Optimal stability of complement value problems for p-Lévy operators

Technical Summary: Optimal Stability of Complement Value Problems for p-Lévy Operators

Problem Statement
This work investigates the asymptotic behavior of weak solutions to nonlocal integro-differential equations (IDEs) governed by symmetric $p$-Lévy operators as the nonlocal interaction range vanishes. Specifically, the paper studies the convergence of solutions to the nonlocal complement value problem $(P_{\nu_\varepsilon, \tau})$ to the solutions of the corresponding local $p$-Laplace boundary value problem $(P_\tau)$ as the parameter $\varepsilon \to 0$.

The nonlocal problem is defined on an open set $\Omega \subset \mathbb{R}^d$ with its complement $\Omega^c$, governed by the operator:
$$\mathcal{L}_\varepsilon u(x) = 2 \text{ p.v.} \int_{\mathbb{R}^d} |u(x) - u(y)|^{p-2}(u(x) - u(y)) \nu_\varepsilon(x-y) \, dy$$
subject to nonlocal Dirichlet ($\tau=0$) or Neumann ($\tau=1$) complement conditions on $\Omega^c$. The kernel sequence $(\nu_\varepsilon)_\varepsilon$ is a family of radial $p$-Lévy kernels approximating the identity, satisfying specific normalization conditions (e.g., $\int_{\mathbb{R}^d} (1 \wedge |h|^p)\nu_\varepsilon(h) dh = 1$). The goal is to establish optimal convergence, meaning strong convergence in the natural nonlocal energy spaces $W^p_{\nu_\varepsilon}(\Omega)$ and $W^p_{\nu_\varepsilon}(\Omega|\mathbb{R}^d)$, rather than merely in $L^p(\Omega)$.

Methodology
The analysis relies on a combination of variational methods, asymptotic analysis of nonlocal forms, and the theory of trace spaces.

1. Energy Form Analysis: The authors analyze the convergence of nonlocal bilinear forms $E_\varepsilon(\cdot, \cdot)$ (and their variants $E_\varepsilon^\Omega$, $E_\varepsilon^+$, and the cross-boundary form $E_\varepsilon^{cr}$) to the local Dirichlet form $E_0(u, v) = \int_\Omega |\nabla u|^{p-2}\nabla u \cdot \nabla v \, dx$. A crucial step is establishing the asymptotic demi-convergence of these forms, which characterizes the convergence of solutions even when the sequence of functions depends on $\varepsilon$.
2. Uniform Linearization: To handle the nonlinearity of the $p$-Lévy operators (for $1 < p < \infty$), the paper employs uniform linearization estimates (Theorem 2.4, 2.6) that relate the difference of forms to the energy of the difference of functions. This allows the authors to control the convergence of the nonlinear terms.
3. Robustness of Trace Spaces: A significant portion of the work is dedicated to proving the robustness of nonlocal trace spaces $T^p_{\nu_\varepsilon}(\Omega^c)$ as $\varepsilon \to 0$. The authors demonstrate that these spaces converge to the local trace space $W^{1-1/p, p}(\partial\Omega)$ in a strong sense, including the convergence of lifting operators. This is essential for handling Dirichlet boundary conditions where data is prescribed on $\Omega^c$.
4. $\Gamma$-Convergence and Compactness: The convergence of solutions is framed through the $\Gamma$-convergence of the associated energy functionals $J_\varepsilon$ to the local functional $J$. Due to the domain mismatch (nonlocal functions are defined on $\mathbb{R}^d$, local on $\Omega$), the authors utilize asymptotic compactness results (Theorem 2.18) and uniform Poincaré inequalities to ensure the existence of convergent subsequences and the identification of the limit.

Key Contributions and Results

• Optimal Convergence of Solutions: The primary result (Section 5) establishes that under suitable asymptotic weak/strong convergence assumptions on the data ($f_\varepsilon, g_\varepsilon$), the sequence of weak solutions $(u_\varepsilon)_\varepsilon$ converges strongly to the local solution $u$ in the optimal nonlocal norms:
  $$\lim_{\varepsilon \to 0} \|u_\varepsilon - u\|_{W^p_{\nu_\varepsilon}(\Omega|\mathbb{R}^d)} = 0$$
  This implies the collapse of the cross-boundary energy term: $\lim_{\varepsilon \to 0} E_\varepsilon^{cr}(u_\varepsilon, u_\varepsilon) = 0$.
• Robustness of Nonlocal Trace Spaces: The paper proves that the nonlocal trace spaces $T^p_{\nu_\varepsilon}(\Omega^c)$ converge to the local trace space $W^{1-1/p, p}(\partial\Omega)$. Specifically, if a sequence of boundary data converges in the nonlocal trace norm, the limit of the norms coincides with the local trace norm (Theorem 3.5). Furthermore, robust right-lifting operators exist that converge strongly to the local lifting operator.
• Asymptotic Demi-Convergence: The authors establish a generalized form of demi-convergence for the energy forms $E_\varepsilon$ to $E_0$ (Theorem 2.16, 2.17). This result holds under mild regularity assumptions on the domain (e.g., $W^{1,p}$-extension domains) or when one of the functions vanishes on the boundary, extending previous results on form convergence.
• Fractional $p$-Laplacian Case: The general theory is applied to the fractional $p$-Laplacian $(-\Delta)^s_p$ as $s \to 1^-$. The paper shows that solutions to the fractional complement problems converge strongly to the local $p$-Laplace solutions (Section 6). This result is noted as new even for the fractional $p$-Laplacian.
• Dual Space Convergence: The dual spaces of the nonlocal Sobolev spaces are shown to converge to their local counterparts (Theorem 3.9), ensuring the stability of the variational formulations.

Significance and Claims
The paper claims to provide the optimal convergence result for nonlocal-to-local limits of $p$-Lévy operators. Previous works (including the author's prior research [24]) established strong convergence only in $L^p(\Omega)$ under relaxed assumptions. This work strengthens the result to the energy norm, which is the natural topology for these variational problems.

The authors emphasize that the choice of the kernel sequence $(\nu_\varepsilon)_\varepsilon$ satisfying condition (1.1) is "essentially optimal" because it is intrinsic to the characterization of the local gradient norm via nonlocal differences (Eq. 1.3), extending seminal work by Brezis, Bourgain, and Mironescu.

The robustness of the trace spaces is highlighted as a crucial component of the approach, allowing for a rigorous treatment of nonlocal boundary conditions that converge to local boundary conditions. The paper explicitly states that no stronger convergence can be expected in general, as the nonlocal solutions $u_\varepsilon$ belong to $W^p_{\nu_\varepsilon}(\Omega)$ while the limit $u$ belongs to $W^{1,p}(\Omega)$, and the norms are distinct.

The results unify the treatment of Dirichlet and Neumann complement problems and apply to both the full nonlocal operator and regional operators (where interactions are restricted to $\Omega$). The findings are presented as a generalization of existing literature on $p=2$ and fractional cases, providing a comprehensive framework for $1 < p < \infty$.