非线性偏微分方程与几何分析研讨会

偏微分方程与微分几何是现代数学重要与活跃的分支。为提供一个相互交流、学习与合作的平台，我们定于2019年12月26日--30日 在浙江大学数学科学院举办“非线性偏微分方程与几何分析”研讨会。希望能借此共同探讨相关的前沿课题和最新的研究进展。

会议时间：2019年12月26日报到，12月27-29日会议报告，12月30日离会。

会议地点：浙江大学 紫金港圆正–启真酒店

会议主办单位：星空体育官方网站

组委会成员： 盛为民（浙江大学）             麻希南（中国科学技术大学）

                    张 希（中国科学技术大学）   李奇睿（浙江大学）

日程安排

报告摘要

(按报告人姓名音序排列)

报告人：陈传强 (宁波大学)

题目： The Neumann problems of some fully nonlinear elliptic PDEs

摘要： In this talk, we will introduce the Neumann problems of Hessian quotient equations and Special Lagrangian equations. These are joint works with Xinan Ma, Wei Wei and Dekai Zhang.

报告人：陈世炳 (中国科学技术大学)

题目：  Regularity of optimal transport map between hypercubes

摘要：  I will discuss our recent result on the global $C^{2,\alpha}$ regularity of optimal transport map between two hypercubes. The two dimensional case was previously proved by Yash Jhaveri using partial Legendre transform. Our approach is completely different and works for all dimensions.

报告人：韩小利 (清华大学)

题目： Stability of line bundle mean curvature flow

摘要：  I will talk the stability of line bundle mean curvature flow. That means the line bundle mean curvature flow exists for long time and converges to the deformed Hermitian Yang-Mills metric if the initial data is close to the deformed Hermitian Yang-Mills metric.

报告人：蒋飞达 (南京信息工程大学)

题目： Neumann problem for degenerate Monge-Ampere equation

摘要： In this talk, we will talk about the Neumann problem for degenerate Monge-Ampere equation. By establishing the global C⁰, C¹ and C² estimates, the existence and uniqueness of C^1,1 solutions to the problem are proved.

报告人：李东升 (西安交通大学)

题目： Interior $L^p$ Estimates of Partial Differential Equations and Systems

摘要：  We will talk about interior $L^p$ estimates for partial differential equations and systems. As for equations, we will mainly discuss $L^p$ estimates for elliptic partial differential equations. As for systems, we will mainly discuss $L^p$ estimates for non-stationary Stokes system. Since the key idea of the proofs is the same, we can put these two topics together.

报告人：李海刚 (北京师范大学)

题目：  Babuska Problem in Composite Materials and its Applications

摘要： In high-contrast composite materials, the stress concentration is a common phenomenon when inclusions are densely distributed and may possibly touch. It always causes damage initiation and even ultimate failure. This problem was proposed mathematically by Ivo Babuska, concerning the system of linear elasticity, modeled by a class of second order elliptic system of divergence form with discontinuous coefficients.

I will first review some of our results on upper and lower bounds of the gradients by developing an iteration technique with respect to the energy integral to overcome the essential difficulty from the lack of maximal principle for elliptic systems, then present our very recent results, improving the previous inequality results on the gradients’ estimates to asymptotics to close this problem and show the key role of the geometry of the inclusions played in such blow-up analysis.

报告人：刘佳堃 (澳大利亚卧龙岗大学)

题目：  Regularity of free boundaries in optimal transportation

摘要： In this talk, we introduce some recent regularity results of the free boundary in optimal transportation with the quadratic cost. Assuming the densities $f, g$ are bounded away from zero and infinity, and supported on convex domains $\Omega, \Omega^*$, respectively, our first result shows that the free boundaries are $C^{1,\alpha}$ inside the domains. Furthermore, when $f, g \in C^\alpha$, and $\Omega, \Omega^*$ are far apart, by adopting our recent results on boundary regularity of Monge-Ampere equations, we show that the free boundaries are $C^{2,\alpha}$. More recently, in dimension two we obtain higher order regularity without assuming the far-apart condition, which is a joint work with Shibing Chen and Xu-Jia Wang.

报告人：邱国寰 (香港中文大学)

题目： On degenerate case of prescribed curvature measure problems.

摘要： In this talk,  the C^{1,1} estimate of degenerate case of prescribed curvature measure problems will be considered.

One observation is that by a simple comparison principle we can reduce the key estimate into the corresponding estimate to a homogeneous mean curvature equation. Then by analysis the existence of homogeneous mean curvature equation, we can give a sufficient condition such that the prescribed function may touch zero somewhere.

