Optimal stability of complement value problems for p-L\'evy operators

TL;DR

This paper proves the optimal convergence of solutions to nonlocal integro-differential equations governed by symmetric p-Lévy operators as the nonlocal parameter approaches the local limit, including boundary condition considerations.

Contribution

It establishes the optimal convergence rates of solutions and trace spaces for p-Lévy operators, bridging nonlocal and local boundary value problems.

Findings

Solutions converge strongly in Sobolev spaces as the nonlocal parameter approaches the local case.

Nonlocal trace spaces converge to local trace spaces in an appropriate sense.

The results apply to fractional p-Laplacian operators with boundary conditions.

Abstract

We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-L\'evy operators, $1 < p < \infty$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-\Delta)^s_p$ with $0 < s \le 1$. If $(-\Delta)^s_p u_s = f_s $ in $\Omega \subset \mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\Omega$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \to 1^-$ in the the optimal, that is, $\|u_s - u_1\|_{W^{s,p}(\Omega)} \to 0$. \smallskip Another subsequent goal underpinning our approach is the robustness of the nonlocal trace spaces; specifically, we also show that the nonlocal trace spaces converge, in an appropriate sense, to the local trace space.