The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations.It is a brief presentation given by the author at a round table discussion.The s...The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations.It is a brief presentation given by the author at a round table discussion.The second part is a supplement of various details provided by several outstanding researchers on subjects.展开更多
This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating...This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients. We show that the(d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L_ε(u_ε) = 0 in a ball in Rdare bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u_ε by solutions of the homogenized equation.展开更多
For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C...For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.展开更多
In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known...In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known conjecture due to Polya, its connections to Weyl's asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.展开更多
The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems.This approach is based on the monotonicity of several variational integrals,the Fede...The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems.This approach is based on the monotonicity of several variational integrals,the Federer-Almgren dimension reduction and stratification theorems,and some simple PDE arguments.展开更多
Professor Andrew J. Majda is one of the most influential mathematicians of our time. Hiscontributions to theoretical partial differential equations and many applied areas are seminaland fundamental. In the course of h...Professor Andrew J. Majda is one of the most influential mathematicians of our time. Hiscontributions to theoretical partial differential equations and many applied areas are seminaland fundamental. In the course of his phenomenal scientific career, Professor Majda has writtenmore than 300 papers and 7 books, which have been cited more than 10000 times.展开更多
This is in the sequel of authors’ paper [Lin, F. H., Pan, X. B. and Wang, C. Y.,Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6),2012, 833–888] in which the authors had set up...This is in the sequel of authors’ paper [Lin, F. H., Pan, X. B. and Wang, C. Y.,Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6),2012, 833–888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition.The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg’s work(in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.展开更多
We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type estimates and boundary ...We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type estimates and boundary Harnack inequalities on a class of nodal domains.展开更多
In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,a...In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,and the constant c 0 is independent of the shape ofΩ.Here,l^(1)_(m)(Ω)denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition ofΩ.展开更多
基金This article is initiated during a round table discussion at International Center for Mathematics in Southern University of Science and Technology(SUSTech)held in the January 2020.I am grateful to the wonderful organization and hospitality provided by Professors Tao Tang,Xiaoming Wang and Linlin Su.In the course of the preparation of this work,several colleagues have contributed to the scientific part of this review article.Here I would like to thank particularly:Chenjie Fan from University of Chicago,Jun Geng from Lanzhou University,Yanlin Liu from The Chinese Academy of Sciences,Xinan Ma from University of Sciences and Technology of China,Shuang Miao and Kelei Wang from Wuhan UniversityPing Zhang from The Chinese Academy of Sciences and Zhifei Zhang from Beijing University.Their enthusiasm and passion have greatly improved the presentation and the quality of this work.
文摘The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations.It is a brief presentation given by the author at a round table discussion.The second part is a supplement of various details provided by several outstanding researchers on subjects.
基金supported in part by NSF(Grant No.DMS-1501000)supported in part by NSF(Grant No.DMS-1600520)
文摘This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients. We show that the(d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L_ε(u_ε) = 0 in a ball in Rdare bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u_ε by solutions of the homogenized equation.
基金supported by the National Science Foundations (Nos. 0700517, 1001115)
文摘For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.
文摘In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known conjecture due to Polya, its connections to Weyl's asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.
基金Project supported by the National Science Foundation (No.DMS 0700517)
文摘The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems.This approach is based on the monotonicity of several variational integrals,the Federer-Almgren dimension reduction and stratification theorems,and some simple PDE arguments.
文摘Professor Andrew J. Majda is one of the most influential mathematicians of our time. Hiscontributions to theoretical partial differential equations and many applied areas are seminaland fundamental. In the course of his phenomenal scientific career, Professor Majda has writtenmore than 300 papers and 7 books, which have been cited more than 10000 times.
文摘This is in the sequel of authors’ paper [Lin, F. H., Pan, X. B. and Wang, C. Y.,Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6),2012, 833–888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition.The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg’s work(in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.
基金supported by National Science Foundation of USA(Grant No.DMS 1955249)。
文摘We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type estimates and boundary Harnack inequalities on a class of nodal domains.
基金supported by National Science Foundation of USA(Grant Nos.DMS1501000 and DMS-1955249)。
文摘In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,and the constant c 0 is independent of the shape ofΩ.Here,l^(1)_(m)(Ω)denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition ofΩ.
