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A sharp Sobolev trace inequality involving the mean curvature on Riemannian manifolds


Authors: Tianling Jin and Jingang Xiong
Journal: Trans. Amer. Math. Soc. 367 (2015), 6751-6770
MSC (2010): Primary 46E35
DOI: https://doi.org/10.1090/S0002-9947-2014-06429-5
Published electronically: November 12, 2014
MathSciNet review: 3356953
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Abstract: Let $ (M,g)$ be a smooth compact $ n$-dimensional Riemannian manifold with smooth boundary $ \partial M$ for $ n\ge 5$. We prove a trace inequality, that is

$\displaystyle \Vert u\Vert^2_{L^q(\partial M)}\leq S\left (\int _{M}\vert\nabla... ...{\partial M}h_g u^2\,\textup {d} s_g\right )+A\Vert u\Vert^2_{L^r(\partial M)} $

for all $ u\in H^1(M)$, where $ S=\frac {2}{n-2}\omega _n^{-1/(n-1)}$ with $ \omega _n$ the volume of the unit sphere in $ \mathbb{R}^n$, $ q=\frac {2(n-1)}{n-2}$, $ r=\frac {2(n-1)}{n}$, $ h_g$ is the mean curvature of $ \partial M$, $ \textup {d} v_g$ is the volume form of $ (M,g)$, $ \textup {d} s_g$ is the induced volume form on $ \partial M$, and $ A$ is a positive constant depending only on $ (M, g)$. This inequality is sharp in the sense that $ S$ cannot be replaced by any smaller constant, $ h$ in general cannot be replaced by any smooth function which is smaller than $ h$ at some point on $ \partial M$, and $ r$ cannot be replaced by any smaller number.

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Additional Information

Tianling Jin
Affiliation: Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: tj@math.uchicago.edu

Jingang Xiong
Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
Email: jxiong@math.pku.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06429-5
Received by editor(s): December 30, 2013
Received by editor(s) in revised form: March 3, 2014
Published electronically: November 12, 2014
Additional Notes: The second author was supported in part by the First Class Postdoctoral Science Foundation of China (No. 2012M520002).
Article copyright: © Copyright 2014 American Mathematical Society