Lin, Fanghua
Overview
Works:  38 works in 78 publications in 2 languages and 1,814 library holdings 

Genres:  Textbooks Academic theses Conference papers and proceedings 
Roles:  Author, Editor 
Classifications:  QA377, 515.353 
Publication Timeline
.
Most widely held works by
Fanghua Lin
The analysis of harmonic maps and their heat flows by
Fanghua Lin(
Book
)
15 editions published in 2008 in English and Undetermined and held by 162 WorldCat member libraries worldwide
"This book provides a broad yet comprehensive introduction to the analysis of harmonic maps and their heat flows. The book can be used as a textbook for the topic course of advanced graduate students and for researchers who are interested in geometric partial differential equations and geometric analysis."Jacket
15 editions published in 2008 in English and Undetermined and held by 162 WorldCat member libraries worldwide
"This book provides a broad yet comprehensive introduction to the analysis of harmonic maps and their heat flows. The book can be used as a textbook for the topic course of advanced graduate students and for researchers who are interested in geometric partial differential equations and geometric analysis."Jacket
Elliptic partial differential equations by
Qing Han(
Book
)
1 edition published in 2011 in English and held by 122 WorldCat member libraries worldwide
1 edition published in 2011 in English and held by 122 WorldCat member libraries worldwide
Geometric measure theory : an introduction by
Fanghua Lin(
Book
)
12 editions published between 2002 and 2010 in English and Spanish and held by 89 WorldCat member libraries worldwide
12 editions published between 2002 and 2010 in English and Spanish and held by 89 WorldCat member libraries worldwide
Lectures on the analysis of nonlinear partial differential equations(
Book
)
in English and held by 13 WorldCat member libraries worldwide
in English and held by 13 WorldCat member libraries worldwide
Nonlinear partial differential equations : proceedings of International Conference, Zhejiang University, June 1992(
Book
)
4 editions published in 1993 in English and held by 13 WorldCat member libraries worldwide
4 editions published in 1993 in English and held by 13 WorldCat member libraries worldwide
Lectures on the analysis of nonlinear partial differential equations(
Book
)
4 editions published in 2016 in English and held by 10 WorldCat member libraries worldwide
4 editions published in 2016 in English and held by 10 WorldCat member libraries worldwide
Universal symplectic forms in the soliton theory by Fedor Soloviev(
)
1 edition published in 2010 in English and held by 7 WorldCat member libraries worldwide
Using KricheverPhong's universal formula, we show that a multiplicative representation linearizes Sklyanian quadratic brackets for a multipole Lax function with a spectral parameter. The spectral parameter can be either rational or elliptic. As a byproduct, we obtain an extension of a Sklyanin algebra in the elliptic case. We discuss applications of these results to isospectral and isomonodromic equations with discrete time. KricheverPhong's formula provides a hierarchy of symplectic structures, and we show that there exists a nontrivial cubic bracket in Sklyanin's case. Also, we consider a generalization of the universal formula to variable base curves (suggested in [22]) in the simplest possible case of a rational base curve with moving marked points
1 edition published in 2010 in English and held by 7 WorldCat member libraries worldwide
Using KricheverPhong's universal formula, we show that a multiplicative representation linearizes Sklyanian quadratic brackets for a multipole Lax function with a spectral parameter. The spectral parameter can be either rational or elliptic. As a byproduct, we obtain an extension of a Sklyanin algebra in the elliptic case. We discuss applications of these results to isospectral and isomonodromic equations with discrete time. KricheverPhong's formula provides a hierarchy of symplectic structures, and we show that there exists a nontrivial cubic bracket in Sklyanin's case. Also, we consider a generalization of the universal formula to variable base curves (suggested in [22]) in the simplest possible case of a rational base curve with moving marked points
On the NavierStokes equations by Hantaek Bae(
)
1 edition published in 2009 in English and held by 7 WorldCat member libraries worldwide
This paper consists of two parts. The first part is the harmonic analysis methods for partial differential equations. We introduce the LittlewoodPaley decomposition, paradifferential calculus, and its application to the NavierStokes equations. Using the same idea, we prove that the dissipative quasigeostrophic equations are globally wellposedness in critical spaces
1 edition published in 2009 in English and held by 7 WorldCat member libraries worldwide
This paper consists of two parts. The first part is the harmonic analysis methods for partial differential equations. We introduce the LittlewoodPaley decomposition, paradifferential calculus, and its application to the NavierStokes equations. Using the same idea, we prove that the dissipative quasigeostrophic equations are globally wellposedness in critical spaces
