Fredholm alternative for a general class of nonlocal operators

1. Problem Statement

The paper addresses the solvability and uniqueness of solutions for a broad class of nonlocal elliptic operators of mixed order. Specifically, the authors aim to extend the classical Fredholm Alternative (a fundamental dichotomy in linear operator theory stating that either a unique solution exists, or the homogeneous equation has non-trivial solutions) to operators that:

• Involve a superposition of fractional gradients of varying orders $s \in (0, 1]$.
• Possess measurable, unbounded, and discontinuous coefficients.
• Are defined via a measure $\mu$ that weights different fractional orders, allowing for continuous, discrete, or mixed distributions of orders.

This generalizes the seminal work of Neil Trudinger (1973) on classical second-order elliptic operators with measurable coefficients to the nonlocal, fractional setting. The motivation stems from applications in mathematical biology (e.g., modeling animal dispersal strategies with different Lévy exponents) and the need for a unified framework for classical and nonlocal phenomena.

2. Methodology and Framework

The Operator

The core object of study is the operator $L$ defined formally as:
$$Lu := \int_{(0,1]} \left( -D^s_i (a_{ij}(s, x) D^s_j u + a_i(s, x)u) + b_i(s, x) D^s_i u \right) d\mu(s) + a(x)u$$
where:

• $D^s$ is the fractional gradient of order $s$.
• $\mu$ is a non-negative $\sigma$-finite measure on $[0, 1]$ with support bounded away from 0.
• The coefficients $a_{ij}, a_i, b_i, a$ are measurable and can be unbounded or discontinuous.
• The matrix $A = [a_{ij}]$ is positive definite but not necessarily uniformly elliptic; it satisfies specific growth and integrability conditions (e.g., $\Lambda(x)$ grows at most polynomially at infinity, and $\lambda^{-1}$ is locally integrable).

Functional Analytic Setting

To handle the lack of uniform regularity and the mixed-order nature of the operator, the authors construct a bespoke functional framework:

1. Fractional Gradient Spaces: They utilize Bessel-type spaces $H^{s,p}_0(\Omega)$, defined via the norm $\|u\|_{H^{s,p}} = (\|u\|_{L^p}^p + \|D^s u\|_{L^p}^p)^{1/p}$.
2. The Space $H_0(A, g, \Omega)$: A Hilbert space constructed as the completion of $C^\infty_c(\Omega)$ under the scalar product:
   $$\langle u, v \rangle_{H_0(A,g,\Omega)} = \int_{\mathbb{R}^n} \int_{(0,1]} a_{ij}^S(s,x) D^s_i u D^s_j v \, d\mu(s) dx + \int_\Omega g(x)uv \, dx$$
   where $a_{ij}^S$ is the symmetric part of the coefficient matrix.
3. Compact Boundedness: A crucial concept introduced is compact boundedness. A function $f$ is compactly bounded on $H_0(A, g, \Omega)$ if for every $\epsilon > 0$, there exists $K_\epsilon$ such that:
   $$\| \phi \|_{L^2(f, \Omega)}^2 \leq \epsilon \| \phi \|_{H_0(A,g,\Omega)}^2 + K_\epsilon \| \phi \|_{L^1(\Omega)}^2$$
   This condition is weaker than standard boundedness but sufficient to ensure compact embeddings into weighted $L^2$ spaces, which is a prerequisite for applying the Fredholm Alternative.

Key Technical Tools

• Fractional Calculus: Detailed analysis of the fractional gradient $D^s$, including its convergence to the classical gradient as $s \to 1$, its relation to the Riesz potential, and Fourier transform properties.
• Embedding Theorems: New proofs and extensions of Sobolev-type embeddings for $H^{s,p}_0(\Omega)$, specifically controlling the dependence on the fractional parameter $s$ and extending results to the case $p=1$.
• Variational Formulation: The operator $L$ is treated via a bilinear form $(Lu, v)$ defined on $H_0(A, \Omega) \times H_0(A, \Omega)$.

3. Key Contributions

1. Generalization of Trudinger's Result: The paper successfully extends the Fredholm Alternative from classical divergence-form elliptic operators with measurable coefficients to a highly general class of nonlocal, mixed-order operators.
2. Unified Framework: It provides a single theoretical structure that encompasses:
   • Purely fractional operators (discrete or continuous measures).
   • Classical operators (Dirac mass at $s=1$).
   • Mixed operators (superposition of fractional and classical terms).
3. Handling Singular Coefficients: The framework accommodates coefficients that are unbounded and discontinuous, provided they satisfy specific integrability conditions relative to the measure $\mu$ and the domain geometry.
4. Introduction of Compact Boundedness: The authors define and utilize "compact boundedness" to overcome the lack of compactness in standard embeddings for these generalized spaces, enabling the use of Riesz-Schauder theory.

4. Main Results

Theorem 1.5 (Fredholm Alternative):
Let $f$ be a function defined by the coefficients of the operator (specifically related to the lower-order terms) that is compactly bounded on $H_0(A, \Omega)$. Then:

• There exists a countable, isolated set of real numbers $\Sigma$ (the eigenvalues) such that $\Sigma \subset (-\infty, \sigma_0)$ for some coercivity constant $\sigma_0$.
• Case 1 (Non-resonance): If $\sigma \notin \Sigma$, the equation $L_\sigma(f)u = T$ (where $L_\sigma = L + \sigma I(f)$) has a unique solution $u \in H_0(A, \Omega)$ for any $T$ in the dual space.
• Case 2 (Resonance): If $\sigma \in \Sigma$, the equation has a solution if and only if $T$ is orthogonal to the kernel of the adjoint operator $L_\sigma^*$. In this case, if a solution exists, there are infinitely many solutions.

Corollary/Remark 1.6:
If the measure $\mu$ has an atom at $s=1$ (i.e., $\mu(\{1\}) > 0$), the results hold even if the integrability condition on the ellipticity bound $\lambda$ is relaxed (specifically, $\lambda^{-1} \in L^1(\Omega)$ is sufficient, recovering Trudinger's classical result).

5. Significance

• Theoretical Advancement: This work bridges a significant gap between classical elliptic theory and modern nonlocal analysis. It demonstrates that the structural dichotomy of the Fredholm Alternative is robust enough to survive the transition to operators with variable fractional orders and rough coefficients.
• Biological Modeling: The ability to handle superpositions of different fractional orders is directly applicable to modeling complex biological systems (e.g., animal foraging) where populations exhibit heterogeneous dispersal strategies (a mix of Gaussian and Lévy flights).
• Analytical Rigor: By constructing a specific Hilbert space $H_0(A, g, \Omega)$ and proving necessary embedding and compactness properties, the paper lays the groundwork for future studies on existence, regularity, and numerical approximation of solutions to mixed-order nonlocal PDEs.

In summary, the paper establishes a rigorous functional analytic foundation for solving linear equations involving mixed-order nonlocal operators with minimal regularity assumptions on coefficients, successfully generalizing a cornerstone result of classical PDE theory to the fractional realm.