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非线性椭圆和非线性抛物型方程
  • 项目名称：非线性椭圆和非线性抛物型方程
  • 项目类别：重点项目
  • 批准号：10631020
  • 申请代码：A0108
  • 项目来源：国家自然科学基金
  • 研究期限：2007-01-01-2010-12-31
  • 项目负责人：洪家兴
  • 负责人职称：教授
  • 依托单位：复旦大学
  • 批准年度：2006
中文摘要：

自然科学基金会重点项目“非线性椭圆和非线性抛物型方程”（10631020），执行时间2007年1月至今1010年12。该项目主要研究几何、物理、材料科学、生命科学与医学中所提出的非线性椭圆与非线性抛物型方程（组）的理论。课题组在国际偏微分方程及相关学科的重要期刊上共发表论文82篇，其中SCI(包括SCIE)78篇，包括比较著名的Comm. Math. P., J. Func. Anal., Arch. Rational Mech. Anal.等重要的数学杂志。在这期间培养博士生26名，硕士生33名，博士后5名。课题组曾邀请包括法国、澳大利亚和加拿大等科学院院士在内的二十余名国际著名数学家来访，从事合作研究。同时，课题组成员出访从事合作研究或参加国际学术会议四十余次。主要的成果如下关于完全非线性椭圆、蜕化椭圆Monge-Ampere方程的研究，对国际著名数学家Trudingers所提出的问题在2维的情形给出了肯定的回答，对Caffarelli等人关于Monge-Ampere方程解的经典C2估计给出了一个统一新证明，并发现了解的梯度Log-Lipshing估计....详见主报告。

中文主题词： Monge-Ampere 方程；Chemotaxis 方程组；Ricci 流；Ginsburg-Landau方程组；
结论摘要：

英文主题词Monge-Ampere equations；Chemotaxis system；Ricci flow；Ginsburg-Landau system；

成果综合统计
成果类型
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  • 期刊论文
  • 会议论文
  • 专利
  • 获奖
  • 著作
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期刊论文
Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians
GRADIENT ESTIMATES FOR POSITIVE SMOOTH f-HARMONIC FUNCTIONS
Symmetry results for classical solutions of Monge-Ampere systems on bounded planar domains
On the conditions to control curvature tensors of Ricci flow
Optimal $L_p$ Riemannian Gagliardo-Nirenberg inequalities,
INEQUALITIES OF EIGENVALUES FOR BI-KOHN LAPLACIAN ON HEISENBERG GROUP
Inhomogeneous boundary value problem for Hartree-type equation
On energy stability for the coupled nonlinear wave and Schrodinger systems
Chern-Simons invariants and isometric immersions of warped products
Stability for the time-dependent Hartree equation with positive energy
REMARKS ON COMPLETE NON-COMPACT GRADIENT RICCI EXPANDING SOLITONS
New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems
Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg–Landau equat
Finite Morse index steady states of van der Waals force driven thin film equations
Inhomogeneous Problem for the Ginzburg-Landau Equation on Two Dimensional Compact Manifolds
Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation
Q-curvature flow with indefinite nonlinearity
Analytic smoothness effect of solutions for spatially homogeneous Landau equation
Curvature flow to the Nirenberg problem
Symmetry results for decay solutions of elliptic systems in the whole space
UNIVERSAL BOUNDS FOR EIGENVALUES OF LAPLACIAN OPERATOR OF ANY ORDER
A squeeze flow problem with a Navier slip condition
Existence of Global Smooth solutions to Dirichlet problem for Degenerate Elliptic Monge-Ampere equat
关于可压液晶组能量有界性
The Gevrey hypoellipticity for a class of kinetic equations
Regularity for positive weak solutions to semi-linear elliptic equations
Uniqueness for the solution of semi-linear elliptic Neumann problems in R-3
Properties of positive solutions to an elliptic equation with negative exponent
CONVERGENCE TO EQUILIBRIUM FOR PARABOLIC-HYPERBOLIC TIME-DEPENDENT GINZBURG-LANDAU-MAXWELL EQUATIONS
Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrodinger system
the local and global existence of the solutions of hyperbolic-parabolic system modeling biological p
ASYMPTOTIC BEHAVIOR OF A NONISOTHERMAL GINZBURG-LANDAU MODEL
Gradient estimate for the degenerate parabolic equation u(t)=Delta F(u)+H(u) on manifolds
Regularity of the p-harmonic maps with potential
Uniqueness of ground states of some coupled nonlinear Schrodinger systems and their application
UNIQUENESS OF POSITIVE BOUND STATES TO SCHRODINGER SYSTEMS WITH CRITICAL EXPONENTS
The Dirichlet problem of the quantum drift-diffusion model
Radial symmetry and monotonicity for an integral equation
UNIQUENESS FOR THE BREZIS-NIRENBERG PROBLEM ON COMPACT EINSTEIN MANIFOLDS
Long-time existence of mean curvature flow with external force fields
PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS
Global existence and asymptotic behavior for a semiconductor drift-diffusion-poisson model
液晶系统流问题的存在性
Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds
CONVEX EIGENFUNCTION OF A DRIFTING LAPLACIAN OPERATOR AND THE FUNDAMENTAL GAP
Nonsingular Ricci flow on a noncompact manifold in dimension three
Existence, semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion mo
Non-local heat flows and gradient estimates on closed manifolds
THE LANDAU-LIFSHITZ- MAXWELL EQUATION IN DIMENSION THREE
Partial regularity of weak solutions for Landau-Lifshitz system with potential
The regularity of a class of degenerate elliptic Monge-Ampère equations
Radial symmetry and uniqueness for positive solutions of a Schrodinger type system
GEVREY REGULARITY FOR SOLUTION OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION
The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equ
SEMILINEAR ELLIPTIC EQUATIONS ON FRACTAL SETS
On energy stability for the coupled nonlinear Schrodinger system
Spacelike hypersurfaces of prescribed Gauss-Kronecker curvature in exterior domains
ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRODINGER SYSTEM
Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations
Continuity estimates for the Monge-Ampere equation
Global attractor for the 1-D thin film equation
Convergence to equilibrium for a parabolic-hyperbolic phase-field system with dynamical boundary con
The existence and long-time behavior of weak solution to bipolar quantum drift-diffusion model
Asymptotic behavior of positive solutions of some quasilinear elliptic problems
Eigenvalues and lambda constants on Riemannian submersions
Curve shortening in a Riemannian manifold
Convergence to equilibrium for a parabolic-hyperbolic phase-field system with neumann boundary condi
Evolution of hypersurfaces by the mean curvature minus an external force field
Blow up of solutions to the nonlinear Schrodinger equations on manifolds
Monotone solutions to a class of elliptic and diffusion equations
Blow-up criterion for 2-D boussinesq equations in bounded domain
On existence of solutions for some hyperbolic-parabolic-type chemotaxis systems
一类Monge-Ampère方程的特征值问题
关于可压液晶方程组的能量有界性
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