On a nonlocal superconductivity problem

1. Problem Statement and Context

The paper investigates degenerate nonlocal free boundary problems arising in the context of superconductivity and nonlocal diffusion. The study extends the local work of Caffarelli, Salazar, and Shahgholian [9, 10] to the nonlocal setting involving the fractional Laplacian.

The core problem involves finding an unknown pair $(u, \Omega)$, where $u$ is a function defined on $\mathbb{R}^n$ and $\Omega \subset \mathbb{R}^n$ is an open set, satisfying the following system in the viscosity sense:
$$\begin{cases} (-\Delta)^s u = f & \text{in } B_1 \cap \Omega, \\ |Du| = 0 & \text{in } B_1 \setminus \Omega. \end{cases}$$
Here, $(-\Delta)^s$ denotes the fractional Laplacian with $s \in (1/2, 1)$.

• Physical Interpretation: In the set $\Omega$, the solution behaves like a solution to a nonlocal PDE (diffusion occurs). In the complement $B_1 \setminus \Omega$, the gradient vanishes ($|Du|=0$), representing critical points where diffusion is "blocked" or degenerate.
• Applications: These models are relevant to finance (jump processes) and superconductivity (stationary equations for superconducting vortices).

Key Challenge: Unlike the local case (where Imbert and Silvestre showed that harmonicity on non-critical points implies global harmonicity), the nonlocal setting does not allow for such a reduction. Solutions to the homogeneous problem are not necessarily $s$-harmonic globally because the fractional Laplacian requires integration over the entire space $\mathbb{R}^n$. The behavior of $u$ outside the domain $B_1$ significantly influences the interior, preventing a simple extension of the PDE to the whole domain.

2. Methodology

The authors develop a De Giorgi-type gradient oscillation method tailored specifically for nonlocal degenerate equations. The strategy relies on two main pillars:

A. Nonlocal Positivity Argument

The authors establish that if the gradient is small in a set of positive measure, it oscillates in a controlled manner.

• They construct a subsolution for the derivatives of the solution. By using a smooth cut-off function $\eta$ and difference quotients, they show that the positive part of the derivative, $v = \eta(\partial_e u - \mu)_+$, satisfies a nonlocal inequality:
  $$\Delta_s v \geq -C \|u\|_{L^1_s(\mathbb{R}^n)}$$
  This allows them to apply nonlocal maximum principles and measure estimates (Lemma 2.1) to control the size of sublevel sets of the gradient.

B. Iterative Oscillation Control (Dichotomy)

The proof of regularity relies on a dichotomy argument applied to dyadic balls:

1. Case 1 (Gradient is large in measure): If the gradient is "large" (non-vanishing) in a significant portion of a ball, the authors use a rescaling and perturbation argument. They show that the solution is close to an affine function, allowing them to apply interior regularity estimates for $s$-harmonic functions to improve the oscillation of the gradient.
2. Case 2 (Gradient is small in measure): If the gradient remains small in measure across infinitely many iterations, this implies the presence of critical points. The authors iteratively refine the oscillation estimates in this scenario, overcoming the lack of PDE information at critical points by leveraging the nonlocal positivity argument.

The method carefully handles the growth at infinity (using the $L^1_s$ norm) to ensure that rescaled functions maintain the necessary properties for the nonlocal operator.

3. Key Contributions

• Extension to Nonlocal Setting: The paper successfully generalizes the regularity theory for degenerate free boundary problems from local elliptic PDEs to nonlocal fractional operators.
• Handling Degeneracy: It addresses the specific difficulty where the PDE degenerates ($|Du|=0$) and no equation governs the set of critical points, a scenario where standard elliptic regularity fails.
• Stability as $s \to 1$: The results are shown to be stable as the fractional order $s$ approaches 1, seamlessly recovering the classical local results of Caffarelli et al.
• Fully Nonlinear Extension: The methodology is extended to a broader class of fully nonlinear nonlocal operators (elliptic with respect to the class $\mathcal{L}_1(s)$), demonstrating the robustness of the approach.

4. Main Results

Theorem 1.1 (Homogeneous Case)

For the homogeneous problem ($f=0$), if $(u, \Omega)$ solves the system with $u \in C(B_1) \cap L^\infty(\mathbb{R}^n)$, then $u$ is locally $C^{1,\alpha}$ for some universal $\alpha \in (0, 1)$.

• Estimate: $\|u\|_{C^{1,\alpha}(B_{1/2})} \leq C \|u\|_{L^\infty(\mathbb{R}^n)}$.

Theorem 1.2 (Non-Homogeneous Case)

For the non-homogeneous problem with a sufficiently smooth source term $f$ ($f \in C^{0,1}$), if $u \in C(B_1) \cap L^1_s(\mathbb{R}^n)$, then $u$ is locally $C^{1,\alpha}$.

• Estimate: The bound is refined to use the natural nonlocal norm:
  $$\|u\|_{C^{1,\alpha}(B_{1/2})} \leq C \left( \|u\|_{L^\infty(B_1)} + \|u\|_{L^1_s(\mathbb{R}^n)} + \|f\|_{C^{0,1}(B_1)} \right)$$
  This replaces the global $L^\infty$ bound with the weaker $L^1_s$ norm, which is more appropriate for nonlocal problems.

Corollary 1.1 (Obstacle-Type Problems)

The results are applied to obstacle-type problems of the form $\min\{-( -\Delta)^s v, v - \phi\} = 0$, establishing $C^{1,\alpha}$ regularity for solutions under similar conditions.

Theorem 6.1 (Fully Nonlinear Operators)

The $C^{1,\alpha}$ regularity is extended to solutions of fully nonlinear nonlocal equations $Iu = f$ where $I$ is elliptic with respect to the class $\mathcal{L}_1(s)$. The constants depend on the ellipticity constant $\Lambda$.

5. Significance

This work is significant because it bridges a gap in the regularity theory of nonlocal free boundary problems.

1. Theoretical Advancement: It proves that despite the nonlocal nature preventing the reduction to a global PDE, solutions still possess high regularity ($C^{1,\alpha}$) up to the free boundary.
2. Methodological Innovation: The development of a "nonlocal positivity argument" for derivatives provides a new toolkit for analyzing degenerate nonlocal equations where the operator is not defined or is singular at critical points.
3. Applicability: By establishing these estimates, the paper provides a rigorous foundation for modeling complex physical phenomena like superconducting vortices and financial jump processes using nonlocal models, ensuring that the mathematical solutions are well-behaved and physically meaningful.