Variable-Order Fractional Laplacian

• Variable-order fractional Laplacian is a nonlocal operator allowing spatially varying differentiation orders to model anomalous diffusion and complex media.
• It is defined using Fourier, hypersingular integral, and variational formulations, accommodating both linear and nonlinear variants.
• Applications include adaptive image processing, phase transitions, and heterogeneous material modeling, underscoring its theoretical and practical significance.

The variable-order fractional Laplacian (VOFL) is a class of nonlocal operators that generalize the classical and constant-order fractional Laplacian by allowing the order of differentiation to vary with position, or for nonlinear variants, with both position and the function itself. VOFLs model spatially heterogeneous anomalous diffusion, phase transitions, image processing with adaptive regularization, material interface phenomena, and a range of complex media where local regularity, memory, or dispersion change in space. VOFLs can be defined as linear operators of spectral, integral, or extension type, or as nonlinear nonlocal $p(\cdot)$-Laplacian variants, and possess rich analytical, variational, and numerical properties, including emerging frameworks for efficient high-dimensional computation on general geometries.

1. Mathematical Definitions and Operator Forms

The linear VOFL of a function $u:\mathbb{R}^d\to\mathbb{R}$, with pointwise order $\alpha(x)\in(0,2)$, is typically defined by:

• Pseudo-differential (Fourier) form:

$$\mathcal{F}\big\{(-\Delta)^{\alpha(x)/2}u\big\}(\xi) = |\xi|^{\alpha(x)}\,\widehat u(\xi)$$

for sufficiently smooth $u$ and $\alpha(x)$ (Wu et al., 2024, Hao et al., 2024).

• Hypersingular integral form:

$$(-\Delta)^{\alpha(x)/2}u(x) = C_{d,\alpha(x)}\,\,\mathrm{P.V.}\int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha(x)}}dy$$

with $C_{d,\alpha}$ a normalization constant (Hao et al., 2024, Wu et al., 2024). Equivalence with the Fourier form holds under mild decay.

For nonlinear, variable-exponent analogues, the variable-order fractional $p(\cdot)$-Laplacian is given by:

$$(-\Delta)^{s(\cdot)}_{p(\cdot)} u(x) = \mathrm{P.V.} \int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^{p(x,y)-2}(u(x)-u(y))}{|x-y|^{N + s(x,y)p(x,y)}}\,dy$$

where $s:\mathbb{R}^N\times\mathbb{R}^N\to(0,1)$, $p:\mathbb{R}^N\times\mathbb{R}^N\to(1,\infty)$ are continuous and symmetric (Hamdani et al., 2023, Allaoui et al., 2023, Liu et al., 8 Jan 2025).

Extensions to the 1-Laplacian limit, degenerate Kirchhoff-type problems, and spatially variant forms on bounded domains arise in image processing, nonlocal mechanics, and PDE theory (Li et al., 2023, Ceretani et al., 2021, D'Elia et al., 2021).

2. Functional Setting, Sobolev Spaces, and Regularity

The analytical framework for VOFLs is built on variable-order fractional Sobolev spaces. For $$p(\cdot):\mathbb{R}^N\times\mathbb{R}^N\to(1,\infty)$$ and $s(\cdot):\mathbb{R}^N\times\mathbb{R}^N\to(0,1)$:

$$[u]_{s(\cdot),p(\cdot)}^{p(\cdot)} = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}}\,dx\,dy$$

The corresponding function space

$$W^{s(\cdot),p(\cdot)}(\Omega) = \left\{ u\in L^{p(\cdot)}(\Omega) : [u]_{s(\cdot),p(\cdot)}<\infty \right\}$$

is reflexive and separable if the exponents are bounded away from the endpoints, with compact embeddings $$W^{s(\cdot),p(\cdot)}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega)$$ for $q(x)<p^*_s(x)$ (Hamdani et al., 2023, Allaoui et al., 2023, Liu et al., 8 Jan 2025).