On the wellposedness of boundary layer equations by Tak Kwong Wong(
)
1 edition published in 2010 in English and held by 7 WorldCat member libraries worldwide
In this thesis we are concerned with partial differential equations arising from boundary layer problems, namely, hydrostatic Euler equations and Prandtl equations. After a brief survey of the current progress of the hydrostatic Euler equations, we establish the local existence and uniqueness of Hs solutions under the local Rayleigh condition. The proof is based on new weighted Hs estimates. Moreover, we provide a finite time blowup result as well. For the Prandtl equations, we first give a brief introduction of the known existence and illposedness results. Then, under Oleinik's monotonicity assumption, we prove the local existence of weak solutions to the Prandtl equations by the viscoussplitting algorithm
1 edition published in 2010 in English and held by 7 WorldCat member libraries worldwide
In this thesis we are concerned with partial differential equations arising from boundary layer problems, namely, hydrostatic Euler equations and Prandtl equations. After a brief survey of the current progress of the hydrostatic Euler equations, we establish the local existence and uniqueness of Hs solutions under the local Rayleigh condition. The proof is based on new weighted Hs estimates. Moreover, we provide a finite time blowup result as well. For the Prandtl equations, we first give a brief introduction of the known existence and illposedness results. Then, under Oleinik's monotonicity assumption, we prove the local existence of weak solutions to the Prandtl equations by the viscoussplitting algorithm
On the Boltzmann equation: Hydrodynamic limit with longrange interactions and mild solutions by Diogo Arsenio(
)
1 edition published in 2009 in English and held by 7 WorldCat member libraries worldwide
Secondly, we develop a new theory of existence of global solutions to the Boltzmann equation for small initial data, which we name mild solutions in analogy with the mild solutions for the NavierStokes equations. The existence comes as a result of the study of the competing phenomena of dispersion, due to the transport operator, and of singularity formation, due to the nonlinear Boltzmann collision operator. It is the joint use of the socalled dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions
1 edition published in 2009 in English and held by 7 WorldCat member libraries worldwide
Secondly, we develop a new theory of existence of global solutions to the Boltzmann equation for small initial data, which we name mild solutions in analogy with the mild solutions for the NavierStokes equations. The existence comes as a result of the study of the competing phenomena of dispersion, due to the transport operator, and of singularity formation, due to the nonlinear Boltzmann collision operator. It is the joint use of the socalled dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions
The lattice Faddeev Model by Hans K Shmidheiser(
)
1 edition published in 2009 in English and held by 7 WorldCat member libraries worldwide
We describe a lattice version of the Faddeev Model for knotted solitons. We briefly mention the known results for the original Faddeev Model, including the 3/4energy growth law and the existence results of Lin and Yang. Then we discuss numerical implementations of the model by Faddeev, Battye and Sutcliffe and Ward, motivating the theoretical study of the Lattice Faddeev Model. In particular, we will focus on how one computes a Hopf number for a nice map from R3 to S2 by sampling at integer lattice points Z3. Alternatively, we discuss how one can use this to assign Hopf numbers to a map from Z3 to S2. We then describe how this technique could be generalized to compute general topological invariants for maps between manifolds, given in terms of differential forms, by sampling at a discrete set of points. We characterize some of the function spaces involved in the Lattice Faddeev Model, and derive some energy estimates. Finally, we apply the techniques of Lin and Yang to the Lattice Faddeev Model, to obtain existence results
1 edition published in 2009 in English and held by 7 WorldCat member libraries worldwide
We describe a lattice version of the Faddeev Model for knotted solitons. We briefly mention the known results for the original Faddeev Model, including the 3/4energy growth law and the existence results of Lin and Yang. Then we discuss numerical implementations of the model by Faddeev, Battye and Sutcliffe and Ward, motivating the theoretical study of the Lattice Faddeev Model. In particular, we will focus on how one computes a Hopf number for a nice map from R3 to S2 by sampling at integer lattice points Z3. Alternatively, we discuss how one can use this to assign Hopf numbers to a map from Z3 to S2. We then describe how this technique could be generalized to compute general topological invariants for maps between manifolds, given in terms of differential forms, by sampling at a discrete set of points. We characterize some of the function spaces involved in the Lattice Faddeev Model, and derive some energy estimates. Finally, we apply the techniques of Lin and Yang to the Lattice Faddeev Model, to obtain existence results
