Robust interpolation inequalities via Chebyshev-type integral inequalities

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. Our primary motivation for establishing these robust interpolation inequalities stems from the study of the asymptotic nonlocal-to-local stability of weak solutions to the boundary Dirichlet problem associated with the regional fractional $p$-Laplacian. More precisely, if $u_s\in W^{s,p}(\hat{I}\text{c◯})$ weakly satisfies $(-\hat{I}”{)}_{p,\hat{I}\text{c◯}}^s{u}_s=f_s$ in $\hat{I}\text{c◯}$ and $ γ^s_0(u_s) = g_s$ on $\partial \hat{I}\text{c◯},$ with $\frac{1}{p}<s\le 1$ and $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s\to 1^{-}$, we establish that $\Vert u_s-u_1{\Vert }_{W^{\hat{I}\cdot ,p}(\hat{I}\text{c◯})}\stackrel{s\to 1^{-}}{\to }0$ for all $0\le \hat{I}\cdot <1$.

59 pages, 1 figure