Publications (42)
Large asymptotics for minimal partitions of the Dirichlet eigenvalue
Zhiyuan Geng, Fanghua Lin
In this paper, we study large asymptotics of the minimal -partition problem for Dirichlet eigenvalue. For any smooth domain such that , we…
Remarks on the heat flow of harmonic maps into CAT(0)-spaces
Fanghua Lin, Changyou Wang
In this paper, we present an alternate, elementary proof of the local Lipschitz regularity of the suitable weak solution of heat flow of harmonic maps into CAT(0)-metric spaces, wh…
Regularity and existence of global solutions to the Ericksen-Leslie system in
Jinrui Huang, Fanghua Lin, Changyou Wang
In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establi…
Global Existence for Two Dimensional Incompressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity
Xianpeng Hu, Fanghua Lin
The existence of global-in-time classical solutions to the Cauchy problem of incompressible Magnetohydrodynamic flows with zero magnetic diffusivity is considered in two dimensions…
Isotropic-Nematic Phase Transition and Liquid Crystal Droplets
Fanghua Lin, Changyou Wang
Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the…
Nodal Sets and Doubling Conditions in Elliptic Homogenization
Fanghua Lin, Zhongwei Shen
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 $\{ \mathcal{…
Regularity for Shape Optimizers: The Degenerate Case
Dennis Kriventsov, Fanghua Lin
We consider minimizers of \[ F(λ_1(Ω),\ldots,λ_N(Ω)) + |Ω|, \] where is a function nondecreasing in each parameter, and is the -th Dirichlet eigenvalue of $΅
Space of ancient caloric functions on some manifolds beyond volume doubling
Fanghua Lin, Hongbing Qiu, Jun Sun +1
Under a condition that breaks the volume doubling barrier, we obtain a time polynomial structure result on the space of ancient caloric functions with polynomial growth on manifold…
Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions
Yuan Cai, Zhen Lei, Fanghua Lin +1
This paper studies the inviscid limit of the two-dimensional incompressible viscoelasticity, which is a system coupling a Navier-Stokes equation with a transport equation for the d…
Pólya's conjecture up to -loss and quantitative estimates for the remainder of Weyl's law
Renjin Jiang, Fanghua Lin
Let be a bounded Lipschitz domain. For any we show that for any Dirichlet eigenvalue , it holds \begin{align*} k&\le (1+ε…
Liquid crystals and topological vorticity: smoothness of mild solutions
Fanghua Lin, Yannick Sire, Yantao Wu +1
We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffus…
Harmonic maps in connection of phase transitions with higher dimensional potential wells
Fanghua Lin, Changyou Wang
This is in the sequel of authors' paper \cite{LPW} in which we had set up a program to verify rigorously some formal statements associated with the multiple component phase transit…
Finite time singularity of the nematic liquid crystal flow in dimension three
Tao Huang, Fanghua Lin, Chun Liu +1
In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singulari…
On Brezis Open Problem 3.1
Qi Guo, Xueping Huang, Yi C. Huang +2
Let be the unit disk in . We consider the harmonic map equation subject to the Dirichlet boundary condition $ u(e^{iθ})=(R\cosθ,R\si…
Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem
Mingwen Fei, Fanghua Lin, Wei Wang +1
In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form: $$\partial_t A=ÎA-\varepsilon^{-2}( A A^{\mathrm{T}}A- A)~~\text…
Global small solutions to 2-D incompressible MHD system
Fanghua Lin, Li Xu, Ping Zhang
In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with small and smooth initial data. It is a coupled system between the Navie…
Critical Sets of Solutions of Elliptic Equations in Periodic Homogenization
Fanghua Lin, Zhongwei Shen
In this paper we study critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. We show that the…
On a Problem Posed by Brezis and Mironescu
Fanghua Lin, Malkeil Shoshan, Changyou Wang
The purpose of this note is to present a positive answer to an open problem proposed in the recent book \cite{Brezis-Mironescu} by H. Brezis and P. Mironescu. It has been stated in…
Homogenization of Green and Neumann Functions
Carlos E. Kenig, Fanghua Lin, Zhongwei Shen
For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet…
Regularity for Shape Optimizers: The Nondegenerate Case
Dennis Kriventsov, Fanghua Lin
We consider minimizers of \[ F(λ_1(Ω),\ldots,λ_N(Ω)) + |Ω|, \] where is a function strictly increasing in each parameter, and is the -th Dirichlet eigenvalue…
