Variable Fractional Laplacian

• Fractional Order Laplacian is a nonlocal operator where the differentiation order varies spatially to model heterogeneous interactions.
• The analysis utilizes weighted Sobolev spaces and specialized trace operators to address challenges posed by non-Muckenhoupt weights.
• Applications span image processing, geophysical inversion, and inhomogeneous diffusion, highlighting its practical and theoretical importance.

The spatially variant fractional Laplacian is a generalization of the classical fractional Laplacian operator in which the order of differentiation becomes a function of space, denoted $s(x): \Omega \to [0,1]$ on a domain $\Omega \subset \mathbb{R}^n$. This formulation enables modeling and analysis of nonlocal phenomena with heterogeneity in the local intensity or range of nonlocal interactions, as required in applications such as image processing, regularization, geophysical inversion, and inhomogeneous diffusion. Its rigorous mathematical characterization requires significant departures from the constant-order case, especially in the identification of appropriate function spaces and analytical tools for establishing well-posedness.

1. Definition and Structural Properties

The spatially variant fractional Laplacian $(-\Delta)^{s(\cdot)}$ is defined using an extension problem analogous to the Caffarelli–Silvestre construction for constant order, but with variable $s(x)$. One considers the cylinder $\mathcal{C} = \Omega \times (0,\infty)$ and assigns to each $(x,y)$ the weight

$w(x,y) = G_s(x) y^{1-2s(x)},$

where $G_s(x)$ is a normalizing function chosen to ensure consistency with the constant-order constants when $s(x)$ is constant. Defining two weighted Sobolev spaces on $\mathcal{C}$—one with vanishing lateral boundary data and the other with vanishing data on the entire boundary—one introduces the abstract quotient (“trace”) space

$$\mathcal{X}(\Omega, w) = \mathcal{L}_0^{1,2}(\mathcal{C}, w) / \mathcal{L}_0^{1,2}(\mathcal{C}, w).$$

For $v \in \mathcal{X}(\Omega, w)$, let $S(v)$ denote the minimizer of the energy

$$J(u) = \tfrac{1}{2}\int_{\mathcal{C}} w(x,y)|\nabla u(x,y)|^2\,dX$$

among admissible $u$ with prescribed trace $v$ on $\Omega\times\{0\}$. The operator is then characterized as

$$\langle (-\Delta)^{s(\cdot)} v, \mathrm{tr}\,\psi\rangle = \int_{\mathcal{C}} w(x,y) \nabla S(v)\cdot\nabla \psi\,dX,\quad \forall \psi.$$

Thus, $(-\Delta)^{s(\cdot)}$ acts as a Dirichlet-to-Neumann map and as a Lagrange multiplier enforcing the trace constraint in the variational framework (Ceretani et al., 2021).

2. Analytical Tools and Function Space Framework

Allowing $s(x)$ to vary and touch the endpoints $0$ and $1$ pushes the framework beyond classical settings, particularly as the weights $w(x,y)=G_s(x)y^{1-2s(x)}$ can fall outside the Muckenhoupt $A_2$ class, invalidating classical weighted Sobolev space results (such as the “$H = W$” theorem and standard Poincaré inequalities).

Key analytical advances include:

• Formulating the Poisson problem as a penalized minimization in the weighted space, using functionals $J_{\mu, v}(u, v)$ that relax the strict trace constraint and recover it in the $\mu\to\infty$ limit.
• Introducing quotient spaces and trace spaces adapted to variable $s(x)$ and the associated degeneracies in the weight.
• Employing weighted Hardy inequalities to establish boundedness and surjectivity of the trace operator $$\mathrm{tr}: \mathcal{H}_0^{1,p}(\mathcal{C}, w) \rightarrow L^p(\Omega,\tilde w)$$, with $\tilde w(x) = G_s(x)(p-2+2s(x))^p$.
• Developing alternative approaches where, under a Poincaré-type inequality, the domain of $(-\Delta)^{s(\cdot)}$ is embedded in variable exponent weighted Sobolev spaces, mirroring the local smoothness dictated by $s(x)$.

This analysis underscores that, in the variable-order case, almost every classical regularity and compactness result must be custom-tailored to the spatially inhomogeneous context.

