Homoclinic solutions for nonlocal equations and applications to the theory of atom dislocation

Technical Summary: Homoclinic Solutions for Nonlocal Equations and Applications to the Theory of Atom Dislocation

Problem Statement
This paper investigates the existence of homoclinic solutions for systems of nonlocal equations driven by the fractional Laplacian. The study focuses on two distinct structural scenarios for the potential energy term, motivated by the Peierls–Nabarro model for atomic edge dislocations in crystals.

1. Spatially Homogeneous Potentials: The authors consider the equation $(-\Delta)^s q = \nabla V(q)$ on $\mathbb{R} \setminus [a, b]$, where $V$ is a spatially homogeneous function with a strict global maximum at the origin. The solution is constrained to a fixed profile $q_0$ on the interval $[a, b]$. The goal is to find a solution $q$ that decays to zero at infinity (a homoclinic orbit), representing a crystal configuration where atoms at infinity are in a rest position, while the atoms within $[a, b]$ are "pinched" away from equilibrium.
2. Spatially Dependent Potentials: The authors analyze the system $(-\Delta)^s q(x) + L(x)q(x) = \nabla_q W(x, q(x))$, where $L(x)$ is a symmetric, uniformly positive definite matrix that grows at infinity, and $W$ satisfies the Ambrosetti–Rabinowitz condition. This setting models dislocations under a confinement potential.

A central technical challenge addressed is the lack of a convenient Hamiltonian formalism in the nonlocal setting, which prevents the direct application of classical Ordinary Differential Equation (ODE) techniques used in local counterparts (e.g., Rabinowitz and Tanaka).

Methodology
The authors employ a variational approach combined with nonlocal elliptic regularity theory.

• Variational Framework: For the homogeneous case, the problem is formulated as a minimization of an energy functional $I(q)$ over a set $\Gamma$ of functions satisfying the boundary condition on $[a, b]$. For the spatially dependent case, the Mountain Pass Theorem is utilized to find critical points of the associated energy functional in a Hilbert space $\tilde{H}^s$.
• Regularity and Bootstrap Arguments: Due to the nonlocal nature of the fractional Laplacian, the authors cannot rely on ODE arguments to establish the decay of solutions or their regularity. Instead, they develop a "regularity bootstrap" strategy. This involves:
  • Establishing interior and boundary regularity estimates for weak solutions using results from the literature on fractional Sobolev spaces (e.g., Ros-Oton, Serra, Abatangelo).
  • Proving that bounded weak solutions possess Hölder continuity, which allows for the application of decay lemmas.
  • Using energy estimates to locate critical levels and ensure the existence of nontrivial minimizers or critical points.
• Barrier Construction: For the case where $s \in (0, 1/2]$ in the spatially dependent problem, standard energy bounds are insufficient to guarantee decay at infinity due to the unbounded coefficients of $L(x)$. The authors construct an ad-hoc barrier function based on the derivative of a monotone solution to a related fractional equation to control the behavior of the solution at infinity.
• Optimality Analysis: The paper includes a rigorous discussion on the necessity of the assumptions, specifically demonstrating that for $s \in (0, 1/2]$, the minimization problem trivializes (infimum is zero) if the constraint interval degenerates to a single point ($a=b$), whereas this is not the case for $s \in (1/2, 1)$.

Key Contributions and Results

• Existence of Homoclinic Solutions (Homogeneous Case):

  • Theorem 1.2: Proves the existence of a homoclinic solution for $s \in (0, 1)$ when the constraint is an interval $[a, b]$ with $a < b$. The solution is shown to be in $C^\beta(\mathbb{R})$ and decays to zero at infinity.
  • Theorem 1.3: Extends the result to the case $a = b$ (a single point constraint) but only for the range $s \in (1/2, 1)$. The solution is symmetric with respect to the constraint point.
  • Proposition 2.2: Demonstrates that for $s \in (0, 1/2]$, the case $a=b$ is not admissible for non-trivial homoclinic solutions, as the infimum of the energy is attained at the zero level.

• Existence of Homoclinic Solutions (Spatially Dependent Case):

  • Theorem 1.4: Establishes the existence of nontrivial homoclinic solutions for $s \in (1/2, 1)$ under the Ambrosetti–Rabinowitz condition and a specific growth condition on the potential $W$. The solution is shown to be in $C^{s-1/2}$ and decays to zero.
  • Theorem 1.5: Proves existence for $s \in (0, 1/2]$. This requires additional growth conditions on $W$ (polynomial bounds) and, crucially, a structural assumption on the matrix $L(x)$ (diagonal dominance at infinity) to construct the necessary barrier. The solution is bounded and decays to zero.

• Regularity and Decay:

  • The paper provides detailed regularity results, showing that solutions belong to specific Hölder spaces ($C^\beta$, $C^{s-1/2}$, or $C^{2s}$ locally) depending on the fractional exponent and the regularity of the potential.
  • It rigorously proves that these solutions satisfy $\lim_{x \to \pm \infty} q(x) = 0$, confirming their homoclinic nature.

Significance and Claims
The paper claims to provide a "natural fractional counterpart" to classical results by Rabinowitz and Tanaka regarding homoclinic orbits in local systems. The primary significance lies in overcoming the technical difficulties inherent to nonlocal operators, specifically the absence of a Hamiltonian structure and the different capacity theories associated with fractional Sobolev spaces for different values of $s$.

In the context of the Peierls–Nabarro model, the results offer a mathematical justification for the existence of specific crystal configurations:

1. For the homogeneous potential, the results suggest that a crystal can support a configuration where atoms are in an unstable rest position at infinity, provided the potential is slightly modified and a "pinch" (displacement) is applied at a specific point or interval.
2. For the spatially dependent potential, the results indicate that equilibrium configurations at infinity are possible under small superquadratic perturbations of the classical potential, provided the confinement potential grows sufficiently at infinity.

The authors emphasize that these findings are derived strictly through variational methods and nonlocal regularity theory, avoiding the ODE techniques that fail in the nonlocal setting. The paper does not propose new experimental setups but rather provides the theoretical existence of solutions that correspond to physically relevant dislocation patterns.