tends to infinity, capturing extreme directional increments.
• It arises from formulations in dynamic programming and game theory, linking classical Aronsson-type equations with modern nonlocal variational problems.
• Its analysis via viscosity solutions and asymptotic expansions underpins existence, uniqueness, and regularity results in a broad class of nonlocal obstacle, Dirichlet, and evolution problems.

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 (Dweik et al., 6 Jul 2025).

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 $\nabla u$ (Bjorland et al., 2010, Teso et al., 2022).

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 $s$ (Bonder et al., 2018).

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 $p \to \infty$ (Dweik et al., 6 Jul 2025, Bonder et al., 2018).

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 (Dweik et al., 6 Jul 2025, Bonder et al., 2018).

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 (Dweik et al., 6 Jul 2025).

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 (Bjorland et al., 2010).

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 $\Delta^s_\infty u = f$ (Dweik et al., 6 Jul 2025, Bjorland et al., 2010).

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 (Dweik et al., 6 Jul 2025) 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 (Dweik et al., 6 Jul 2025, Bjorland et al., 2010).
• Comparison Principle: Strong comparison principles hold in suitable domains, providing uniqueness (in strip-like geometries for the tug-of-war operator) (Bjorland et al., 2010).

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 (Teso et al., 2020).

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 (Bjorland et al., 2010).
• 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 (Teso et al., 2022).

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 (Teso et al., 2022).

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 (Bonder et al., 2018).

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:

• (Dweik et al., 6 Jul 2025) Dweik, Sabra: "Fractional Infinity Laplacian with Obstacle"
• (Bjorland et al., 2010) Bjorland, Caffarelli, Figalli: "Non-Local Tug-of-War and the Infinity Fractional Laplacian"
• (Teso et al., 2020) del Teso, Endal, Lewicka: "On asymptotic expansions for the fractional infinity Laplacian"
• (Teso et al., 2022) del Teso, Endal, Jakobsen, Vázquez: "Evolution Driven by the Infinity Fractional Laplacian"
• (Bonder et al., 2018) Demengel, Sire, Valdinoci: "A Hölder Infinity Laplacian obtained as limit of Orlicz Fractional Laplacians"