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

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- by Tianling Jin and Jingang Xiong PDF
- Trans. Amer. Math. Soc.
**367**(2015), 6751-6770 Request permission

## 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 \[ \|u\|^2_{L^q(\partial M)}\leq S\left (\int _{M}|\nabla _g u|^2 \mathrm {d} v_g+ \frac {n-2}{2}\int _{\partial M}h_g u^2 \mathrm {d} s_g\right )+A\|u\|^2_{L^r(\partial M)} \] 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 $\partial M$, $\mathrm {d} v_g$ is the volume form of $(M,g)$, $\mathrm {d} s_g$ 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.## References

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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
- 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).
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 6751-6770 - MSC (2010): Primary 46E35
- DOI: https://doi.org/10.1090/S0002-9947-2014-06429-5
- MathSciNet review: 3356953