椭圆型偏微分方程及相关专题讲习（I）
Some Recent Results on Conformally Invariant Equations
授课人: 李岩岩 教授
时 间：27/12/23--12/01/24，
每周三、五 9：30--11：30
地 点：北京师范大学后主楼 1124
（1.5 在1220）
摘要：In this mini-course, we will present some recent advancements on conformally invariant nonlinear elliptic equations. This includes findings related to Liouville-type theorems, the analysis of isolated singularities in solutions, as well as the investigation into the existence and nonexistence of smooth solutions. Our course materials will be primarily drawn from the following two papers: [1] Baozhi Chu, Yanyan Li, and Zongyuan Li, Liouville theorems for conformally invariant fully nonlinear equations. I, arXiv:2311.07542, Nov 2023. [2] Yanyan Li, Luc Nguyen, and Jingang Xiong, Regularity of viscosity solutions of the $\sigma_k$-Loewner-Nirenberg problem, Proc. Lond. Math. Soc. (3) 127 (2023), no. 1, 1- 34. |
授课人简介：李岩岩，美国罗格斯大学Distinguished Professor、非线性分析中心主任。在非线性分析和偏微分方程领域做出了杰出的研究成果， - 2002年受邀在国际数学家大会（ICM）上做45分钟报告 - 连续获美国NSF资助——1989年至2026年，两次为5年期特别资助 - 现担任国际重要期刊Adv. Math, CVPDE, IMRN等编委 |
椭圆型偏微分方程及相关专题讲习（II）
Some Progress on Hot Spots Conjecture on Triangles and Symmetry Domains
授课人: 姚若飞 (华南理工大学)
时 间：2024 年1月15--17日9：30—11：30
地 点：北京师范大学后主楼 1223
摘要：In this talk, we will present some results related to Hot Spots Conjecture, which states that the second Neumann eigenfunctions attain their global extrema only on the boundary of the domain. The focus is primarily on the properties of the second Neumann eigenfunction, denoted as ${u}$, within specific planar domains, including arbitrary triangles and certain symmetry domains. For triangles, we establish the uniqueness (if exists) of non-vertex critical point, the sufficient and necessary condition for the existence of the non-vertex critical point, and the exactly location of global extrema. This confirms the original theorem and Conjecture 13.6 posed by Judge and Mondal in [Ann. Math., 2020], and successfully achieves the objective of Polymath 7 research thread 1 by Terence Tao. Additionally, some monotonicity results are also presented for some symmetry domains. The proof is based on continuity methods, eigenvalue inequalities, and the moving plane method. Notably, a portion of the content is taken from the preprint titled ``Uniqueness of Critical Points of the Second Neumann Eigenfunctions on Triangles" (arXiv:2311.12659, Nov 2023). |
授课人简介：姚若飞, 博士毕业于西安交通大学，曾在中南大学做博士后，现为华南理工大学数学学院副教授。主要研究兴趣是椭圆方程的定性性质, 包括半线性椭圆方程正解的对称性和单调性，以及Neumann边值的第二特征函数的定性性质。在CVPDE、Nonlinearity、JDE等数学期刊发表SCI论文多篇。 |
椭圆型偏微分方程及相关专题讲习（III）
Problems Related to The Torsion Function
授课人: 李沁峰 (湖南大学)
时 间：2024 年1月11日9：30—11：30
地 点：北京师范大学后主楼 1124
摘要：In the first lecture, we study the classical torsion problem on convex domains, focusing on determining the location of maximal gradient of torsion function, which physically means maximal stress of material. A long-standing belief, which is the modified Saint-Venant conjecture, states that for an evenly symmetric convex domain, points of maximal stress must be either on the shorter axis or location of minimal curvature. We disprove this folklore via domain variational approach and harmonic extension argument. Nevertheless, we prove that the original Saint-Venant’s conjecture is valid if the domain is sufficiently narrow. Moreover, for triangles and rectangles, we can exactly determine the position at which maximal stress occurs, and we also prove uniqueness and nondegeneracy of critical points on each side, via moving plane method and nodal line analysis. Open questions will be given along the talk. |
Rigidity and Stability of Serrin’s Over-Determined System
授课人: 李沁峰 (湖南大学)
时 间：2024 年1月12日13：30—15：30
地 点：北京师范大学后主楼 1124
摘要：In the second lecture, we review the proof of soap bubble theorem and rigidity of Serrin’s overdetermined system, which corresponds to uniqueness of critical shape to perimeter functional and classical torsional energy under volume constraints. Then we will introduce the Neumann and Robin torsion problem, proving the corresponding shape functional inequalities and rigidity results. We also discuss some stability problems on critical domains. Open questions will be given along the talk. |
An Introduction to Boundary Mean Zero Laplacian Eigenvalue Theory
授课人: 李沁峰 (湖南大学)
时 间：2024 年1月13日14：00—16：00
地 点：北京师范大学后主楼 1124
摘要：In the third lecture, I will first review some basic results for Dirichlet and Neumann Laplacian eigenvalue theory, particular the minimax principle, Weyl law, nodal domain theorem, eigenvalue comparison results, etc. Then I will introduce a new eigenvalue problem, and report the background, recent research progress and open questions. |
授课人简介：李沁峰，本科毕业于南开大学，博士毕业于普渡大学，曾在德州大学圣安东尼奥分校做博士后，现为湖南大学数学系副教授。主要研究方向为几何测度论及其在几何变分问题中的应用、椭圆特征值问题、几何和物理中的半线性椭圆方程。代表作发表在JMPA、中国科学（数学）英文版、Math. Z.、IMRN、IUMJ、CVPDE、PAMS等期刊上。 |
