Logarithmic Laplacian: Spectrum & Analysis

• The logarithmic Laplacian is a nonlocal operator emerging as the first-order expansion of the fractional Laplacian at s=0, featuring a hypersingular kernel and far-field tail contributions.
• It adopts a distinct Dirichlet framework on bounded domains that yields a critical energy space with compact L² embedding, leading to a discrete spectrum and cases of negative first eigenvalues.
• Recent studies extend its applications to nonlinear resonance, Fučík spectrum analysis, maximum principles, extension problems, and numerical approximations, underscoring its innovative role in spectral theory.

Searching arXiv for recent and foundational papers on the logarithmic Laplacian. Search query: "logarithmic Laplacian Chen Weth extension problem Fučík spectrum graphs" The logarithmic Laplacian, usually denoted $L_\Delta$ in the Euclidean setting and $\log(-\Delta)$ in spectral formulations, is a borderline nonlocal operator obtained as the first-order expansion of the fractional Laplacian at $s=0$. In $\mathbb R^N$, it is the pseudodifferential operator with Fourier symbol $2\ln|\xi|$, equivalently $\ln(|\xi|^2)$, and it admits a real-space representation with a near-field hypersingular kernel of order $|x-y|^{-N}$, a far-field tail term, and a local renormalization term. This combination makes it qualitatively distinct from both the classical Laplacian and the fractional Laplacian of positive order: it is a zero-order nonlocal operator, it is not scale invariant in the usual sense, and on bounded domains its first Dirichlet eigenvalue may be negative (Chen et al., 2017, Arora et al., 7 Jan 2026).

1. Definition and operator structure

For $s\in(0,1)$, the fractional Laplacian satisfies

$$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$

in $L^p(\mathbb R^N)$, $\log(-\Delta)$0, for the regularity classes considered in the Euclidean theory. Accordingly, $\log(-\Delta)$1 is the pseudodifferential operator with Fourier multiplier $\log(-\Delta)$2, or equivalently $\log(-\Delta)$3 (Chen et al., 2017, Arora et al., 7 Jan 2026).

A standard Euclidean integral representation is

$\log(-\Delta)$4

with

$\log(-\Delta)$5

where $\log(-\Delta)$6 is the digamma function and $\log(-\Delta)$7 is the Euler–Mascheroni constant. The kernel $\log(-\Delta)$8 is the borderline case of hypersingular kernels, and the tail integral together with the local term $\log(-\Delta)$9 is intrinsic to the operator rather than an auxiliary correction (Arora et al., 7 Jan 2026).

This Euclidean operator is the whole-space counterpart of several related realizations. On bounded domains one usually imposes the nonlocal Dirichlet condition $s=0$0 in $s=0$1, whereas in spectral formulations one writes $s=0$2 through functional calculus. A precise implication is that the same logarithmic symbol organizes Euclidean, bounded-domain, graph, and manifold theories, but the concrete realization depends on the ambient geometry and on whether one works with restricted exterior conditions or with spectral calculus (Chen et al., 2017, Chen, 24 Jun 2025).

2. Dirichlet framework on bounded domains

For a bounded domain $s=0$3 with $s=0$4 boundary, the nonlocal Dirichlet problem is posed with

$s=0$5

Using

$s=0$6

the natural energy space is

$s=0$7

Its basic quadratic form is

$s=0$8

and the logarithmic quadratic form is

$s=0$9

When $\mathbb R^N$0 vanishes outside $\mathbb R^N$1, one also has

$\mathbb R^N$2

with

$\mathbb R^N$3

This formula makes explicit the joint role of the interior interaction, the tail, and the geometry-dependent potential $\mathbb R^N$4 (Arora et al., 7 Jan 2026).

A basic structural fact is that $\mathbb R^N$5 compactly. At the same time, the logarithmic Laplacian is a genuinely borderline operator: the natural energy space does not embed continuously into $\mathbb R^N$6 for $\mathbb R^N$7. Instead, the sharp replacement is Orlicz-type control,

$\mathbb R^N$8

with compact embedding into smaller Orlicz spaces. These embeddings are central in isolation arguments for the Fučík lines and in nonlinear variational analysis (Arora et al., 7 Jan 2026).

3. Spectral theory and eigenvalue asymptotics

The Dirichlet eigenvalue problem is

$\mathbb R^N$9

Its spectrum is discrete: $2\ln|\xi|$0 The first eigenvalue admits the Rayleigh characterization

$2\ln|\xi|$1

and the first eigenfunction can be chosen strictly positive in $2\ln|\xi|$2; moreover, $2\ln|\xi|$3 is simple. A distinguishing feature is that, unlike the Laplacian or fractional Laplacian, $2\ln|\xi|$4 may be negative (Arora et al., 7 Jan 2026).

