Abstract

We establish  robust log-convex interpolation inequalities within the scale of Gagliardo seminorms. We achieve this by  deriving some Chebyshev-type integral inequalities for general non-synchronous functions. An application for establishing these robust interpolation inequalities stems from the study of the asymptotic nonlocal-to-local optimal stability of weak solutions to the boundary  Dirichlet problem associated with the regional fractional $p$-Laplacian. More precisely, if $u_s\in W^{s,p}(\mathrm{\Omega })$ weakly satisfies $(-\mathrm{\Delta }{)}_{p,\mathrm{\Omega }}^s{u}_s=f_s$ in $\mathrm{\Omega }$ and ${\gamma }_0^s(u_s)=g_s$ on  $\partial \mathrm{\Omega },$  with   and  $\mathrm{\Omega }\subset {\mathrm{R}}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s\to 1^{-}$, we establish that  $\Vert u_s-u_1{\Vert }_{W^{s,p}(\mathrm{\Omega })}\stackrel{s\to 1^{-}}{\to }0$.

Zoom Link

https://us02web.zoom.us/j/87303952902

Organizer

Daniel Spector (NTNU)