A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds

Abstract

Given $(M,g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $\rho$ be a defining function of $M$ and $\sigma \in (0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}({\rho }^{1-2\sigma },M)\hookrightarrow L^p(\partial M)$, where $p=\frac{2n}{n-2\sigma }$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u\in W^{1,2}({\rho }^{1-2\sigma },M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|^{p-2}u \,\ud s_{g}\Big|^{2/(p-1)}. $$ This inequality is sharp in the sense that ${\mu }^{-1}$ cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, ${\mu }^{-1}$ does not depend on $n$ and $\sigma$ only, but depends on the manifold.