3. Well-posedness of the Variable-order Poisson Problem

The framework enables the paper of Poisson problems of the form

$$\begin{cases} (-\Delta)^{s(\cdot)}v = h & \text{in }\Omega,\ v = 0 &\text{on }\partial\Omega, \end{cases}$$

with source term $h$ lying in the dual $\mathcal{X}(\Omega, w)'$. Existence and uniqueness of solutions are established via direct minimization of the energy

$$\mathcal{J}(u) = \frac{1}{2}\|u\|_{\mathcal{L}_0^{1,2}(\mathcal{C},w)}^2 - \langle h, \mathrm{tr}\ u\rangle,$$

with the trace of the minimizer $u$ providing the weak solution. When additional Poincaré-type inequalities hold (e.g., when $s(x)$ is a step function with each subdomain’s closure touching $\partial\Omega$), the domain can be characterized more concretely as a (subset of a) weighted Sobolev space with local exponent $s(x)$ (Ceretani et al., 2021).

4. Weighted Sobolev Spaces and Trace Theory

Standard Sobolev spaces are replaced by weighted analogues:

• $L^{1,2}(\mathcal{C},w)$: functions with weighted square-integrable gradients.
• Subspaces vanishing on various portions of the boundary to encode Dirichlet or lateral boundary conditions.
• Quotient “trace” spaces $\mathcal{X}(\Omega, w)$ encapsulating the equivalence relation due to vanishing at $\partial\mathcal{C}$.

Supposing a (variable-order) Poincaré inequality is valid, the trace operator maps into a weighted $L^p$ space or a variable-order Sobolev space $\mathbb{W}^{s(\cdot),p}$:

$$[u]_{\mathbb{W}^{s(\cdot),p}(\Omega)} = \int_{\Omega^2} \frac{|u(x)-u(y)|^p}{|x-y|^{n+p s(x)}}\,dx\,dy,$$

exhibiting local regularity properties that match the spatial profile $s(x)$. Refined Hardy inequalities provide further regularity, especially for trace embeddings and identification of the operator’s domain (Ceretani et al., 2021).

5. Trace Regularity, Extensions, and Examples

The paper proves the existence and boundedness of the trace operator, mapping the appropriate weighted extension space to $L^p(\Omega, \tilde w)$ for an explicit $\tilde w$ that encapsulates both the local scaling and the variation in $s(x)$. In special geometries such as the hypercube $Q_N$, the trace can be identified with a variable-order fractional Sobolev space, with proofs combining Hardy-type inequalities and explicit cut-off constructions.

Constructive examples include:

• Step-function $s(x)=\sum_{i=1}^M s_i \mathbf{1}_{\Omega_i}(x)$, where each $\Omega_i$ meets $\partial\Omega$ in positive measure, enabling a global Poincaré inequality and explicit identification of the domain.
• Truncated extension domains $\mathcal{C}_R$ with finite cylinder height, yielding localized versions of the trace results.

These constructions are crucial for designing domain decompositions and adaptive methods where regularization or diffusion order is tailored by region.

6. Applications and Modeling Implications

Spatially variable order fractional Laplacians address several application scenarios:

• Image processing: Models with $s(x)$ adapting according to local texture, edge, or feature information outperform classical total variation or fixed-order fractional regularizers in denoising and inverse tasks.
• Heterogeneous/nonhomogeneous diffusion: Allowing variation in $s(x)$ equips models to distinguish between regions of anomalous (sub/superdiffusive) propagation and classical (Brownian) diffusion within a unified PDE framework.
• Geophysical and inverse problems: Regularization by spatially adapted fractional Laplacians is advantageous for recovering spatially heterogeneous conductivities or material properties, corresponding to heterogeneous nonlocality in propagation mechanisms.

Extensions to numerical analysis and optimal control for variable-order fractional PDEs follow naturally from this analytical foundation.

7. Broader Impact and Perspectives

The analytical framework for variable-order fractional Laplacians establishes:

• Well-posedness theory for linear Poisson-type problems in nonstandard function spaces, including cases with non-Muckenhoupt weights and minimal assumptions on $s(\cdot)$.
• Trace theorems and embedding results for variable regularity spaces, crucial for both the analysis of PDEs and the development of finite element, spectral, and meshless solvers tailored to nonlocal, spatially heterogeneous differential operators.
• Guidelines for modeling and numerics in applications where the local degree of nonlocality must be tuned spatially, which is increasingly relevant in data-driven PDE modeling and nonlocal continuum theories (Ceretani et al., 2021).

In summary, the spatially variant fractional Laplacian extends the notion of nonlocal, fractional elliptic operators to heterogeneous settings, accommodating spatial adaptation of regularization or diffusion order in both theory and applications. Its rigorous treatment necessitates a combination of weighted extension techniques, novel function spaces, and refined trace and variational analysis to account for the inhomogeneity in the underlying differential order.