Let Ω⊆M be a bounded domain with a smooth boundary ∂Ω,where(M,J,g)is a compact,almost Hermitian manifold.The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on...Let Ω⊆M be a bounded domain with a smooth boundary ∂Ω,where(M,J,g)is a compact,almost Hermitian manifold.The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on Ω.Under the existence of a C^(2)-smooth strictly J-plurisubharmonic(J-psh for short)subsolution,we can solve this Dirichlet problem.Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.展开更多
The author applies the arguments in his PKU Master degree thesis in 1988 to derive a third derivative estimate, and consequently, a C^(2,α)-estimate, for complex MongeAmpere equations in the conic case. This C^(2,α)...The author applies the arguments in his PKU Master degree thesis in 1988 to derive a third derivative estimate, and consequently, a C^(2,α)-estimate, for complex MongeAmpere equations in the conic case. This C^(2,α)-estimate was used by Jeffres-MazzeoRubinstein in their proof of the existence of K¨ahler-Einstein metrics with conic singularities.展开更多
The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampere equation by a C0 interior penalty method with Lagrang...The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampere equation by a C0 interior penalty method with Lagrange finite elements.We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton’s method on a finer mesh.Thus both steps are inexpensive.We give quasi-optimal W1,1 error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution.Numerical experiments confirm the computational efficiency of the approach compared to Newton’s method on the fine mesh.展开更多
In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the followin...In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the following evolution equation δ^2F /δt^2 （u, t） = k（u, t）N（u, t）-▽ρ（u, t）, ∨（u, t） ∈ S^1 × [0, T ） with the initial data F （u, 0） = F0（u） and δF/δt （u, 0） = f（u）N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F （u, t）, f（u） and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=（δ^2F /δsδt ,δF/δt） T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax） and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.展开更多
Hildebrand classified all semi-homogeneous cones in R3 and computed their cor- responding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand＇s new examples. After giving their a...Hildebrand classified all semi-homogeneous cones in R3 and computed their cor- responding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand＇s new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand＇s examples. The generic member of these affine spheres is given by Weierstrass f,ζ and a functions. In general any regular convex cone in R^3 has a natural associated S^1-family of such cones, which deserves further studies.展开更多
In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampere equation with Dirichlet boundary conditions. We formulate the Monge-Ampere equation as an optimization problem. Th...In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampere equation with Dirichlet boundary conditions. We formulate the Monge-Ampere equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.展开更多
Considering the hyperbolic affine sphere equation in a smooth strictly convex bounded domain with zero boundary values, the sharp derivative estimates of any order for its convex solution are obtained.
For the more general parabolic Monge-Ampère equations defined by the operator F （D2u ＋ σ（x））, the existence and uniqueness of the admissible solution to the third initial-boundary value problem for the equa...For the more general parabolic Monge-Ampère equations defined by the operator F （D2u ＋ σ（x））, the existence and uniqueness of the admissible solution to the third initial-boundary value problem for the equation are established. A new structure condition which is used to get a priori estimate is established.展开更多
In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds.We derive C∞a priori estimates under the existence of an admissible C-subsolution.Finally,we obtain an existence r...In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds.We derive C∞a priori estimates under the existence of an admissible C-subsolution.Finally,we obtain an existence result if there exists an admissible supersolution.展开更多
基金supported by the National Key R and D Program of China(2020YFA0713100).
文摘Let Ω⊆M be a bounded domain with a smooth boundary ∂Ω,where(M,J,g)is a compact,almost Hermitian manifold.The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on Ω.Under the existence of a C^(2)-smooth strictly J-plurisubharmonic(J-psh for short)subsolution,we can solve this Dirichlet problem.Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.
文摘The author applies the arguments in his PKU Master degree thesis in 1988 to derive a third derivative estimate, and consequently, a C^(2,α)-estimate, for complex MongeAmpere equations in the conic case. This C^(2,α)-estimate was used by Jeffres-MazzeoRubinstein in their proof of the existence of K¨ahler-Einstein metrics with conic singularities.
基金Gerard Awanou was partially supported by NSF DMS grant#1720276Hengguang Li by NSF DMS grant#1819041 and by NSF China Grant 11628104.
文摘The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampere equation by a C0 interior penalty method with Lagrange finite elements.We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton’s method on a finer mesh.Thus both steps are inexpensive.We give quasi-optimal W1,1 error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution.Numerical experiments confirm the computational efficiency of the approach compared to Newton’s method on the fine mesh.
基金Kong and Wang was supported in part by the NSF of China (10671124)the NCET of China (NCET-05-0390)the work of Liu was supported in part by the NSF of China
文摘In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the following evolution equation δ^2F /δt^2 （u, t） = k（u, t）N（u, t）-▽ρ（u, t）, ∨（u, t） ∈ S^1 × [0, T ） with the initial data F （u, 0） = F0（u） and δF/δt （u, 0） = f（u）N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F （u, t）, f（u） and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=（δ^2F /δsδt ,δF/δt） T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax） and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.
基金supported by the NSF of China(10941002,11001262)the Starting Fund for Distinguished Young Scholars of Wuhan Institute of Physics and Mathematics(O9S6031001)
文摘Hildebrand classified all semi-homogeneous cones in R3 and computed their cor- responding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand＇s new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand＇s examples. The generic member of these affine spheres is given by Weierstrass f,ζ and a functions. In general any regular convex cone in R^3 has a natural associated S^1-family of such cones, which deserves further studies.
文摘In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampere equation with Dirichlet boundary conditions. We formulate the Monge-Ampere equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.
文摘Considering the hyperbolic affine sphere equation in a smooth strictly convex bounded domain with zero boundary values, the sharp derivative estimates of any order for its convex solution are obtained.
基金The NSF (10401009) of ChinaNCET (060275) of China
文摘For the more general parabolic Monge-Ampère equations defined by the operator F （D2u ＋ σ（x））, the existence and uniqueness of the admissible solution to the third initial-boundary value problem for the equation are established. A new structure condition which is used to get a priori estimate is established.
基金Supported by the project“Analysis and Geometry on Bundle”of Ministry of Science and Technology of the People’s Republic of China(Grant No.SQ2020YFA070080)by China Postdoctoral Science Foundation(Grant No.290612)。
文摘In this paper we consider the Monge–Ampère type equations on compact almost Hermitian manifolds.We derive C∞a priori estimates under the existence of an admissible C-subsolution.Finally,we obtain an existence result if there exists an admissible supersolution.
基金supported by NSFC(No.11071171)the Beijing Natural Science Foundation(No.1122010)the Science and Technology Project of Beijing Municipal Commission of Education(No.KM201210028007)
