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

Abstract

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

26 pages, No Figure