Restricted Fractional Laplacian Overview

• The restricted fractional Laplacian is a nonlocal operator defined on functions that vanish outside a domain, characterized via quadratic forms and singular integrals.
• It employs variational and microlocal techniques to establish spectral properties, including discrete eigenvalue distributions and Weyl-type asymptotics.
• Applications include modeling stable processes and nonlocal PDEs, with analytic advancements via the Caffarelli–Silvestre extension and boundary regularity theory.

The restricted fractional Laplacian, also known as the regional or Dirichlet fractional Laplacian, is a fundamental nonlocal operator that acts on functions supported in a domain while imposing a "hard" Dirichlet-type exterior condition. Its rich analytic, spectral, and geometric structure provides a nonlocal generalization of the classical Dirichlet Laplacian and is central in the theory of stable processes, nonlocal PDEs, and mathematical physics.

1. Operator Definition and Analytical Framework

Given a bounded or unbounded open set $\Omega \subset \mathbb{R}^n$ and order $s \in (0,1)$, the restricted fractional Laplacian $(-\Delta)^s|_\Omega$ is most canonically defined via quadratic forms or singular integrals. The operator acts on functions $u$ vanishing outside $\Omega$ (i.e., $u(x) = 0$ for almost every $x \notin \Omega$), with its natural form domain the subspace

$$\widetilde{H}^s(\Omega) := \{ u \in H^s(\mathbb{R}^n) : \operatorname{supp}\ u \subset \overline{\Omega} \}$$

of the global fractional Sobolev (Slobodeckii) space.

The quadratic form is given by

$$a_s^\Omega[u] = \int_{\mathbb{R}^n} |\xi|^{2s} |\mathcal{F}u(\xi)|^2\, d\xi$$

where $\mathcal{F}$ is the $n$-dimensional Fourier transform. Equivalently, a nonlocal integral representation holds:

$$(-\Delta)^s u(x) = C_{n,s}\, \mathrm{P.V.} \int_\Omega \frac{u(x) - u(y)}{|x-y|^{n + 2s}}\, dy$$

with a normalization so that, in the whole space, the operator symbol is $|\xi|^{2s}$ (D'Elia et al., 2013).

A variational characterization as a self-adjoint positive operator is obtained through the spectral theorem, using the above quadratic form on $\widetilde{H}^s(\Omega)$ and realizing $(-\Delta)^s|_\Omega$ as the unique operator satisfying $( (-\Delta)^s u, v ) = a_s^\Omega[u,v]$ (Bakharev et al., 2022, Bakharev et al., 27 May 2024).

2. Key Functional Analytic and Boundary Properties

The boundary behavior of the restricted fractional Laplacian is governed by the $\mu$-transmission property: functions in its domain exhibit a leading singularity of order $\delta(x)^s$ near $\partial \Omega$, where $\delta(x)$ is the distance to the boundary (Grubb, 2013). For $u$ near $\partial\Omega$, expansions of the form

$$u(x) = \delta(x)^s\, v(x), \quad v \in C^\infty(\overline\Omega)$$

hold, with full regularity in tangential directions up to the boundary.

For intervals or more general bounded domains, the maximal domain of the (restricted) operator can be characterized in transmission pseudodifferential terms, often involving intricate boundary traces, as in the Hörmander-Grubb theory (Behrndt et al., 30 Apr 2025, Grubb, 2013). On $(0,L)$ and $s \in (\frac12,1)$, every self-adjoint realization (with suitable boundary conditions relating weighted Dirichlet and Neumann traces) is unbounded, has compact resolvent, and a purely discrete spectrum (Behrndt et al., 30 Apr 2025).

3. Extension Principle and Localized Equivalents

A major analytic advantage is provided by the Caffarelli–Silvestre extension, which realizes the restricted fractional Laplacian as a Dirichlet-to-Neumann map for a degenerate elliptic equation in one higher dimension:

$$\left\{ \begin{array}{rl} \operatorname{div}_{x,y}\big( y^{1-2s}\nabla U \big) = 0 & \text{in}\ \Omega \times (0,\infty) \ U(x,0) = u(x),\quad U|_{\mathbb{R}^n \setminus \Omega \times (0,\infty)} = 0 \end{array} \right.$$

and the quadratic form equivalence

$$a_s^\Omega[u] = C_s \int_0^\infty \int_{\mathbb{R}^n} y^{1-2s} |\nabla U(x,y)|^2\, dx\,dy$$

where $C_s = 4^s \Gamma(s+1)/(2s\Gamma(1-s))$ (Bakharev et al., 2022, Bakharev et al., 27 May 2024, Bakharev et al., 24 Nov 2025). This representation enables precise energy estimates, spectral variational constructions, and boundary regularity theory.

4. Spectral Structure and Variational Principles

4.1. Spectral Problem and Weyl-Law Asymptotics

On bounded or unbounded domains with sufficient regularity, the spectrum of $(-\Delta)^s|_\Omega$ subject to Dirichlet boundary conditions is discrete when $\Omega$ is bounded, and contains essential spectrum associated with "waveguide"-type products or tubes (Bakharev et al., 2022, Bakharev et al., 27 May 2024, Bakharev et al., 24 Nov 2025).