Partially regular and singular solutions to the LandauLifshitz (Gilbert) equations by Yueh Joy Ko(
)
1 edition published in 2004 in English and held by 6 WorldCat member libraries worldwide
The LandauLifshitz and LandauLifshitz Gilbert equations are the basic evolution equations in micromagnetics, a continuum model for magnetic behavior in ferromagnetic materials. In the setting where the magnetic behavior is determined by the Dirichlet energy, these equations are a hybrid Schrodinger map flow and harmonic map heat flow into the unit sphere S2. In this thesis, more general geometries involving targets that are compact smooth hypersurfaces are considered. We address the questions of whether global, partially regular solutions exist in the space of finite energy for these equations and whether singularities form from smooth finite energy data when evolved by these equations. The domain is always taken to be flat and twodimensional. A general framework is given to construct partially regular solutions to the LLG. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution into the discrete setting. The question of singularity formation for these equations is motivated by the fact that under a suitable transformation, these equations are reminiscent of the cubic nonlinear Schrodinger equation for which singular solutions abound. Analytical attempts have met impasses but our numerical investigations here demonstrate the existence of singular solutions for the LLG. Issues related to the numerical computation of the LL, which only involves the Schrodinger term, are also discussed
1 edition published in 2004 in English and held by 6 WorldCat member libraries worldwide
The LandauLifshitz and LandauLifshitz Gilbert equations are the basic evolution equations in micromagnetics, a continuum model for magnetic behavior in ferromagnetic materials. In the setting where the magnetic behavior is determined by the Dirichlet energy, these equations are a hybrid Schrodinger map flow and harmonic map heat flow into the unit sphere S2. In this thesis, more general geometries involving targets that are compact smooth hypersurfaces are considered. We address the questions of whether global, partially regular solutions exist in the space of finite energy for these equations and whether singularities form from smooth finite energy data when evolved by these equations. The domain is always taken to be flat and twodimensional. A general framework is given to construct partially regular solutions to the LLG. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution into the discrete setting. The question of singularity formation for these equations is motivated by the fact that under a suitable transformation, these equations are reminiscent of the cubic nonlinear Schrodinger equation for which singular solutions abound. Analytical attempts have met impasses but our numerical investigations here demonstrate the existence of singular solutions for the LLG. Issues related to the numerical computation of the LL, which only involves the Schrodinger term, are also discussed
Free boundary problems with volume constraint: Zero set of Sobolev functions by Huiqiang Jiang(
)
1 edition published in 2003 in English and held by 6 WorldCat member libraries worldwide
In the second part, we study the zero set of Sobolev functions with negative power of integrability. We give a general Hausdorff dimension estimate on the size of zero set. After obtaining some basic regularity result for the model on van der Waals driven thin film, as an application of our result, we give an estimate on the size of rupture set of thin film
1 edition published in 2003 in English and held by 6 WorldCat member libraries worldwide
In the second part, we study the zero set of Sobolev functions with negative power of integrability. We give a general Hausdorff dimension estimate on the size of zero set. After obtaining some basic regularity result for the model on van der Waals driven thin film, as an application of our result, we give an estimate on the size of rupture set of thin film
Regularity of the free boundary of American options by
Yŏngsŏk Yi(
)
1 edition published in 2011 in English and held by 6 WorldCat member libraries worldwide
We introduce pricing of American options on several assets that follow a geometric Brownian motion as a free boundary problem. A crucial assumption is that payoff functions are smooth in the region where the option is exercised. This assumption is true for various options traded in the derivatives market. A key ingredient for analysis of the free boundary is a monotonicity formula. Existence and uniqueness of homogeneous blowup limits at a fixed point on the free boundary are guaranteed by the monotonicity formula. Then the free boundary can be classified into high energy points and low energy points by nondegeneracy of the solution. The time projection of the set of high energy point is analyzed measure theoretically. We prove that the set of low energy points is locally spacetime C1,alpha. Finally, we can obtain C1,alpha regularity of the free boundary for some particular payoff functions
