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

We establish the optimal stability of Dirichlet boundary value problem for the regional (fractional) $p$-Laplacian $(-\mathrm{\Delta }{)}_{p,\mathrm{\Omega }}^s$ with $0<s\le 1$, $\frac{1}{s}<p<\mathrm{\infty }$ and $\mathrm{\Omega }\subset {\mathbb{R}}^d$ bounded Lipschitz. More precisely, if $u_s\in W^{s,p}(\mathrm{\Omega })$ satisfies $(-\mathrm{\Delta }{)}_{p,\mathrm{\Omega }}^s{u}_s=f_s$ in $\mathrm{\Omega }$ and $u_s=g_s$ on $\mathrm{\partial }\mathrm{\Omega }$, then under appropriate condition on the date $f_s$ and $g_s$ we show that $\mathrm{\parallel }{u}_s-u_1{\mathrm{\parallel }}_{W^{s,p}(\mathrm{\Omega })}\to 0\text{as }s\to 1^{-}\mathrm{.}$ We also obtain an analogous optimal stability of the normalized Dirichlet eigenpairs $({\lambda }_s,{\phi }_s)$ associated with $(-\mathrm{\Delta }{)}_{p,\mathrm{\Omega }}^s$.