Optimal stability of Dirichlet problem for the regional fractional $p$-Laplacian

Technical Summary: Optimal Stability of the Dirichlet Problem for the Regional Fractional $p$-Laplacian

Problem Statement
This paper investigates the asymptotic behavior of the Dirichlet boundary value problem associated with the regional fractional $p$-Laplacian, denoted as $(-\Delta)^s_{p,\Omega}$, as the fractional parameter $s$ approaches 1 from below ($s \to 1^-$). The problem is defined on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ with $1 < p < \infty$ and $sp > 1$. The equation is given by:
$$\begin{cases} (-\Delta)^s_{p,\Omega} u_s = f_s & \text{in } \Omega, \\ \gamma^s_0(u_s) = g_s & \text{on } \partial\Omega, \end{cases}$$
where $\gamma^s_0$ is the trace operator mapping $W^{s,p}(\Omega)$ to the fractional trace space $W^{s-1/p,p}(\partial\Omega)$. The operator is normalized by a constant $C_{d,p,s}$ to ensure consistent asymptotic behavior as $s \to 1^-$ (recovering the classical $p$-Laplacian $-\Delta_p$) and as $s \to 0^+$.

The primary objective is to establish the optimal stability (strong convergence in the natural energy space) of the weak solutions $u_s$ to the limit solution $u_1$ as $s \to 1^-$. A secondary objective is to analyze the asymptotic stability of the associated Dirichlet eigenpairs $(\lambda_s, \phi_s)$.

Methodology and Framework
The analysis relies on several key technical components:

1. Asymptotic Convergence Definitions: Since the underlying spaces $W^{s,p}_0(\Omega)$ and trace spaces vary with $s$, standard weak convergence is insufficient. The author introduces:

   • Asymptotic Weak Convergence: A sequence of functionals $(f_s)$ converges asymptotically weakly to $f_1$ if it remains bounded and satisfies a weak-strong convergence property against sequences converging strongly in $L^p$.
   • Asymptotic Strong Convergence: Boundary data $(g_s)$ converges strongly to $g_1$ in the appropriate trace norms.

2. Robust Interpolation and Compactness: The proof utilizes robust interpolation inequalities (borrowed from Foghem [19]) to establish compactness of the solution sequence $(u_s)$ in $W^{\eta,p}(\Omega)$ for $1/p < \eta < 1$. This compactness is crucial for handling the boundary behavior, which is noted as a significant difficulty.

3. Trace Space Asymptotics: A substantial portion of the work is dedicated to the asymptotic properties of the trace spaces $W^{s-1/p,p}(\partial\Omega)$. The author proves that these spaces converge to the classical trace space $W^{1-1/p,p}(\partial\Omega)$ as $s \to 1^-$. This includes establishing the asymptotic compactness of bounded families in these trace spaces and providing a characterization of $W^{1-1/p,p}(\partial\Omega)$ via the liminf of norms.

4. Robust Poincaré-Friedrichs Inequalities: The paper establishes several variants of robust Poincaré-type inequalities that hold uniformly for $s$ close to 1. These inequalities are essential for proving the coercivity of the energy functionals and the well-posedness of the problem.

5. $\Gamma$-Convergence: The energy functionals associated with the Dirichlet problem are shown to $\Gamma$-converge in the strong topology of $W^{\eta,p}(\Omega)$. This provides a variational framework for the convergence of minimizers.

Key Contributions and Results

• Theorem 1.1 (Optimal Stability of Solutions): Under the assumptions of asymptotic weak convergence for the source term $f_s$ and asymptotic strong convergence for the boundary data $g_s$, the sequence of weak solutions $u_s$ converges strongly to the unique weak solution $u_1$ of the limit problem in the optimal sense:
  $$\lim_{s \to 1^-} \|u_s - u_1\|_{W^{s,p}(\Omega)} = 0.$$
  This result improves upon previous findings (e.g., Foghem [19]) which only established convergence in $W^{\eta,p}(\Omega)$ for $\eta < 1$.

• Theorem 1.2 (Stability of Eigenpairs): For the normalized Dirichlet eigenpairs $(\lambda_s, \phi_s)$ of the regional fractional $p$-Laplacian, there exists a subsequence $s_n \to 1^-$ such that the eigenvalues converge ($\lambda_{s_n} \to \lambda_1$) and the eigenfunctions converge strongly in the energy norm:
  $$\lim_{n \to \infty} \|\phi_{s_n} - \phi_1\|_{W^{s_n,p}(\Omega)} = 0.$$

• Trace Space Characterization (Theorems 3.3 & 3.4): The paper establishes the asymptotic compactness of the trace spaces $W^{s-1/p,p}(\partial\Omega)$ and characterizes the limit space $W^{1-1/p,p}(\partial\Omega)$ via the boundedness of the liminf of the fractional norms.

• Robust Inequalities (Section 4): The author derives robust Poincaré and Friedrichs inequalities that hold uniformly for $s \in (s_0, 1)$, including cases with partial boundary conditions on a subset $\Gamma_0 \subset \partial\Omega$ with positive Hausdorff measure.

Significance and Claims
The paper claims that the convergence of weak solutions for the Dirichlet problem associated with the regional fractional $p$-Laplacian is a new result. The significance lies in achieving optimal convergence (strong convergence in the $W^{s,p}$ norm) rather than the weaker convergence in $W^{\eta,p}$ for $\eta < 1$ established in prior literature.

The author emphasizes that the main difficulty in this analysis stems from the behavior of weak solutions near the boundary $\partial\Omega$. By establishing the robustness of the trace operator and the asymptotic properties of the trace spaces, the paper overcomes this obstacle. The results serve as a bridge between nonlocal variational problems and classical local problems, providing a rigorous foundation for the limit $s \to 1^-$ in the context of regional operators with inhomogeneous boundary conditions. The work also contributes to the literature on nonlocal-to-local asymptotics by extending results from the full fractional $p$-Laplacian (defined on $\mathbb{R}^d$) to the regional case (defined on a bounded domain).