Modulus of continuity for solutions of non-local heat equations

Technical Summary: Modulus of Continuity for Solutions of Non-Local Heat Equations

1. Problem Statement and Context
This paper investigates the preservation of the modulus of continuity for solutions to non-local heat equations. The authors extend the "modulus of continuity" method, a technique historically utilized to prove the Fundamental Gap Conjecture for classical parabolic equations, to the non-local setting. The primary objective is to determine under what conditions an initial modulus of continuity $\phi(\cdot, 0)$ for a solution $u(\cdot, 0)$ is preserved at all subsequent times $t > 0$. Specifically, the authors seek to establish that if the initial data satisfies $|u(y, 0) - u(x, 0)| \le 2\phi(|y-x|/2, 0)$, then $|u(y, t) - u(x, t)| \le 2\phi(|y-x|/2, t)$ holds for all $t \ge 0$.

The study addresses three distinct cases:

1. The non-local heat equation on the entire space $\mathbb{R}^n$.
2. The "regional" non-local heat equation on a bounded convex domain $\Omega \subset \mathbb{R}^n$ (where interactions are restricted to the domain).
3. A non-linear non-local heat equation on $\mathbb{R}^n$.

2. Methodology
The proofs across all sections rely on a proof by contradiction utilizing a maximum principle argument adapted for non-local operators. The core methodology involves the following steps:

• Auxiliary Function Construction: An auxiliary function $Z_\epsilon(x, y, t)$ is defined as:
  $$Z_\epsilon(x, y, t) = u(y, t) - u(x, t) - 2\phi\left(\frac{|y-x|}{2}, t\right) - \epsilon e^{Ct}(\Psi(x) + \Psi(y))$$
  where $\Psi$ is a radially symmetric regularization function that grows to infinity at the boundary (or infinity in $\mathbb{R}^n$), and $\epsilon, C$ are positive constants. The goal is to show $Z_\epsilon < 0$ everywhere.

• First Contact Point Argument: Assuming a contradiction (i.e., $Z_\epsilon \ge 0$ somewhere), the authors consider the first time $t_0$ and distinct points $x_0, y_0$ where $Z_\epsilon$ attains a global maximum of zero. At this point, the time derivative $\partial_t Z_\epsilon \ge 0$.

• Coupling-by-Reflection: To handle the non-local integrals, the authors employ a "coupling-by-reflection" technique. They introduce a reflection map $R$ across the hyperplane perpendicular to the chord connecting $x$ and $y$. By changing variables and utilizing the rotational symmetry and monotonicity of the kernel $\rho$, they pair the integration domains to cancel terms or establish inequalities.

• One-Dimensional Reduction: A critical component of the method is the requirement that the modulus function $\phi$ (and its odd extension $\tilde{\phi}$) must satisfy a corresponding one-dimensional non-local heat equation. This allows the non-local terms in the $n$-dimensional operator to be matched against the time derivative of the modulus function.

• Handling Singular Kernels: For singular kernels (e.g., the fractional Laplacian), the authors replace absolute integrals with principal-value limits. They introduce a truncated reflection identity (Lemma 2.13) to manage the singularity at the origin, showing that the remainder terms vanish as the truncation parameter $\delta \to 0$.

3. Key Results

• Case 1: Non-Local Heat Equation on $\mathbb{R}^n$
  Theorem 2.3 establishes that for solutions on $\mathbb{R}^n$ with Lebesgue integrable kernels, the modulus of continuity is preserved provided the kernel is non-negative, non-increasing, and rotationally symmetric. Theorem 2.16 extends this result to singular kernels satisfying the Lévy integrability condition (including the fractional Laplacian $-( -\Delta)^s$), provided the solution is bounded and the regularization function $\Psi$ satisfies specific growth and boundedness conditions.

• Case 2: Regional Heat Equation (Bounded Convex Domains)
  Theorem 3.3 proves that the modulus of continuity property holds for the regional heat equation in one dimension (on a compact interval). The proof requires careful handling of boundary cases where the integration intervals intersect the domain boundary.
  However, the authors demonstrate that this result does not generalize to higher dimensions. In Section 3.2, they construct a counterexample using a thin rectangular domain $[0, L] \times [-\epsilon, \epsilon]$. They show that the first non-trivial eigenvalue $\lambda_2$ of the regional fractional Laplacian on such a domain can be made arbitrarily small by varying the aspect ratio, even if the diameter remains fixed. This contradicts the implication of a higher-dimensional modulus of continuity theorem, which would suggest $\lambda_2$ is bounded below by a quantity depending only on the domain's diameter (via the Payne-Weinberger inequality analogy). Consequently, the authors conclude that a non-local analogue of the Payne-Weinberger inequality depending solely on diameter is false in dimensions $n > 1$.

• Case 3: Non-Linear Non-Local Heat Equation
  Theorem 4.2 extends the modulus of continuity results to equations of the form $u_t = L_{\mathbb{R}^n}u + q(|Du|)$, where $q$ is a non-decreasing, Lipschitz continuous non-linearity. The proof adapts the linear case by bounding the non-linear term $T_q = q(|Du(y)|) - q(|Du(x)|)$ using the Lipschitz constant and the gradient of the regularization function $\Psi$.

4. Significance and Claims
The paper claims to successfully generalize the modulus of continuity method from classical local parabolic equations to non-local settings on $\mathbb{R}^n$ and in one-dimensional bounded domains. The primary significance lies in the rigorous establishment of these preservation properties for both regular and singular kernels (including the fractional Laplacian).

Crucially, the paper provides a negative result regarding the generalization to higher-dimensional bounded domains. By linking the potential validity of a higher-dimensional modulus of continuity theorem to a non-local version of the Payne-Weinberger inequality, and subsequently disproving the latter via a counterexample, the authors clarify the geometric limitations of non-local spectral bounds. They assert that unlike the local case, the spectral gap for non-local operators on convex domains depends on more than just the diameter.

The work is presented as a foundational step toward understanding non-local spectral geometry, specifically noting that the failure of the higher-dimensional modulus of continuity theorem suggests that proving a non-local Payne-Weinberger inequality will require more complex geometric invariants than the diameter alone.