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A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds (2205.07272v1)

Published 15 May 2022 in math.AP

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-Poincar\'e type trace inequality corresponding to the embedding of $W{1,2}(\rho{1-2 \sigma}, M) \hookrightarrow L{p}(\partial M)$, where $p=\frac{2 n}{n-2 \sigma}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W{1,2}(\rho{1-2\sigma}, M)$, $$ \Big(\int_{\partial M}|u|{p} \,\ud s_{g}\Big){2/p} \leq \mu{-1} \int_{M} \rho{1-2 \sigma}|\nabla_{g} u|{2} \,\ud v_{g}+B \Big|\int_{\partial M} |u|{p-2}u \,\ud s_{g}\Big|{2/(p-1)}. $$ 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.

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