Regional Fractional Laplacian

• Regional fractional Laplacian is a bounded-domain nonlocal operator defined by restricting the fractional Laplacian’s singular integral to interactions within the domain.
• It is formulated within a Sobolev framework using the Gagliardo seminorm and reveals varying boundary behaviors, especially as the order parameter crosses 1/2.
• This operator exhibits distinct spectral, variational, and numerical properties compared to other fractional Laplacians, impacting applications in PDEs and stochastic models.

The regional fractional Laplacian is a bounded-domain nonlocal operator defined by restricting the singular integral of the fractional Laplacian to interior interactions. For a bounded open or Lipschitz domain $\Omega\subset\mathbb{R}^n$ and $0

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

with the normalization chosen so that on $\mathbb{R}^n$ one recovers the Fourier symbol $|\xi|^{2s}$ (Frank et al., 2024). In probabilistic terms it is the generator of the censored $2s$-stable process, and analytically it differs from the whole-space, restricted, and spectral fractional Laplacians by the way the boundary enters both the operator and the admissible data (Duo et al., 2017).

1. Definition and functional framework

The operator is naturally paired with the Gagliardo seminorm

$$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$

and with the Sobolev-type spaces

$$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$

which support the bilinear form associated with $(-\Delta)^s_\Omega$ (Chen, 2015). In variational form, the operator is characterized by

$$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$

and weak solutions are defined by duality against $0Fall et al., 2021).

Because the integral is restricted to $0Frank et al., 2024). In the linear elliptic setting, coercivity follows from a Hardy–Sobolev inequality and a Poincaré inequality on $0Chen, 2015).

A recurrent point in the literature is that the boundary formulation is regime-dependent. Some treatments regard the appropriate Dirichlet condition as the local one $0$$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$0 (Duo et al., 2017). Variational treatments instead encode Dirichlet data by requiring $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$1; for $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$2 in bounded Lipschitz domains, one has $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$3, so the Dirichlet and Neumann variational formulations coincide at the level of the energy space (Fall, 2020).

2. Position among bounded-domain fractional operators

On bounded domains, the regional fractional Laplacian is one of several inequivalent fractional Laplace operators. Their distinction is structural rather than notational.

Operator	Integral domain / prescription	Distinctive feature
Regional fractional Laplacian	Integral over $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$4 only	Generator of the censored $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$5-stable process
Restricted fractional Laplacian	Integral over $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$6 with $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$7 on $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$8	Exterior volume constraint
Spectral fractional Laplacian	Functional calculus of the Dirichlet Laplacian	Uses Laplace eigenpairs on $$(-\Delta)^s_{\Omega}u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\qquad x\in\Omega,$$9

For the spectral operator, if $\mathbb{R}^n$0 is the $\mathbb{R}^n$1th Dirichlet eigenpair of $\mathbb{R}^n$2, then $\mathbb{R}^n$3 and $\mathbb{R}^n$4 (Duo et al., 2017). By contrast, the regional operator has its own spectrum and its own boundary regularity class. In one dimension on $\mathbb{R}^n$5, numerical experiments showed

$\mathbb{R}^n$6

and all three families converge to the classical Dirichlet Laplacian eigenvalues as $\mathbb{R}^n$7 (Duo et al., 2017).

The discrepancy is especially pronounced in the strongly nonlocal regime. For $\mathbb{R}^n$8, the regional operator neglects jumps from $\mathbb{R}^n$9 to $|\xi|^{2s}$0, while the restricted fractional Laplacian still “sees” all of $|\xi|^{2s}$1; in the one-dimensional formula recorded for $|\xi|^{2s}$2, the difference between the two operators contains an explicit boundary-singular factor $|\xi|^{2s}$3 (Duo et al., 2017). This suggests that replacing the restricted operator by the regional one is benign only near the local limit $|\xi|^{2s}$4, not uniformly over the full nonlocal range.

A further variant appears when homogeneous nonlocal Neumann conditions are imposed on the whole-space fractional Laplacian. Under the condition $|\xi|^{2s}$5 on $|\xi|^{2s}$6, the whole-space operator can be rewritten on $|\xi|^{2s}$7 as a purely regional operator with a modified kernel $|\xi|^{2s}$8 that encodes the exterior jumps; the correction term has logarithmic blow-up near the boundary, so the resulting kernel is not stable-like in the sense of kernels bounded above and below on $|\xi|^{2s}$9 (Abatangelo, 2017).

3. Boundary regimes and regularity theory

The boundary behavior of the regional operator depends sharply on whether $2s$0 is above or below $2s$1. For $2s$2, the censored symmetric $2s$3-stable process is conservative and never approaches the boundary, and the corresponding PDE theory exhibits strong nonexistence phenomena (Chen et al., 2020). In particular, for bounded $2s$4 domains, the equation

$2s$5

admits no viscosity solution bounded from above or bounded from below, and the Lane–Emden problem

$2s$6

admits no nonnegative nontrivial solution for any $2s$7 (Chen et al., 2020).

For $2s$8, the boundary rate $2s$9 becomes fundamental. A Hopf boundary lemma shows that if $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$0 is a pointwise or weak super-solution of

$$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$1

then either $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$2 or

$$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$3

at every boundary point $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$4; the same paper identifies the torsion function as having two-sided bounds of order $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$5, so the exponent $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$6 is sharp (Abatangelo et al., 2021).

Boundary regularity has been developed in both Hölder and Sobolev scales. For bounded $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$7 domains and $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$8, zero-Neumann solutions satisfy $$[u]_{H^s(\Omega)}^2=\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$9, and if $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$0 then $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$1; for the Dirichlet problem with $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$2 and $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$3, one obtains

$$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$4

up to the boundary (Fall, 2020). The proofs combine half-space Liouville classifications, blow-up arguments, compactness, and kernel-freezing.