For linear VOFLs, the functional calculus accommodates both constant and variable exponents, but nontrivial trace and Hardy-type inequalities are needed when $s(x)$ approaches $0$ or $1$ (Ceretani et al., 2021). Weighted and quotient Sobolev spaces arise in the extension framework. Regularity, embedding, and density results for smooth functions are available for various cases, including variable-order $1$-Laplacian and extension-type definitions (Li et al., 2023, Ceretani et al., 2021).

3. Variational, Spectral, and Extension Formulations

Several variational and spectral characterizations underlie the modern theory:

• Spectral/extension method: For $s(\cdot):\Omega\to[0,1]$, the operator $(-\Delta)^{s(\cdot)}$ on a bounded domain can be realized as the Dirichlet-to-Neumann map of a weighted extension problem in $\mathcal{C} = \Omega\times(0,\infty)$ with weight $w(x,y)=G_{s}(x)y^{1-2s(x)}$ (Ceretani et al., 2021). The solution is found by minimizing the energy

$$J(u) = \frac{1}{2}\int_{\mathcal{C}} w(x,y)|\nabla u|^2 dxdy - \langle h, \operatorname{Tr} u\rangle$$

with $(-\Delta)^{s(\cdot)} v = h$ for $v = \operatorname{Tr} u$.

• Generalized Riesz potential and Fourier symbol inversion: For $s(x)\in(0,n/2)$,

$$\mathcal{F}\left[(−\Delta)^{s(\cdot)}f\right](k) = \frac{\hat f(k)}{\widehat K_{s(\cdot)}(k)}$$

with $K_{s(\cdot)}(x)$ a generalized Riesz kernel (Darve et al., 2021). This allows for explicit Green's functions $G(x,y) = -K_{s(\cdot)}(|x-y|)$ and a formal inverse property.

• Energy and weak formulation: Nonlinear problems with variable-order $p(\cdot)$-Laplacians, possibly with nonlocal Kirchhoff or Choquard terms, are tackled by variational minimization of functionals such as

$$I(u) = a\sigma_{p(\cdot)}(u) - b\frac{1}{\gamma+1}[\sigma_{p(\cdot)}(u)]^{\gamma+1} + [\text{nonlinear reactions}]$$

with weak solutions in the appropriate Sobolev class (Hamdani et al., 2023, Biswas et al., 2020, Allaoui et al., 2023).

4. Numerical Discretization and Computational Methods

VOFLs present unique computational challenges due to the hypersingular, nonlocal, and nonhomogeneous kernel:

• Finite difference methods: Advanced schemes based on the symbol of the Laplacian are implemented to produce quasi-Toeplitz discrete operators:

$$(-\Delta_h)^{\alpha(x_j)/2} u_j = \sum_{k\neq j} w_{jk} (u_j - u_k)$$

with FFT-based application for $O(N\log N)$ complexity and spectral convergence in $h$ under smoothness (Hao et al., 2024).

• Radial basis function methods: Meshfree RBF approaches approximate $u$ by

$$\hat u(x) = \sum_{i=1}^{\bar N} \lambda_i \varphi(\|x - x_i\|)$$

where analytic formulas for $(-\Delta)^{\alpha(x)/2}\varphi(\cdot)$ are derived for Gaussian, multiquadric, or Bessel-type basis, avoiding singular quadrature and achieving high accuracy with few centers (Wu et al., 2024).

• Finite element methods: For general nonlocal kernels (possibly truncated), Galerkin discretizations assemble the associated stiffness matrix via adaptive quadrature and (for piecewise-constant orders) hierarchical clustering or panel methods. $h$-convergence rates match the constant-order theory with energy error $\mathcal{O}(h^{1/2})$ and $L^2$-error $\mathcal{O}(h^{\min\{1,1/2+\min s(x,y)\}})$ (D'Elia et al., 2021).
• Specialized iterative solvers: Krylov subspace, contour integral matrix function-vector products, and two-level preconditioning enable efficient time-stepping and solution of large-scale variable-order PDEs (Farquhar et al., 2018).

