Nonlocal eigenvalue problems and superposition operators

1. Problem Statement and Motivation

The paper investigates the spectral theory of mixed local and nonlocal operators where the right-hand side of the eigenvalue equation involves a superposition of lower-order operators with a "wrong sign."

• The Operator: The authors consider linear superpositions of fractional Laplacians defined by a signed measure $\mu = \mu_+ - \mu_-$ over the interval $[0, 1]$. The operator is given by:
  $$L_\mu u := \int_{[0,1]} (-\Delta)^s u \, d\mu(s)$$
  where $(-\Delta)^s$ is the fractional Laplacian.
• The Equation: The core problem is the "doubly-nonlocal" eigenvalue problem:
  $$\int_{[0,1]} (-\Delta)^s u \, d\mu_+(s) = \lambda \int_{[0,s)} (-\Delta)^s u \, d\mu_-(s) \quad \text{in } \Omega$$
  with $u=0$ in $\mathbb{R}^N \setminus \Omega$.
• Significance: Unlike classical eigenvalue problems where the right-hand side is simply $\lambda u$ (a zero-order term), here the RHS involves lower-order nonlocal operators. This structure arises in modeling the overlap of diffusive phenomena (fractional diffusion) with singularity formation or concentration effects. The "wrong sign" of the lower-order terms (relative to classical elliptic theory) creates unique analytical challenges and structural differences compared to standard local or single-fractional problems.

2. Methodology and Functional Framework

The authors develop a rigorous functional analytic framework to handle the mixed nature of the operators.

• Functional Spaces: They define a Hilbert space $X_+(\Omega)$ as the completion of $C_c^\infty(\Omega)$ with respect to a norm involving the positive measure $\mu_+$:
  $$\|u\|_{X_+} = \left( \int_{[0,1]} [u]_s^2 \, d\mu_+(s) \right)^{1/2}$$
  where $[u]_s$ is the Gagliardo seminorm (or $L^2$ norm for $s=0$, and $H^1$ seminorm for $s=1$).
• Variational Characterization: The first eigenvalue $\lambda_\mu(\Omega)$ is characterized variationally as the minimum of a Rayleigh quotient:
  $$\lambda_\mu(\Omega) = \min_{u \in X_+(\Omega) \setminus \{0\}} \frac{\int_{[0,1]} [u]_s^2 \, d\mu_+(s)}{\int_{[0,s)} [u]_s^2 \, d\mu_-(s)}$$
• Regularity Theory: A crucial component of the methodology is the development of a regularity theory for equations of the form $-\Delta u + \int (-\Delta)^s u \, d\mu = f$. The authors prove that solutions belong to $W^{2,p}(\Omega)$ under specific conditions on the measure and the domain, utilizing interpolation inequalities and classical elliptic regularity as a base.
• Perturbation Analysis: To study convergence, they consider a sequence of measures $\mu_n^-$ supported in shrinking intervals $[0, \epsilon_n]$ near zero. This allows them to analyze the limit as the "wrong sign" term localizes to a standard $L^2$ term.

3. Key Contributions and Results

A. Convergence to Classical Cases (Theorems 1.2 & 1.3)

The authors establish that as the support of the negative measure $\mu_-$ shrinks to zero (localizing the RHS), the nonlocal eigenvalue problem converges to a classical local eigenvalue problem.

• Convergence: The first eigenvalue $\lambda_{\epsilon_n}$ converges to the first eigenvalue $\lambda_0$ of the limiting local problem. The associated eigenfunctions converge strongly in $L^2$ and weakly in $X_+$.
• Simplicity and Sign-Definiteness: In the limit (and for sufficiently small $\epsilon_n$), if the domain $\Omega$ is connected and the positive part of the measure is a Dirac delta at $s=1$ (recovering the standard Laplacian), the first eigenvalue is simple, and the corresponding eigenfunction is strictly positive (or negative). This recovers the classical properties of the Laplacian.