A more refined Sobolev-scale boundary theory is available in flat geometry. On the upper half-space $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$5, if $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$6, $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$7, and $$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$8 solves

$$H^s(\Omega)=\{u\in L^2(\Omega):[u]_{H^s(\Omega)}<\infty\},\qquad H_0^s(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^s},$$9

then for any $(-\Delta)^s_\Omega$0 with $(-\Delta)^s_\Omega$1 one has a global tangential Calderón–Zygmund estimate

$(-\Delta)^s_\Omega$2

and the $(-\Delta)^s_\Omega$3-dimensional boundary regularity problem can be reduced to a one-dimensional regional fractional Laplace equation on $(-\Delta)^s_\Omega$4 (Khomrutai et al., 2022).

4. Spectral, asymptotic, and variational structure

The regional fractional Laplacian has a discrete Dirichlet spectrum on bounded Lipschitz domains, with eigenvalues characterized by min–max principles in $(-\Delta)^s_\Omega$5 (Temgoua et al., 2021). As $(-\Delta)^s_\Omega$6, the operator admits an expansion

$(-\Delta)^s_\Omega$7

where

$(-\Delta)^s_\Omega$8

is the regional logarithmic Laplacian (Temgoua et al., 2021). In the same limit, for each fixed $(-\Delta)^s_\Omega$9, $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$0 and $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$1 in $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$2 and even in $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$3 (Temgoua et al., 2021).

The dependence on the order parameter is differentiable at the level of weak solutions. For the free Poisson problem with zero-average datum,

$$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$4

the map $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$5 belongs to $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$6 (Temgoua, 2021). For the first nontrivial eigenvalue $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$7 on the zero-mean space, right differentiability is established, and the derivative is expressed through a logarithmic correction to the energy form (Temgoua, 2021).

Critical variational problems reveal a strong domain dependence absent in the full-space theory. For

$$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$8

the best constant genuinely depends on $$Q_{n,s,\Omega}(u,v)=\frac{C_{n,s}}{2}\int_\Omega\int_\Omega \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$$9 (Frank et al., 2024). If $0Frank et al., 2024).

Existence of minimizers has also been obtained through concentration-compactness and geometric test-function arguments. For bounded $0Fall et al., 2021). At the shape-optimization level, a Rayleigh–Faber–Krahn-type problem for the regional energy admits a compactly supported nonnegative minimizer, but symmetrization techniques may fail, and the regularity and geometry of the positivity set remain open (Jin et al., 2021).

5. Boundary-value problems, nonlinear equations, and identities

For the linear Dirichlet problem on bounded $0

$0

existence and uniqueness are available at three levels of source data when $0Chen, 2015). The same work provides an integration by parts formula involving the nonlocal normal derivative

$0

for classical solutions with general boundary data (Chen, 2015).

Nonlinear equations display boundary phenomena specific to the regional operator. For boundary blow-up problems

$0

with $0Chen et al., 2016). In the existence regime, two-sided boundary estimates of power type are obtained, and the proof uses Perron’s method, Green-operator estimates, and barriers of the form $0Chen et al., 2016).

The low-order regime $0

$0

when the nonlinearity is either $0Chen et al., 2022). The resulting solutions are positive, radial, and strictly decreasing (Chen et al., 2022).

For nonlinear nonlocal operators modeled on the regional fractional $0

$0

with measurable symmetric kernel comparable to $0Lee et al., 25 Jun 2026). The proof uses nonlinear commutator estimates, localization, and a Gehring-type iteration (Lee et al., 25 Jun 2026).

A recent development is a Pohozaev-like identity for the regional operator on bounded $0

$0

satisfy an identity with an explicit remainder term and a boundary contribution involving $0Djitte, 29 Jul 2025). The same work establishes a new integration-by-parts formula for $0Djitte, 29 Jul 2025).

6. Stochastic, numerical, and data-driven realizations

The stochastic interpretation is central. The regional fractional Laplacian is the generator of the censored $0Duo et al., 2017). Under the nonlocal Neumann condition of Dipierro–Ros-Oton–Valdinoci, the induced regional kernel corresponds instead to a “reflected-and-redistributed” process: when a particle lands outside $0Abatangelo, 2017).

The operator also appears as a scaling limit in interacting particle systems. On $0stochastic Burgers equation with characteristic operator given by the regional fractional Laplacian $0Cardoso et al., 2024). In that framework, the energy form

$0

makes $0Cardoso et al., 2024).

Numerically, the regional operator is not a universally accurate surrogate for the restricted fractional Laplacian. Extensive experiments comparing the fractional, spectral, regional, and peridynamic operators show that all collapse to the classical Laplacian as $0Duo et al., 2017).

In data analysis, nonlocal diffusion maps with polynomial-tail kernels provide a discrete approximation of the regional fractional Laplacian on manifolds with boundary. The construction uses graph distances to approximate geodesic distances, a two-stage normalization to remove sampling bias, and a graph generator

$0

which converges in the large-data, small-bandwidth limit to $0Antil et al., 2018). Because the point cloud is confined to $0Antil et al., 2018).

Taken together, these results place the regional fractional Laplacian at the intersection of censored jump processes, boundary-sensitive nonlocal PDE, domain-dependent critical phenomena, and computational schemes that exploit the operator’s strictly in-domain character. The present body of work also isolates several persistent open problems: precise Green’s function estimates, sharper boundary regularity in low-order regimes, Weyl-type asymptotics beyond currently known cases, the geometry of shape minimizers, and the analytic status of the logarithmically corrected kernels arising from nonlocal Neumann reformulations (Duo et al., 2017).