1        Information

•   Purpose: This event gathers some top scholars to discuss nonlinear analysis progress and geometric applications, fostering academic exchange and cooperation.

•   Organizing Committee

 Liming Sun (Chinese Academy of Sciences)

 Yi Zhang (Chinese Academy of Sciences) 

 Yuan Zhou (Beijing Normal University)

•   Contact: Yi Zhang (yzhang@amss.ac.cn).

•   Place: N202, South Building,

       Academy of Mathematics and Systems Science, CAS,

       55 Zhongguancun East Road,

       Haidian District, Beijing, China.

•   Accommodation: Liaoning International Hotel.

•   Meal: Wu-Ke Restaurant

2        Schedule

3        Titile and abstract

•     – Thierry De Pauw (Westlake University)

–  Title: Partial regularity of almost minimizing rectifiable Z-chains in infinite dimensional Hilbert space

–  Abstract: I will present a proof of partial regularity valid in infinite dimensional spaces. It is based on moments calculations introduced by D. Preiss and the epiperimetric inequality introduced by E.R. Reifenberg. Joint work with R. Züst.

•     – Hongjie Dong (Brown University)

–  Title: Recent results on Hopf type lemma (Online)

–  Abstract: The classical Hopf lemma asserts that if a positive harmonic function in a sufficiently smooth domain vanishes at a boundary point, then its inner normal derivative at that point is strictly positive. This foundational result extends to elliptic equations in non-divergence form with measurable coefficients, provided the domain is C^1,α or C^1,Dini. In this talk, I will discuss some recent related work for divergence form equations and double divergence form equations (adjoint equations for non-divergence form equations). Some applications will also be mentioned.

•     – Nicola Fusco (Universitá degli Studi di Napoli Federico II)

–  Title: Consistency for the surface diffusion flow in three dimensions

–  Abstract: We will discuss the flat flow solution for the surface diffusion equation via a discrete minimizing movements scheme proposed in1994 in a celebrated paper by J.W. Cahn and J.E. Taylor.

We will show that in dimension three the scheme converges to the unique smooth solution of the equation, provided the initial set is sufficiently regular.

•     – Changfeng Gui (University of Macau)

–  Title: On a classification of steady solutions to two-dimensional Euler equations

–  Abstract: In this talk, I shall provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. A further classification of this type of solutions will also be discussed. As consequences, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines.

This talk is based on joint works with David Ruiz, Chunjing Xie and Huan Xu.

•     – Yeyao Hu (Central South University)

–  Title: Mean field type equations and the applications in Trudinger-Moser type inequalities

–  Abstract: In this talk, we will first review the resolution of the Chang-Yang conjecture on a sharp Trudinger-Moser inequality on the two-dimensional sphere. Then higher dimensional analogues will be presented. We will also mention some new inequalities involving the second order momentum in the spirit of the Szego’s limit theorem. The talk is based on joint works with Changfeng Gui (University of Macau) and Weihong Xie (Central South University).

•     – Renjin Jiang (Capital Normal University)

–  Title: Doubling index, frequency function and their applications

–  Abstract: The frequency function, observed by Almgren, together with the doubling index, has played important roles in the study of nodal sets and unique continuation. In this talk, we shall review recent developments of these tools, and their applications to Hausdorff measure of nodal sets and decay estimate of solutions to Schrödinger equations.

•     – Yong Liu (Beijing Technology and Business University)

–  Title: From KP lumps to 3D traveling water waves

–  Abstract: We consider gravity-capillary water waves in 3D. Based on its relation with the classical KP equation, we show that each nondegenerated lump type solutions of the KP equation gives rise to a family of traveling water waves.

•     – Hayk Mikayelyan (University of Nottingham Ningbo China)

–  Title: A missing Euler Lagrange condition for Mumford-Shah functional

–  Abstract: Some old results about the asymptotics of the Mumford-Shah minimizers near the crack-tip, as well as the curvature will be presented, which leads to new Euler-Lagrange condition. We also develop a new numerical method to compute the minimizers in 2D, which is very accurate near the crack-tip.

In 3D we construct a continuous family of stationary solutions which are not global minimizers. We make a conjecture that the dependence of the function on the point running over the crack-front is related to the behavior of the co-normal tangent at the crack-front. (joint work with John Andersson, Antoine Lemenant and Zhilin Li)

•     – Guohuan Qiu (Chinese Academy of Sciences)

–  Title: A priori interior estimates for special Lagrangian curvature equations

–  Abstract: We establish a priori interior curvature estimates for the special Lagrangian curvature equations in both the critical phase and convex cases. The supercritical case, however, is distinct from the special Lagrangian equations. In dimension two, we observe that this curvature equation is equivalent to the equation arising in the optimal transportation problem with a "relative heat cost" function, as discussed in Brenier’s paper. When 0 < Θ < π/2 (supercritical phase), the equation violates the MaTrudinger-Wang condition. However, Loeper’s counterexample for general optimal transport problems does not directly apply here, as this concerns a specific optimal transport problem with fixed density functions. Moreover, the interior gradient estimates for this curvature equation are simpler than those for the special Lagrangian equations. We have demonstrated that these gradient estimates also hold for subcritical phases. It is worth noting that for the special Lagrangian equation, particularly in subcritical phases, the interior gradient estimate remains an open problem. This is joint work with Xingchen Zhou.

•     – Bin Zhou (Peking University)

–  Title : Hölder regularity of the complex Monge-Ampere equation

–  Abstract: In this talk, we discuss the global Hölder regularity of solutions to Dirichlet problem for the complex Monge-Ampere equation with L^pvdensity and Hölder boundary value.