李翀(中國科學院數學與系統科學研究院副研究員)

李翀，副研究員。研究方向為非線性泛函分析、變分方法及臨界點理論、偏微分方程。

基本介紹

• 中文名：李翀
• 外文名：Chong Li 
• 職業：副研究員
• 主要成就：建立C^1光滑的Poincare-Hopf定理
  證明開集稠子集上的隱函式定理 
  給出Fucík譜(II)型區域存在性的一個充分條件 
  建立一類與Nehari流形有關的新型環繞定理 
• 代表作品：關於臨界點理論的幾個註記 

研究方向

非線性泛函分析、變分方法及臨界點理論、偏微分方程

代表論文

Chong Li, Shujie Li,The Fučík spectrum of Schrödinger operator and the existence of four solutions of Schrödinger equations with jumping nonlinearities.

Chong Li, Shujie Li, Gaps of consecutive eigenvalues of Laplace operator and the existence of multiple solutions for superlinear elliptic problem, Journal of Functional Analysis, Vo.271, 2016, 245-263.

Chong Li, Generalized Poincaré-Hopf theorem and application to nonlinear elliptic problem, Journal of Functional Analysis, Vo.267, 2014, 3783-3814.

Chong Li, Shujie Li, Zhaoli, Liu, Bifurcation surfaces stemming from the Fucik spectrum，Journal of Functional Analysis, Vo.263, Issue 12(2012), 4059-4080.

Chong Li, Shujie Li, Jiaquan Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems, Journal of Functional Analysis, Vo.221, 2005, No.2, 439-455.

Chong Li,Shujie Li, Zhaoli Liu, Jianzhong Pan, On the Fucik spectrum, Journal of Differential Equations, Vo.244, 2008, 2498-2528.

Chong Li, Shujie Li, Zhaoli Liu, Existence of type(II) region and convexity and concavity of potential functional corresponding jumping nonlinear problems, Calculus of Variations and Partial Differential Equations, Vo.32, 2008, 237-251.

Chong Li, The quantitative analysis on the Fucik spectrum and solvability of jumping nonlinear problems, Communications in Contemporary Mathematics, Vo.13, No.2(2011), 293-308.

Chong Li, Some remarks on the critical point theory, Topological Methods in Nonlinear Analysis, Vo.30, No.2, 2007.

Chong Li, The existence of solutions of elliptic equations with Neumann boundary condition for superlinear problems, Acta Mathematica Sinica, English Series, Dec, 2004, Vo.20, No.6, 965-976.

Chong Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlinear Analysis, 54(2003), 431-443.

Chong Li, Shujie Li, Multiple solutions and sign changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition, JMAA, 298(2004), No.1, 14-32.

李翀，李樹傑，關於臨界點理論的幾個註記，中國科學，Vo.46, No.5, 1-9, 2016.

Chong Li, Yanheng Ding, Shujie Li, Multiple solutions of nonlinear elliptic equations for oscillation problems, JMAA, Vo.303, 2005, 477-485.

Chong Li, Shibo Liu, Homology of saddle point reduction and applications to resonant elliptic systems, Nonlinear Analysis, Vo.81(2013), 236-246.

Li Ma, Chong Li, Zhao Lin, Monotone solutions to a class of elliptic and diffusion equations, Communication on Pure and Applied Analysis，Vo.6, 2007, No.1, 237-246.

Jing Zhang, Chong Li, Xiaoping, Xue, Nontrivial Solutions for a Class of Quasilinear Problems with Jumping Nonlinearities via a Co-homological Local Splitting, Nonlinear Analysis, Vo.75, Issue 13(2012), 4896–4903.

Aixia Qian, Chong Li, Infinitely Many Solutions for a Robin Boundary Value Problem, International Journal of Differential Equations, Vo. 2010, Article ID 548702, 9 pages.

主要工作

（1）建立C^1光滑的Poincare-Hopf定理；對無窮維形式在下述方面予以改進：（i）將有界Gromoll-Meyer對替換為泛函水平集對；（ii）去掉Morse不等式關於泛函水平集之間只有孤立臨界點的限制； (iii)拓廣著名的Marino-Prodi擾動定理；

（2）擺脫經典隱函式定理在開集上可微的限制，證明開集稠子集上的隱函式定理，這一進展導致兩方面突破，一是利用它建立全新的分支定理（包含經典的Crandall-Rabinbowitz分支定理），二是利用它得到弱光滑條件下局部Morse理論，特別是減弱著名數學家Helmut Hofer相關工作的泛函光滑性條件。

（3）研究高維情況下Fucík譜基本性質，解決人們長期關注的從Fucík譜分出分支曲面的問題。特別是給出Fucík譜(II)型區域存在性的一個充分條件；還證明了Conley同倫指標平凡與臨界群平凡之間的等價關係，這一等價關係對跳躍非線性問題可解性起至關重要的作用。

（4）建立一類與Nehari流形有關的新型環繞定理並發現超線性問題的第四個解與橢圓運算元譜間隙的緊密聯繫。

學術報告

7月20日上午，中科院數學與系統科學研究院研究員李樹傑、副研究員李翀在我院求知樓數學系會議室作了關於統計專業如何建設、專業教師如何開展高水平科學研究以及Morse理論的精彩報告。哈爾濱師範大學數學科學學院王玉文教授出席並主持了報告會，數學系全體教師及優秀學生代表參加了報告會。

報告結束後，李樹傑研究員、李翀副研究員就數學系專業建設，學生如何選擇繼續深造專業及教師現有的科研項目中存在的問題等方面與參會教師及學生進行了互動交流並作了精彩的回答。兩位研究員的講座，對我院統計專業的建設，對我院教師增強科研意識，提高科研水平具有十分重要的指導意義。