The second eigenvalue admits a mountain-pass characterization on the unit $2\ln|\xi|$5-sphere

$2\ln|\xi|$6

namely

$2\ln|\xi|$7

where $2\ln|\xi|$8 is the family of continuous paths on $2\ln|\xi|$9 connecting $\ln(|\xi|^2)$0 to $\ln(|\xi|^2)$1. This characterization is obtained through the construction of the first nontrivial Fučík curve and implies, in particular, that all eigenfunctions associated with $\ln(|\xi|^2)$2 are sign-changing (Arora et al., 7 Jan 2026).

Beyond the principal spectral structure, sharp asymptotic information is available for the spectral realization $\ln(|\xi|^2)$3 on open sets of finite measure. One has a Weyl law

$\ln(|\xi|^2)$4

and equivalently

$\ln(|\xi|^2)$5

together with Berezin–Li–Yau-type upper bounds on Riesz means and Faber–Krahn minimization of the first eigenvalue by balls at fixed volume (Laptev et al., 2020). Complementary bounds for the Dirichlet logarithmic Laplacian include Li–Yau-type lower bounds and Kröger-type upper bounds for sums of the first $\ln(|\xi|^2)$6 eigenvalues, as well as the asymptotic relation

$\ln(|\xi|^2)$7

in the normalization used there (Chen et al., 2020).

4. Fučík spectrum and nonlinear resonance

For the logarithmic Laplacian, the Fučík spectrum is

$\ln(|\xi|^2)$8

with $\ln(|\xi|^2)$9. The recent analysis of this spectrum identifies the two “trivial” lines

$|x-y|^{-N}$0

as subsets of $|x-y|^{-N}$1, and proves that they are isolated in the spectrum (Arora et al., 7 Jan 2026).

The central variational object is the constrained functional

$|x-y|^{-N}$2

on the unit $|x-y|^{-N}$3-sphere $|x-y|^{-N}$4. If

$|x-y|^{-N}$5

then for every $|x-y|^{-N}$6,

$|x-y|^{-N}$7

and these points form the first nontrivial Fučík curve

$|x-y|^{-N}$8

This curve intersects the diagonal at

$|x-y|^{-N}$9

so the second eigenvalue is recovered as the diagonal crossing of the first nontrivial Fučík branch (Arora et al., 7 Jan 2026).

The function $s\in(0,1)$0 is Lipschitz and non-increasing, with

$s\in(0,1)$1

and the parametrized branch $s\in(0,1)$2 is strictly decreasing in the sense established there. Moreover,

$s\in(0,1)$3

The same framework yields a nonresonance theorem: under asymptotic slope conditions lying between $s\in(0,1)$4 and a point $s\in(0,1)$5, the nonlinear problem

$s\in(0,1)$6

admits at least one nontrivial solution by the Mountain Pass Theorem, using the Palais–Smale condition and the geometry induced by the Fučík curve (Arora et al., 7 Jan 2026).

5. Maximum principles, symmetry, and critical semilinear equations

For bounded Lipschitz domains, the maximum principle for the Dirichlet logarithmic Laplacian is governed by the sign of the first eigenvalue: $s\in(0,1)$7 satisfies a maximum principle in $s\in(0,1)$8 if and only if $s\in(0,1)$9. This is qualitatively different from the Laplacian and from positive-order fractional Laplacians, because $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$0 may be nonpositive on sufficiently large sets, while positivity is recovered for domains of sufficiently small measure (Chen et al., 2017).

Antisymmetric maximum principles and Hopf-type lemmas have been developed for $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$1. In symmetric sets, these tools support a moving-planes argument showing that if $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$2 is bounded, convex in the direction $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$3, and symmetric with respect to $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$4, then every strictly positive weak solution of

$$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$5

is symmetric with respect to $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$6 and decreasing in the $$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$7-direction; in balls, positive solutions are radial and radially decreasing. The same circle of ideas yields rigidity for a parallel surface problem, forcing the underlying sets to be concentric balls (Pollastro et al., 2024).

On the whole space, a critical semilinear equation involving the logarithmic Laplacian has a complete classification in the normalization stated there. When

$$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$8

the only positive solutions occur at

$$(-\Delta)^s u(x)=u(x)+sL_\Delta u(x)+o(s)\qquad \text{as }s\to0^+$$9

and they are exactly

$L^p(\mathbb R^N)$0

with

$L^p(\mathbb R^N)$1

For $L^p(\mathbb R^N)$2, no positive solution exists (Chen et al., 2024).

A parallel bounded-domain nonlinear theory treats subcritical, critical, and supercritical logarithmic nonlinearities through sharp logarithmic Sobolev inequalities, a Pohozaev identity, and a Díaz–Saa type inequality. In particular, star-shaped domains support nonexistence results at and above the critical logarithmic threshold, while logistic-type problems with $L^p(\mathbb R^N)$3 admit a unique nontrivial nonnegative weak solution together with positivity and boundary estimates (Arora et al., 2024).

