## Selective Preprints

** Currently I shift my attention to flow problems and its application in image analysis. I will try to update this page more frequently. 7-31-2008. **
- Sharp local embedding
inequalities, ( with J. Li), Comm. Pure Appl. Math. 59(2006), pp122-144

A simple proof of Onofri inequality is given.

- Steady states for one dimensional curvature flows, ( with Y. Ni), Comm. Contemp. Math.

We initiate our general one dimensional flow in this paper. Affine flow is one special case in our study.

- One dimensional conformal metric flow, ( with Y. Ni), Adv. in Math.

Existence and exponential convergence of metrics for one-D flows are established here.

- One
dimensional conformal metric flow II, (with Y. Ni), Submitted in March, 2008. If I can not update this in one year, then you guess which journal we submitted the paper to.

We deal with flows involving fourth order derivative. No one knows yet why the filter involving high order derivative is more powerful.

- A sharp inequality and its
applications, (with S. Li), Comm. Contemp. Math.

Analog inequality to Hardy inequality is found. This yields even simple proof of Onofri inequality.

- Parabolic equations related to curve motion, (with W. Guan), to appear in J. Diff. Equs.

**Since 2012, I gradually cam back to pure mathematics **

- Two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents, (joint with J. Dou), to appear in Adv. Math

- Sharp Hardy-Littlewood -Sobolev inequality on the upper half space, (joint with J. Dou), preprint ready in July, 2012, to appear in IMRN

- Reversed Hardy-Littewood-Sobolev inequality, (joint with J. Dou), preprint ready in August, 2012, to appear in IMRN

- Prescribing integral curvature equation, preprint, July, 2014

- Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, (joint with Y. Han), to come, Jan, 2015

**
Results in data analysis 10-31-2010. **
- Microarray image enhancement via nonlinear diffusion, technical report (updated 10-12-2012: interest shifts back to pure mathematics. No further details will come out)