For a bounded domain $\Omega \subset \mathbb{R}^d$, the spectrum is $\{ \lambda^{(s)}_j \}_{j \geq 1}$ with $\lambda^{(s)}_j \to +\infty$. Weyl-type asymptotics hold for eigenvalues:

$$\lambda^{(s)}_k \sim C_{s,d}\, |\Omega|^{-2s/d} k^{2s/d}, \quad k \to \infty$$

with refined multi-term Berezin–Li–Yau type lower bounds established, i.e.,

$$\sum_{j=1}^k \lambda_j^{(s)} \geq A_{d,s}|\Omega|^{-2s/d}k^{1+2s/d} + \cdots$$

where the subleading terms capture corrections due to the domain's moment of inertia and further geometric information (Yolcu et al., 2015).

4.2. Essential Spectrum for Infinite and Composite Domains

On waveguides, multi-tubes, and V- or bent-geometry domains, the essential spectrum is always a half-line $[\Lambda_s, \infty)$, where $\Lambda_s$ is the first eigenvalue of the cross-sectional restricted fractional Laplacian (Bakharev et al., 2022, Bakharev et al., 27 May 2024, Bakharev et al., 24 Nov 2025). Discrete spectrum ("trapped modes") below this threshold can be generated by local shape perturbations, curvature, or junction effects. The number and monotonicity of these bound states can reflect subtle nonlocal phenomena not present in the local case.

5. Domain Geometry and New Spectral Phenomena

The nonlocality of $(-\Delta)^s|_\Omega$ yields several non-classical spectral effects:

• Orientation-sensitivity: In multi-tube domains, the essential spectrum is determined only under a "non co-directionality" condition on the axes, a feature absent for the local Laplacian (Bakharev et al., 2022).
• Curvature-induced localization: In smoothly bent strips, bound states emerge below the essential spectrum threshold precisely when the $L^2$-mass of the curvature exceeds an explicit bound relative to its derivative (Bakharev et al., 24 Nov 2025).
• Junction monotonicity: In V-shaped waveguides, discrete eigenvalues below threshold exist for all opening angles, and each eigenvalue is strictly monotonic with respect to the angle, with detailed formulas based on domain comparison principles (Bakharev et al., 27 May 2024).
• Boundary traces and extension theory: On finite intervals, all self-adjoint realizations (Friedrichs/Dirichlet, Krein–von Neumann, and Neumann-type) can be classified via boundary triplet and Weyl function analysis, with nontrivial negative spectrum for the Neumann realization (Behrndt et al., 30 Apr 2025).

6. Approximation by Truncated/Peridynamic Operators

The restricted fractional Laplacian arises as a scaling limit of truncated or volume-constrained nonlocal diffusion operators:

$$(-\Delta)^s_\delta u(x) = C_{n,s} \;\mathrm{P.V.} \int_{B_\delta(x) \cap \Omega} \frac{u(x) - u(y)}{|x-y|^{n+2s}}\ dy$$

as the interaction horizon $\delta \to \infty$ (Bellido et al., 2020, D'Elia et al., 2013). For each fixed $\delta$ this defines a nonlocal Dirichlet operator with interaction radius $\delta$. As $\delta \to 0^+$, rescaled versions recover the local Dirichlet Laplacian; as $\delta \to \infty$, the spectrum and solutions converge to those of $(-\Delta)^s|_\Omega$. The associated variational and Galerkin-FE frameworks provide both rigorous convergence theorems and useful numerical schemes (D'Elia et al., 2013, Bellido et al., 2020).

7. Regularity, Transmission, and Microlocal Theory

Analytically, the restricted fractional Laplacian is a prototypical $\mu$-transmission pseudodifferential operator (Grubb, 2013). This theory, due to Hörmander and fully developed by Grubb, gives:

• Fredholm solvability on full Sobolev scales: $$(-\Delta)^\mu_\Omega: H^{s+2\mu}_p(\Omega) \to H^s_p(\Omega)$$ is Fredholm for $s > \mu - 1/p'$.
• Precise boundary regularity of solutions, including asymptotic expansions in $\delta(x)^s$.
• Explicit realization of the operator as $r_+ P e^+$ where $P$ is the classical $\mathbb{R}^n$-pseudodifferential operator $|\xi|^{2\mu}$.

This microlocal analysis encompasses all $s > 0$ and all suitable $L_p$ or Hölder targets, capturing both boundary singularities and smoothing inside the domain (Grubb, 2013).

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Table: Main Operator Realizations on Bounded Domains

Realization Type	Boundary Condition	Spectrum Characteristics
Dirichlet/Friedrichs	$u = 0$ at $\partial\Omega$	Discrete, positive, lower bounded
Krein–von Neumann	$\Gamma_1 f = M(0)\Gamma_0 f$	Discrete, semibounded, min$\geq 0$
Neumann-type	$(t^{1-s}f)' = 0$ at $\{0,L\}$	Discrete, lowest eigenvalue $<0$ (simple)

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The restricted fractional Laplacian is thus a central object in the modern theory of nonlocal elliptic operators. Its analysis incorporates nonlocal variational methods, pseudodifferential and microlocal techniques, spectral theory including Weyl law and refining inequalities, and geometric effect studies in domains with complex topologies or boundaries (Bakharev et al., 2022, Bakharev et al., 27 May 2024, Behrndt et al., 30 Apr 2025, Bakharev et al., 24 Nov 2025, Yolcu et al., 2015, D'Elia et al., 2013, Grubb, 2013, Bellido et al., 2020).