报告人：孙俊 (武汉大学)

题目： Variational characterizations of invariant submanifolds in Kahler and Sasaki manifolds

摘要： In this talk, we will first review my joint work with Claudio Arezzo at ICTP on variational characterizations of holomorphic submanifolds in Kahler manifold. Then we will talk about our recent progress on related problems for invariant submanifolds in Sasaki manifold.

报告人：唐岚 (华中师范大学)

题目： Global C^{1,\alpha} regularity for convex solutions to the Dirichlet problems of the degenerate Monge-Ampere equations

摘要： We consider the Dirichlet problem s of the degenerate Monge-Ampere equations and we establish global C^{1,\alpha}regularity for convex solutions with optimal boundary conditions.

报告人：王克磊 (武汉大学)

题目： Regularity of transition layers in Allen-Cahn equation

摘要：  In this talk I will survey a regularity theory about transition layers of solutions to Allen-Cahn equation, from zeroth order regularity to second order one. Some applications of these regularity theories will also be discussed, including problems related to De Giorgi conjecture, the classification of finite Morse index solutions and construction of minimal surfaces by Allen-Cahn approximation.

报告人：汪徐家 (澳大利亚国立大学)

题目： Some curvature related problems

摘要： To be announced

报告人：夏超 (厦门大学)

题目：  Symmetrization with respect to mixed volumes

摘要：  It is well known that the Schwarz symmetrization of a function diminishes the Dirichlet integral. This is the so-called Polya-Szego Principle. An similar symmetrization with respect to quermassintegral, which has been introduced by Talenti and Tso, diminishes the Hessian type integral. In this talk I will introduce my recent joint work with Della Pietra and Gavitone, on new symmetrization with respect to mixed volume or anisotropic curvature integral.

报告人：熊革 (同济大学)

题目： On the applications and generalizations of the cone-volume functional

摘要： The cone-volume functional was originally introduced by E. Lutwak, D. Yang and G. Zhang (LYZ) in 2001 to attack the celebrated longstanding Schneider projection problem in convex geometry. It is closely related to the cone-volume measure of convex bodies and has strong applications to the reverse affine isoperimetric problem and the logarithmic Minkowski problem.

In this talk, we will first review the fundamental properties of the cone-volume functional and its applications to the Schneider projection problem. Then, we will talk on the solved LYZ conjecture for the cone-volume functional (including our work on this conjecture) and its applications to the logarithmic Minkowski problem. Then we will report our very recent results on its generalizations and applications, including the variational formula and the extreme problem on the mixed cone-volume functional.

This talk is based on the joint work with Hu Jiaqi, Lu Xinbao and Sun Qiang.

报告人：徐露 (湖南大学)

题目： Solutions to the equations from the conformal geometry

摘要： We solve the Gursky-Streets equations with uniform $C^{1,1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1,1}$ \emph{a priori estimates} for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time.

报告人：张德凯 (复旦大学)

题目：  On degenerate k-Hessian equations on compact Hermitian manifolds

摘要： We study the regularity problem of degenerate k-Hessian equations on closed Hermitian manifolds. We show the weak $C^{1,1}$ estimates of the solution. This generalizes a result of Dinew-Plis-X.W.Zhang in the Kahler case and also improves our previous result.

报告人：周显潮 (浙江工业大学)

题目：  Scalar curvatures in almost Hermitian geometry and some applications

摘要： In this talk, we will present two formulas on scalar curvatures of canonical Hermitian connections on an almost Hermitian manifold. Then we will show some inequalities of various total scalar curvatures and some characterization results. Finally we will talk about some applications. This is a joint work with Jixiang Fu.

报告人：朱苗苗 (上海交通大学)

题目： Regularity for critical elliptic systems and applications in geometric analysis

摘要：  In this talk, we shall discuss the regularity issue for some critical elliptic systems and explore their applications in some geometric analysis problems.

报告人：朱小华 (北京大学)

题目：  Variation, polytopes and KE metrics on Q-Fano varieties

摘要： In this talk, we introduce a variation method to study the Kaehler-Einstein metrics (possibly singular) on Q-Fano compatifications of reductive Lie groups. On such $Q$-Fano varieties, we can reduce the Euler-Langrange functional of KE equation to a reduced Ding-functional with a weighted Monge-Ampere measure for convex functions on the moment polytope. Then we will derive a similar criterion for the existence of KE metrics on the smooth compatifications proved by Delcroix and Li-Zhou-Zhu.