组织者：李岩岩、熊金钢
地点：后主楼1220
题目一：Free boundary problem of the porous medium equation in one dimension
汇报人：王昊天
时间：1月13、14，14：30—16：30
摘要：In this talk, we consider the waiting time of the porous medium equation in one dimension with specific initial distributions. First, we present some basic results. Then, we focus our discussion on how the flow behaves near the interface during the waiting time. Furthermore, we offer a new proof of the smoothness of the free boundary.
题目二： Regularity of proper harmonic maps between hyperbolic spaces
汇报人：牛婧茹
时间：1月16，14：30—16：30
摘要：Under the assumption that the boundary map has nonwhere-vanishing energy density, I will report some results about boundary regularity of proper harmonic maps. By extending the boundary mapping to construct suitable initial values, we can obtain a harmonic map by solving the parabolic harmonic map. Utilizing the relationship between this harmonic map and its initial values, we derive boundary regularity of harmonic map.
题目三：On the non-degeneracy and existence of nodal solutions to elliptic problem on the Heisenberg group
汇报人：强劼琛
时间：1月17，14：30—16：30
摘要：In this paper we first study the non-degeneracy of the solutions to the critical problem on the Heisenberg group. And as an application of this nondegeneracy, we study the existence of nodal solutions to the slightly sub-critical problem involving the sub-Laplacian on a bounded domain of Heisenberg group. We construct multi-peak solutions as the parameter is sufficiently small under certain assumptions. Moreover, the solutions have precisely two nodal domains.
题目四：Heat Flows with Prescribed Singularities from 3-dimensional Manifold
汇报人：季节
时间：1月19，14：30—15：30
摘要：We study singular heat flows from a 3-dimensional complete bounded Riemannian manifold without boundary into the hyperbolic space with prescribe singularity along a closed curve. We prove the existence and regularity of the singular heat flows. Furthermore, we prove that the singular heat flows converge to a singular harmonic map at an exponential rate.
题目五：Large solutions of elliptic equations with fully nonlinear conformally invariant operators
汇报人：麻悦潇
时间：1月19，15：30—16：30
摘要：Inspired by recent studies on the fully nonlinear Lowner-Nirenberg problem, we investigate large solutions of a class of fully nonlinear elliptic equations involving the conformal Hessian in smooth bounded domains. We establish existence, non-existence and asymptotic behavior of viscosity solutions which blows up at the boundaries. The viscosity solutions are proved to be Lipschitz continuous but they can not be C^1 when the boundaries are disconnected.
