Optimal stability of complement value problems for p-Lévy operators

We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-Lévy operators, $1<p<\mathrm{\infty }$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-\mathrm{\Delta }{)}_p^s$ with $0<s\le 1$. If $(-\mathrm{\Delta }{)}_p^s{u}_s=f_s$ in $\mathrm{\Omega }\subset {\mathbb{R}}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\mathrm{\Omega }$, $f_s$ and $g_s$, we show that $(u_s{)}_s$ strongly converges as $s\to 1^{-}$ in the the optimal, that is, $\mathrm{\parallel }{u}_s-u_1{\mathrm{\parallel }}_{W^{s,p}(\mathrm{\Omega })}\to 0$. \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.