B. Spectral Properties on Disconnected Domains (Theorems 1.4, 1.5, 1.6, 1.7)

The paper reveals profound structural differences between this nonlocal setting and the classical case when the domain $\Omega$ is disconnected (a union of disjoint open sets).

1. Sign-Changing Eigenfunctions (Theorem 1.4):

   • Result: Unlike the classical case where the first eigenfunction is strictly positive, in this mixed nonlocal setting, any eigenfunction associated with the first eigenvalue on a disconnected domain must change sign.
   • Mechanism: If an eigenfunction were of constant sign, one could construct a competitor function (by flipping the sign on one component) that strictly lowers the Rayleigh quotient due to the nonlocal interaction between the disconnected components.

2. Strict Sub-additivity of Eigenvalues (Theorem 1.5):

   • Result: The first eigenvalue of the union of two disjoint domains is strictly smaller than the minimum of the first eigenvalues of the individual components:
     $$\lambda(\Omega_1 \cup \Omega_2) < \min \{ \lambda(\Omega_1), \lambda(\Omega_2) \}$$
   • Contrast: In the classical Laplacian case, $\lambda(\Omega_1 \cup \Omega_2) = \min \{ \lambda(\Omega_1), \lambda(\Omega_2) \}$. The strict inequality arises because the nonlocal operator "feels" the presence of the other component, allowing for a lower energy state via sign-changing combinations.

3. Simplicity of the First Eigenvalue (Theorems 1.6 & 1.7):

   • The simplicity of the first eigenvalue in disconnected domains is not guaranteed and depends on the geometry and the specific nonlocal parameter.
   • Non-Simple Case: For two congruent balls, the first eigenvalue is not simple (multiplicity $\ge 2$) for small fractional orders.
   • Simple Case: For two balls of different sizes (or specific non-congruent geometries), the first eigenvalue can be simple. This contrasts with the classical case where congruent components always lead to non-simple eigenvalues.

C. Regularity Theory (Theorem 1.8)

The authors provide a self-contained proof of $W^{2,p}$ regularity for solutions to mixed local-nonlocal equations with lower-order terms.

• They prove that if $f \in L^p(\Omega)$, the weak solution $u$ belongs to $W^{2,p}(\Omega)$.
• They distinguish cases based on the dimension $N$ and exponent $p$, providing specific estimates that control the $W^{2,p}$ norm by the $L^p$ norm of the source term and the $L^2$ (or $L^p$) norm of the solution. This regularity is essential for proving the convergence and sign properties of the eigenfunctions.

4. Significance and Impact

• Theoretical Novelty: This work fills a gap in the literature regarding eigenvalue problems where the RHS is a superposition of lower-order nonlocal operators with a "wrong sign." It demonstrates that such problems cannot be treated as simple perturbations of classical local problems without accounting for the specific nonlocal interactions.
• Paradigm Shift in Disconnected Domains: The results fundamentally alter the understanding of spectral theory on disconnected sets. The paper proves that the "localization" of eigenfunctions to individual components (a hallmark of classical disconnected domains) does not occur here. Instead, the nonlocal coupling forces sign changes and strictly lowers the energy of the system.
• Modeling Implications: The findings are relevant for mathematical biology and population dynamics (cited in the introduction), where superposition operators model populations with mixed diffusion mechanisms and concentration effects. The "sign-changing" nature of the first eigenmode suggests complex spatial distributions in such models that differ from standard diffusion models.
• Analytical Tools: The developed regularity theory and the specific variational techniques for handling signed measures in the denominator of the Rayleigh quotient provide a robust toolkit for future research in mixed-order nonlocal PDEs.

In summary, the paper establishes that the spectral properties of mixed local-nonlocal operators with "wrong sign" lower-order terms are distinct from both classical local operators and pure nonlocal operators, particularly exhibiting unique behaviors regarding the sign of eigenfunctions and the spectral gap in disconnected domains.