Optimal stability of complement value problems for p-Lévy operators

Abstract

We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-Lévy 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 $(-\hat{I}”{)}_p^s$ with $0<s\le 1$. If $(-\hat{I}”{)}_p^s{u}_s=f_s$ in $\hat{I}\text{c◯}\subset {\mathbb{R}}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\hat{I}\text{c◯}$, $f_s$ and $g_s$, we show that $(u_s{)}_s$ strongly converges as $s\to 1^{-}$ in the the optimal, that is, $\Vert u_s-u_1{\Vert }_{W^{s,p}(\hat{I}\text{c◯})}\to 0$. \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.

49 pages, no figures