Quantization for an evolution equation with critical exponential growth on a closed Riemann surface

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.     

f-Laplace非线性方程的梯度估计和Liouville定理

设(M,g,e~(-f)dv_g)是n维完备光滑的度量测度空间.考虑以下非线性椭圆方程△_f~u+hu~α=0,1α(n+m)/(n+m-2)(n+m≥4)和非线性抛物方程(△_f-?/?t)u+hu~α=0,α0正解的梯度估计.对于经典的Laplace情形,Li (Li J. Gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds [J]. J Funct Anal,1991, 100:233-256.)证明了正解的梯度估计和Liouville定理.在本文中,对于上述的f-Laplace方程,作者将推导出相应的结果.     

Regularity and Compactness of Stationary Map-Varifold Pairs*

The authors introduce the conception of stationary map-varifold pairs and prove a compactness result.As applications,they analyse the asymptotic structure of the pseudo tangent map of stationary harmonic maps.For stationary pair,they also get a strong convergence criterion about the map part and introduce the stratification of the singular set.     

On the pyrolysis of aryl cinnamates a convenient entry to unsymmetrical stilbenes

Symmetrical and unsymmetrical stilbenes can be prepared by pyrolysis of the corresponding aryl cinnamates. The mechanism of product formation is discussed.