On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians

Technical Summary: On the Fundamental Solutions of Two Nonlocal Parabolic Equations Related to Logarithmic Laplacians

Problem Statement
This paper addresses a question posed by V. Maz'ya regarding the analytic continuation of the fundamental solution to a specific nonlocal parabolic equation associated with a logarithmic Laplacian. The study focuses on two integral operators, $A_1$ and $A_2$, which arise in the context of boundary value problems for the ordinary Laplacian on domains with thin cavities.

1. Operator $A_1$: Defined on the unit circle $C \subset \mathbb{R}^2$, this operator is related to the conformal logarithmic Laplacian. Its fundamental solution, $K_1(t, \phi, \theta)$, is represented by a spectral series involving eigenvalues $2\lambda_n$ that grow asymptotically as $2\ln(n)$. The series converges only for $\Re(t) > 1/2$ on the diagonal ($\phi=\theta$). Maz'ya asked whether this series representation could be continued as a meromorphic function of $t$.
2. Operator $A_2$: Defined on the interval $[-1, 1]$, this operator is associated with a "regional" fractional Laplacian or a logarithmic Dirichlet Laplacian. Its fundamental solution, $K_2(t, x, y)$, involves Legendre polynomials and harmonic numbers $H_n$. The convergence properties of its spectral series are more complex, depending on the spatial variables $x$ and $y$.

Methodology
The authors employ a spectral analysis approach combined with asymptotic expansions and complex analysis techniques:

• Spectral Decomposition: Both operators are shown to have purely discrete spectra. The fundamental solutions are expressed as sums over eigenfunctions weighted by $e^{-t\lambda_n}$.
• Asymptotic Expansion of Eigenvalues: The core technical tool is the asymptotic expansion of the eigenvalues (specifically $2\lambda_n$ for $A_1$ and $H_n$ for $A_2$) as $n \to \infty$. The authors utilize the expansion $\mu_n \sim s_{-1}\ln(n) + s_0 + \sum s_k n^{-k}$.
• Bell Polynomials and Exponential Generating Functions: To handle the analytic continuation, the authors expand the term $e^{-t\mu_n}$ using the asymptotic series. This leads to the introduction of polynomials $p_{j,S}(t)$ defined via partial ordinary Bell polynomials. These polynomials capture the coefficients of the asymptotic expansion of the exponential term.
• Integral Representations: For the general case of $K_2$ (where $x, y \in (-1, 1)$), the authors utilize the Mellin transform and the generating function for Legendre polynomials (expressed via the hypergeometric function ${}_2F_1$). This allows the spectral sum to be converted into an integral representation, facilitating the analysis of singularities near $s=0$ (where $s$ is the integration variable in the Mellin transform).
• Singularity Analysis: By analyzing the behavior of the integrand near $s=0$, particularly the interaction between logarithmic terms arising from the hypergeometric function and the Gamma function prefactor, the authors determine the location and nature of the poles of the meromorphic continuation.

Key Contributions and Results

1. Meromorphic Continuation of $K_1$:

   • Theorem 1.2: The fundamental solution $K_1(t, \phi, \theta)$ extends to an entire function of $t$ for all $\phi \neq \theta$.
   • For the diagonal case $\phi = \theta$, $K_1(t, \phi, \phi)$ extends to a meromorphic function with simple poles at $t = 1/2 - m$ for $m \in \mathbb{N}_0$.
   • The residues at these poles are explicitly calculated in terms of the polynomials $p_{j,S_1}(t)$ and the Euler-Mascheroni constant $\gamma_E$.
   • The authors conjecture that these are the only poles, contingent on the non-vanishing of specific polynomials evaluated at these points.

2. Meromorphic Continuation of $K_2$:

   • Theorem 1.4: For $x \neq y$, $K_2(t, x, y)$ extends to an entire function of $t$.
   • For the diagonal case $x = y$, $K_2(t, x, x)$ extends to a meromorphic function.
     • If $x = \pm 1$, poles occur at $t = 2$ and $t = -2m$ ($m \in \mathbb{N}$).
     • If $x \in (-1, 1)$, poles occur at $t = 1$ and potentially at $t = -m$ ($m \in \mathbb{N}$).
   • Theorem 1.5: Explicit formulas for the residues are provided. These involve the polynomials $p_{j,S_2}(t)$, Bernoulli numbers, and coefficients derived from the expansion of the hypergeometric function and the square root term in the integral representation.

3. New Mathematical Identities:

   • The analysis necessitates and proves several identities involving Bell polynomials, Bernoulli numbers, and the digamma function.
   • Specifically, the authors establish that certain polynomials $p_{j,S_2(s)}(1-j)$ are independent of $s$ and vanish for odd $j$.
   • They derive a sequence of identities involving linear combinations of products of Bernoulli polynomials (e.g., $B_1(s)^2/2 - B_2(s)/2 = 1/24$).

4. Open Problems:

   • The paper formulates specific conjectures regarding the non-vanishing of the residue polynomials at the suspected pole locations. Proving these would confirm the exact pole structure (i.e., that no "accidental" cancellations occur to remove the poles).
   • The authors note that while meromorphic continuation is expected, it is not guaranteed for all spectral series (citing examples where logarithmic branch points occur), making their positive results significant.

Significance
The paper provides a rigorous affirmative answer to Maz'ya's question, demonstrating that the fundamental solutions of these nonlocal parabolic equations, despite their initial representation as series with limited convergence, possess rich analytic structures (meromorphic or entire).

The results reveal that these solutions are "reminiscent" of shifted Riemann zeta functions (on the diagonal) and polylogarithms or Dirichlet eta functions (off-diagonal). The work bridges the gap between the spectral theory of logarithmic Laplacians and classical special functions. Furthermore, the derivation of new identities involving Bell and Bernoulli numbers contributes to combinatorial number theory, independent of the primary PDE application. The authors modestly suggest that their methods may be adaptable to higher-dimensional analogues of these operators, though this is left as a direction for future work.