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 $u_s \in W^{s,p}(\Omega)$ weakly satisfies $(-\Delta)_{p, \Omega}^s u_s = f_s $ in $\Omega$ and $ \gamma^s_0(u_s) = g_s$ on $\partial\Omega,$ with $\frac{1}{p} < s \leq 1$ and $\Omega \subset \mathbb{R}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s \to 1^-$, we establish that $\| u_s - u_1 \|_{W^{\eta,p}(\Omega)} \xrightarrow{s \to 1^-} 0 $ for all $0 \leq \eta < 1$.