The seminar usually take place on **Thursday Morning**.
We shall use Zoom or Tencent meeting (TBA)

**Guohuan Qiu**(Academy of Mathematics and Systems Science)**Liming Sun**(Academy of Mathematics and Systems Science)

Date | Speaker | Title |
---|---|---|

Feb 23rd, 9-10am | Shibing Chen (USTC) | The \(L^p\)-Brunn-Minkowski conjecture for \(0<p<1\) |

Mar 1st (Wed), 10-11am | Qidi Zhang (UBC) | Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two |

Mar 16th, 9-10am | Ke Wu (Wuhan) | Monotonicity of the second Neumann eigenfunction in symmetry domain |

Mar 16th, 10-11am | Dongmeng Xi (Shanghai) | On the log-Brunn-Minkowski inequality |

Mar 23rd | Xiaodong Wang (MSU) | A simpler proof of the Frank-Lieb inequality on the Heisenberg group |

Mar 30th,10:30-11:30am on-site | Cheng Zhang (Tsinghua) | Sharp Lp estimates and size of nodal sets of generalized Steklov eigenfunctions |

Apr 6th | Letian Chen (JHU) | Mean curvature flow coming out of cones |

Apr 13th, 9-10am | Zhuolun Yang (Brown) | Some recent progress on the gradient estimates for the insulated conductivity problem. |

Apr 20th, 5-6pm | Pavol Quittner (Comenius) | Liouville theorems for superlinear parabolic problems. |

Apr 26th (Wed),10-11am | Quoc Anh Ngo (VNU) | Integral equations with bubble-type kernel via the method of moving spheres |

May 4th | Yiming Zhao (Syracuse) | The Minkowski problem in Gaussian probability space |

May 11th | Haixia Chen (Central China Normal University) | Nonlinear \(Schr\ddot{o}dinger\) equations on metric graphs. |

This talk will introduce some topics related to the study of nonlinear SchrÃ¶dinger equation on metric graphs, and then show the existence result of concentrated solutions to NLS equations on compact graphs and a brief proof. This work is in cooperation with S. Dovetta, A. Pistoia and E. Serra.

The classical Minkowski problem, which asks for the characterization of surface area measure in Euclidean space with Lebesgue measure, largely motivated the development of elliptic PDEs throughout the last century. In this talk, we will discuss the corresponding problem in Gaussian probability space. Lack of homogeneity and translation invariance make this problem fundamentally different from the classical problem. We will discuss existence results (in all dimensions) as well as uniqueness results (in dimension 2). This is based on joint works with Yong Huang, Dongmeng Xi, and with Shibing Chen, Shengnan Hu, Weiru Liu.

In this talk, I will describe a moving sphere approach to a class of integral equations with bubble-type kernel. This class of equations comes from higher-order elliptic equations on the standard sphere via the steoreographic projection. As a by-product, we obtain a Liouville-type result for higher-order equations on the sphere. This is joint work with Quynh Ngoc Thi Le and Tien-Tai Nguyen.

We first consider positive classical solutions to a semilinear heat equation and discuss the associated Liouville-type theorems and their consequences on a priori estimates of solutions (in particular, blow-up rate estimates). Then we discuss the same issues for several related problems: Sign-changing solutions of the semilinear heat equation, parabolic systems with superlinear gradient nonlinearities (not necessarily homogeneous) and the heat equation with nonlinear boundary conditions.

In this talk, I will describe an elliptic PDE that models electric conduction, and the electric field concentration phenomenon between closely spaced inclusions of high contrast. In the first part, I will present some results on the insulated conductivity problem when the current-electric field relation is linear (jointly with Hongjie Dong and Yanyan Li). We obtained an optimal gradient estimate in terms of the distance between inclusions. This solved one of the major open problems in this area. In the second part, I will present a recent work on the insulated conductivity problem when the current-electric field relation is the power law (jointly with Hongjie Dong and Hanye Zhu). We identify the optimal blow-up rate for the gradient in 2D, and prove some gradient estimates in any dimensions.

We study mean curvature flows (MCFs) coming out of cones. As cones are singular at the origin, the evolution is generally not unique. A special case of such flows is known as the self-expanders. We will construct many non-self-similar MCFs coming out of a given cone provided there are more than one (smooth) self-expanders asymptotic to the cone. We then discuss a classification problem of such flows. In particular, we show that, in low dimensions, and when the cones are not too complex, the above solutions are all one gets.

We prove sharp Lp estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their L2 norms on the boundary. We prove it by establishing Lp bounds for the harmonic extension operators as well as the spectral projection operators on the boundary. Moreover, we derive lower bounds on the size of nodal sets for a variation of the Steklov spectral problem. We consider a generalized version of the Steklov problem by adding a non-smooth potential on the boundary but some of our results are new even without potential. This is a joint work with X. Huang(Maryland), Y. Sire(Johns Hopkins) and X. Wang(Hunan U)

I will discuss the Hardy-Littlewood integral inequality with sharp constant on the Heisenberg goup proved by Frank and Lieb. I will outline a simpler proof which bypasses the sophisticated argument for existence of a minimizer and is based on the study of the 2nd variation of subcritical functionals. This is joint work with Fengbo Hang.

In 2012, Boroczky, Lutwak, Yang, and Zhang posed the log-Brunn-Minkowski conjecture, which represents an enhanced version of the classical Brunn-Minkowski inequality for convex bodies. We will recall some of its development and related studies in recent years, which includes one of our work on a nonsymetric log-Brunn-Minkowski inequality.

In 1974, Jeffrey Rauch proposed the hot spot conjecture for the second eigenfunctions of the Neumann eigenvalue problems. We will recall some of its development and related studies in recent years. We will also introduce our recent results concerning monotonicity properties of symmetric Neumann eigenfunctions of the Laplacian of convex planar domains with one line of symmetry.

Landau-Lifshitz-Gilbert equation (LLG) describes magnetization of ferromagnets. In this paper, we consider the singularity formation of LLG from R^2 to S^2. For any prescribed N distinct points q^{j} in R^2, j=1,2,â€¦,N, and T sufficiently small, there exists initial data u_0 such that the gradient of the solution u to LLG blows up at these N points at finite time t=T. This is joint work with Juncheng Wei and Yifu Zhou.

The Lp-Brunn-Minkowski conjecture for 0 < p < 1 states that the volume functional is concave when origin-symmetric convex bodies are Lp-added, where 0 < p < 1. This conjecture has garnered significant attention in the last decade as it is a stronger form of the classical Brunn-Minkowski inequality. In this talk, we will introduce the conjecture and discuss the recent progress made in the last few years.