6. Extensions, fundamental solutions, and nonlinear analogues

A local extension problem for the Euclidean logarithmic Laplacian is now available. For $L^p(\mathbb R^N)$4, the unique extension $L^p(\mathbb R^N)$5 solves

$L^p(\mathbb R^N)$6

and is represented by the Poisson formula

$L^p(\mathbb R^N)$7

The boundary operator is

$L^p(\mathbb R^N)$8

After doubling the extension variable, the extended function becomes harmonic in $L^p(\mathbb R^N)$9, and this leads to a weak unique continuation principle for $\log(-\Delta)$00 (Chen et al., 2023).

On complete Riemannian manifolds, the logarithmic Laplacian admits a Bochner integral formula

$\log(-\Delta)$01

on the logarithmic Sobolev space $\log(-\Delta)$02. Under a Ricci lower bound, one obtains the pointwise formula

$\log(-\Delta)$03

with

$\log(-\Delta)$04

The discrepancy between spectral and heat-kernel definitions is governed by the mass-loss function and therefore by stochastic completeness (Chen, 24 Jun 2025).

On weighted graphs, the same semigroup formula becomes

$\log(-\Delta)$05

in $\log(-\Delta)$06, and under stochastic completeness one obtains an explicit pointwise form with a short-range symmetric difference term $\log(-\Delta)$07, a long-range linear term $\log(-\Delta)$08, and the constant $\log(-\Delta)$09. For weighted lattice graphs, $\log(-\Delta)$10 has super-fast off-diagonal decay, whereas $\log(-\Delta)$11 decays like $\log(-\Delta)$12; accordingly, $\log(-\Delta)$13 fails to be bounded on $\log(-\Delta)$14 (Chen et al., 8 Jul 2025).

The logarithmic Laplacian also supports further structural extensions. An alternative construction of fundamental solutions, based on a division problem, proves that in dimensions $\log(-\Delta)$15 there exists a fundamental solution and identifies the entire $\log(-\Delta)$16-harmonic class in the Lizorkin setting as those distributions whose Fourier transforms are single-layer distributions supported on $\log(-\Delta)$17; the same approach gives low-dimensional existence results away from the origin (Lee, 25 Jun 2025). Higher logarithmic operators $\log(-\Delta)$18 with symbol $\log(-\Delta)$19 arise as higher $\log(-\Delta)$20-derivatives of $\log(-\Delta)$21 and generate Taylor expansions in the order parameter (Chen, 2023). In the nonlinear direction, the logarithmic $\log(-\Delta)$22-Laplacian $\log(-\Delta)$23 is the $\log(-\Delta)$24 derivative of the fractional $\log(-\Delta)$25-Laplacian, with a variational Dirichlet theory, a Faber–Krahn inequality, a boundary Hardy-type inequality, and a maximum principle whose validity depends on the sign of the first Dirichlet eigenvalue (Dyda et al., 2024). Subsequent work constructs an unbounded minimax sequence of eigenvalues for $\log(-\Delta)$26 and develops sharp $\log(-\Delta)$27-logarithmic Sobolev inequalities and critical Orlicz embeddings for the associated energy space (Arora et al., 26 Dec 2025, Arora et al., 30 Oct 2025).

7. Numerical approximation

Two rather different numerical approaches have been developed for Dirichlet problems involving the logarithmic Laplacian. In one dimension, finite element analysis for the restricted Dirichlet problem uses recently established log-Hölder regularity and $\log(-\Delta)$28-weighted spaces to prove a rigorous error estimate

$\log(-\Delta)$29

where $\log(-\Delta)$30. A distinctive feature is that the logarithmic stiffness matrix can be obtained as the derivative at $\log(-\Delta)$31 of the fractional stiffness matrix, and the discrete eigenvalue problem converges to the continuous one (Hernández-Santamaría et al., 2023).

A Fourier-side sinc-basis method treats $\log(-\Delta)$32 directly as the multiplier with symbol $\log(-\Delta)$33. In this framework, the discrete operator is assembled from the Fourier stencil, with singular-frequency treatment near $\log(-\Delta)$34 by a Duffy transform. On disks $\log(-\Delta)$35, the computed Dirichlet eigenvalues follow the scaling law

$\log(-\Delta)$36

and the numerical study reports the radii $\log(-\Delta)$37 at which the $\log(-\Delta)$38-th eigenvalue vanishes, as well as multiplicities consistent with rotational symmetry. For the Dirichlet problem with $\log(-\Delta)$39, the observed $\log(-\Delta)$40 error decays approximately linearly in $\log(-\Delta)$41 in the presented tests, although a rigorous convergence proof for the logarithmic Laplacian is explicitly stated to remain open (Dondl et al., 15 Sep 2025).

Across these developments, the logarithmic Laplacian appears not as a degenerate limit of more familiar operators, but as a structurally independent zero-order nonlocal object. Its tail contributions, Orlicz-scale embeddings, possible negativity of the principal eigenvalue, and sensitivity to stochastic completeness or exterior geometry are recurrent themes in the current theory.