1 edition published in 2011 in English and held by 6 WorldCat member libraries worldwide
We introduce pricing of American options on several assets that follow a geometric Brownian motion as a free boundary problem. A crucial assumption is that payoff functions are smooth in the region where the option is exercised. This assumption is true for various options traded in the derivatives market. A key ingredient for analysis of the free boundary is a monotonicity formula. Existence and uniqueness of homogeneous blowup limits at a fixed point on the free boundary are guaranteed by the monotonicity formula. Then the free boundary can be classified into high energy points and low energy points by nondegeneracy of the solution. The time projection of the set of high energy point is analyzed measure theoretically. We prove that the set of low energy points is locally spacetime C1,alpha. Finally, we can obtain C1,alpha regularity of the free boundary for some particular payoff functions
Topology of Sobolev mappings, Jacobians and GinzburgLandau type functionals by Fengbo Hang(
)
1 edition published in 2001 in English and held by 6 WorldCat member libraries worldwide
In the first part of the thesis we generalize and improve some recent results obtained by H. Brezis and Y.Y. Li concerning topologies of Sobolev mappings between Riemannian manifolds. We also settle two of their conjectures. In connections with the latter results, we find some global topological obstruction for smooth maps to be weakly sequentially dense or to be dense in the strong topology for Sobolev spaces of mappings. In fact, we obtain a necessary and sufficient condition for smooth maps to be dense in the strong topology, which corrects an earlier theorem of F. Bethuel
1 edition published in 2001 in English and held by 6 WorldCat member libraries worldwide
In the first part of the thesis we generalize and improve some recent results obtained by H. Brezis and Y.Y. Li concerning topologies of Sobolev mappings between Riemannian manifolds. We also settle two of their conjectures. In connections with the latter results, we find some global topological obstruction for smooth maps to be weakly sequentially dense or to be dense in the strong topology for Sobolev spaces of mappings. In fact, we obtain a necessary and sufficient condition for smooth maps to be dense in the strong topology, which corrects an earlier theorem of F. Bethuel
Mappings minimizing the Lp norm of the gradient by
R Hardt(
Book
)
2 editions published in 1986 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 1986 in English and held by 3 WorldCat member libraries worldwide
Regularity at infinity for areaminimizing hypersurfaces in hyperbolic space by
R Hardt(
Book
)
2 editions published in 1986 in English and held by 3 WorldCat member libraries worldwide
2 editions published in 1986 in English and held by 3 WorldCat member libraries worldwide
Nonminimality of minimal graphs by
R Hardt(
Book
)
2 editions published in 1986 in English and held by 2 WorldCat member libraries worldwide
2 editions published in 1986 in English and held by 2 WorldCat member libraries worldwide
Lorentz space estimates and applied boundary current dynamics for GinzburgLandau by Ian Tice(
Book
)
1 edition published in 2008 in English and held by 2 WorldCat member libraries worldwide
This thesis, which is divided into three parts, is concerned with the study of the GinzburgLandau energy functional in two dimensions. In the first part, we prove novel lower bounds for the GinzburgLandau energy with or without magnetic field. These bounds rely on an improvement of the "vortex balls construction" estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results
1 edition published in 2008 in English and held by 2 WorldCat member libraries worldwide
This thesis, which is divided into three parts, is concerned with the study of the GinzburgLandau energy functional in two dimensions. In the first part, we prove novel lower bounds for the GinzburgLandau energy with or without magnetic field. These bounds rely on an improvement of the "vortex balls construction" estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results
Elliptic Partial Differential Equations Second Edition by
Qing Han(
)
1 edition published in 2011 in English and held by 0 WorldCat member libraries worldwide
Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things abo
1 edition published in 2011 in English and held by 0 WorldCat member libraries worldwide
Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things abo
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Related Identities
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