Homogenization of Elliptic Systems with Neumann Boundary Conditions
Carlos E. Kenig, Fanghua Lin, Zhongwei Shen
The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators , arising in the theory of hom…
Recent developments of analysis for hydrodynamic flow of nematic liquid crystals
Fanghua Lin, Changyou Wang
The study of hydrodynamics of liquid crystal leads to many fasci- nating mathematical problems, which has prompted various interesting works recently. This article reviews the stat…
Nematic liquid crystal flow with partially free boundary
Fanghua Lin, Yannick Sire, Juncheng Wei +1
We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes…
Finite time blow-up for the nematic liquid crystal flow in dimension two
Chen-Chih Lai, Fanghua Lin, Changyou Wang +2
We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain . Given any distinct points in the…
Superfluids Passing an Obstacle and Vortex Nucleation
Fanghua Lin, Juncheng Wei
We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \[ε^2 Îu+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash Ω, \ \ \frac{\partial u}{\p…
Convergence Rates in L^2 for Elliptic Homogenization Problems
Carlos E. Kenig, Fanghua Lin, Zhongwei Shen
We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_ε} with rapidly oscillating oscillating coefficients in Lipschitz domains with Di…
Two dimensional liquid crystal droplet problem with tangential boundary condition
Zhiyuan Geng, Fanghua Lin
This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a…
Improved Berezin-Li-Yau inequality and Kröger inequality and consequences
Zaihui Gan, Renjin Jiang, Fanghua Lin
We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kröger inequality, in , . The improvement on Kröger's inequality resolves an o…
Riesz transform under perturbations via heat kernel regularity
Renjin Jiang, Fanghua Lin
Let be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of -boundedness of the Riesz transfo…
Estimates of Eigenvalues and Eigenfunctions in Periodic Homogenization
Carlos E. Kenig, Fanghua Lin, Zhongwei Shen
For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of…
Uniform Boundary Controllability and Homogenization of Wave Equations
Fanghua Lin, Zhongwei Shen
We obtain sharp convergence rates, using Dirichlet correctors, for solutions of wave equations in a bounded domain with rapidly oscillating periodic coefficients. The results are u…
Upper bounds of nodal sets for eigenfunctions of eigenvalue problems
Fanghua Lin, Jiuyi Zhu
The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real a…
Boundary Harnack Principle on Nodal Domains
Fanghua Lin, Zhengjiang Lin
We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions,…
Riesz transform on exterior Lipschitz domains and applications
Renjin Jiang, Fanghua Lin
Let be a uniformly elliptic operator on , . Let be an exterior Lipschitz domain, and let and ${\math…
Global existence of weak solutions of the nematic liquid crystal flow in dimensions three
Fanghua Lin, Changyou Wang
For any bounded smooth domain , we establish the global existence of a weak solution of the initial-bo…
On the Cauchy problem for two dimensional incompressible viscoelastic flows
Xianpeng Hu, Fanghua Lin
We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak so…
Global Small Solutions to a Complex Fluid Model in 3D
Fanghua Lin, Ting Zhang
In this paper, we provide a much simplified proof of the main result in [Lin and Zhang, Comm. Pure Appl. Math.,67(2014), 531--580] concerning the global existence and uniqueness of…
Nodal Sets of Steklov Eigenfunctions
Katarina Bellova, Fanghua Lin
We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in - the eigenfunctions of the Dirichlet-to-Neumann map. Under the as…
On ancient solutions of the heat equation
Fanghua Lin, Qi S. Zhang
An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature,…
On the Two-Dimensional Muskat Problem with Monotone Large Initial Data
Fan Deng, Zhen Lei, Fanghua Lin
We consider the evolution of two incompressible, immiscible fluids with different densities in porous media, known as the Muskat problem [21], which in two dimensions is analogous…
Global solutions of two dimensional incompressible viscoelastic flows with discontinuous initial data
Xianpeng Hu, Fanghua Lin
The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a long standing open problem, and it is studied in this paper. We…
Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions
Fanghua Lin, Zhongwei Shen
In this paper we continue the study of critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. I…