5. Existence, Regularity, and Multiplicity of Solutions

A robust existence and multiplicity theory has been developed for both linear and nonlinear VOFLs:

• Linear VOFLs with variable order satisfying suitable Poincaré or Hardy inequalities in weighted Sobolev spaces yield well-posed Poisson problems, with uniqueness and regularity depending on the weight structure, trace conditions, and domain geometry (Ceretani et al., 2021, Darve et al., 2021).
• For nonlinear variants, under regularity, boundedness, and symmetry of the exponents, as well as growth and coercivity of the nonlinearities and Kirchhoff functions, mountain-pass, Symmetric Mountain Pass, Fountain, and genus theorems yield existence and multiplicity of weak solutions (Hamdani et al., 2023, Allaoui et al., 2023, Biswas et al., 2020). Degenerate cases (e.g., $K(0)=0$ in Kirchhoff) are covered by explicit polynomial bounds on the Kirchhoff coefficient.
• Comparison principles, renormalized and entropy solution frameworks, and $L^1$-data theory exist, underpinning well-posedness even for low regularity and measure data (Liu et al., 8 Jan 2025).
• Explicit formulas for Green's functions are available for certain radially symmetric or constant-coefficient cases, providing analytic control and insight into structural properties (Darve et al., 2021).

6. Applications in Science, Engineering, and Data Analysis

VOFLs are applicable in a wide range of modeling contexts:

• Heterogeneous diffusion and transport: Cardiac electrophysiology models with variable fractional order accurately reproduce propagation anomalies in healthy vs. damaged tissue, including re-entry phenomena in heart meshes (Farquhar et al., 2018).
• Phase transitions and interfacial phenomena: Allen–Cahn and phase-field models with spatially varying order capture interface pinning, asymmetric motion, and multi-modal coalescence, with direct impact from the local value of $\alpha(x)$ (Wu et al., 2024, Hao et al., 2024).
• Image processing and adaptive regularization: VOFL-driven diffusion and $1$-Laplacian flows with texture/adaptive order functions outperform classical models in edge preservation and denoising, as measured by PSNR/SSIM, especially in heterogeneous or textured images (Li et al., 2023).
• Groundwater flow, complex media, and astrophysics: VOFLs model anomalous transport with spatially variable behavior, for instance in groundwater with inhomogeneous conductivity or in galactic dynamics with modified Newtonian gravity (Darve et al., 2021).
• Mathematics of variable-exponent and doubly-nonlocal PDEs: Kirchhoff-Choquard systems and p-Laplacian evolution allow, through VOFLs, the analysis of materials with both nonlocal mechanical response and spatially varying power-law nonlinearity (Hamdani et al., 2023, Biswas et al., 2020, Allaoui et al., 2023, Liu et al., 8 Jan 2025).

7. Open Challenges and Future Directions

Despite rapid progress, several aspects remain active research fronts:

• The full functional-analytic theory for variable-order fractional Sobolev spaces, including optimal embeddings, maximal regularity, and explicit interpolation or trace theorems, is not yet complete for the most general exponents or weight structures (Ceretani et al., 2021, Darve et al., 2021).
• Extension to variable-order, nonlinear, and degenerate operators in higher dimensions requires further development in numerical quadrature, domain decomposition, and operator preconditioning.
• Multiphysics couplings (e.g., variable fractional order with adaptive reaction or memory coefficients) are under-explored, especially regarding parameter identification and inverse problems in heterogeneous domains.
• Efficient computation of VOFLs remains limited by the cost of nonlocality and the loss of translation invariance when order varies; adaptive and hierarchical solvers, as well as RBF meshfree schemes, are active directions (Wu et al., 2024, Hao et al., 2024).
• Theoretical foundations for the limit cases ($s(x)\downarrow0$ or $s(x)\uparrow1$), particularly in the presence of highly non-smooth order functions, are not fully established, especially in the nonlinear $p(\cdot)$-Laplacian setting (Ceretani et al., 2021).

Tables and further technical details for each discretization, operator definition, or functional setting can be found in the referenced works. The rapidly evolving theory and computational practice around the VOFL continue to expand its reach across mathematics, applied science, and engineering.