We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We ob...We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.展开更多
We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous <span style="white-space:nowrap;">Monge-Ampère</span> equation to...We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous <span style="white-space:nowrap;">Monge-Ampère</span> equation to first-order ODEs. Some classes of the invariant solutions are constructed.展开更多
In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent ...In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent global regularity of optimal transportation in convex domains by the authors.展开更多
Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equatio...Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.展开更多
We consider the Monge-Ampère equation det(D^(2)u)=f in R^(n),where f is a positive bounded periodic function.We prove that u must be the sum of a quadratic polynomial and a periodic function.For f≡1,this is the ...We consider the Monge-Ampère equation det(D^(2)u)=f in R^(n),where f is a positive bounded periodic function.We prove that u must be the sum of a quadratic polynomial and a periodic function.For f≡1,this is the classic result by Jörgens,Calabi and Pogorelov.For f∈C^(α),this was proved by Caffarelli and the first named author.展开更多
We propose a numerical method to solve the Monge-Ampère equation which admits a classical convex solution.The Monge-Ampère equation is reformulated into an equivalent first-order system.We adopt a novel reco...We propose a numerical method to solve the Monge-Ampère equation which admits a classical convex solution.The Monge-Ampère equation is reformulated into an equivalent first-order system.We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials.This space allows us to solve the first-order system in two sequential steps.In the first step,we solve a nonlinear system to obtain the approximation to the gradient.A Newton iteration is adopted to handle the nonlinearity of the system.The approximation to the primitive variable is obtained from the approximate gradient by a trivial least squares finite element method in the second step.Numerical examples in both two and three dimensions are presented to show an optimal convergence rate in accuracy.It is interesting to observe that the approximation solution is piecewise convex.Particularly,with the reconstructed approximation space,the proposed method numerically demonstrates a remarkable robustness.The convergence of the Newton iteration does not rely on the initial values.The dependence of the convergence on the penalty parameter in the discretization is also negligible,in comparison to the classical discontinuous approximation space.展开更多
In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder con...In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>1.Our proof uses various PDE techniques but not the pluripotential theory.展开更多
Optimal transportation plays a fundamental role in many fi elds in engineering and medicine,including surface parameterization in graphics,registration in computer vision,and generative models in deep learning.For qua...Optimal transportation plays a fundamental role in many fi elds in engineering and medicine,including surface parameterization in graphics,registration in computer vision,and generative models in deep learning.For quadratic distance cost,optimal transportation map is the gradient of the Brenier potential,which can be obtained by solving the Monge-Ampère equation.Furthermore,it is induced to a geometric convex optimization problem.The Monge-Ampère equation is highly non-linear,and during the solving process,the intermediate solutions have to be strictly convex.Specifi cally,the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure.In this work,we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions.Experimental results demonstrate the efficiency and efficacy of our method.展开更多
In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations.By employing the concept of(a,η) type domain,we emphasize that the bound...In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations.By employing the concept of(a,η) type domain,we emphasize that the boundary regularity depends on the convexity of the domain in nature.The key idea of our proof is to provide more effective global H?lder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.展开更多
基金supported by Shandong Provincial NSF(ZR2022MA020).
文摘We consider the singular Dirichlet problem for the Monge-Ampère type equation■=0,whereΩis a strictly convex and bounded smooth domain in■is positive and strictly decreasing in(0,∞)with■is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.
文摘We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous <span style="white-space:nowrap;">Monge-Ampère</span> equation to first-order ODEs. Some classes of the invariant solutions are constructed.
基金supported by Australian Research Council (Grant Nos. FL130100118 and DP170100929)
文摘In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent global regularity of optimal transportation in convex domains by the authors.
基金supported by the Research Foundation of Beijing Government(Grant No.YB20081002802)National Natural Science Foundation of China(Grant No.10771144)
文摘Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.
基金partially supported by NSF Grants DMS-1501004,DMS2000261Simons Fellows Award 677077partially supported by NSERC Discovery Grant.
文摘We consider the Monge-Ampère equation det(D^(2)u)=f in R^(n),where f is a positive bounded periodic function.We prove that u must be the sum of a quadratic polynomial and a periodic function.For f≡1,this is the classic result by Jörgens,Calabi and Pogorelov.For f∈C^(α),this was proved by Caffarelli and the first named author.
基金This research was supported by the National Natural Science Foundation in China(Nos.12201442,and 11971041).
文摘We propose a numerical method to solve the Monge-Ampère equation which admits a classical convex solution.The Monge-Ampère equation is reformulated into an equivalent first-order system.We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials.This space allows us to solve the first-order system in two sequential steps.In the first step,we solve a nonlinear system to obtain the approximation to the gradient.A Newton iteration is adopted to handle the nonlinearity of the system.The approximation to the primitive variable is obtained from the approximate gradient by a trivial least squares finite element method in the second step.Numerical examples in both two and three dimensions are presented to show an optimal convergence rate in accuracy.It is interesting to observe that the approximation solution is piecewise convex.Particularly,with the reconstructed approximation space,the proposed method numerically demonstrates a remarkable robustness.The convergence of the Newton iteration does not rely on the initial values.The dependence of the convergence on the penalty parameter in the discretization is also negligible,in comparison to the classical discontinuous approximation space.
文摘In this paper,we use the Sobolev type inequality in Wang et al.(Moser-Trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>1.Our proof uses various PDE techniques but not the pluripotential theory.
基金the National Numerical Wind Tunnel Project,China(No.NNW2019ZT5-B13)the National Natural Science Foundation of China(Nos.61907005,61772105,61936002,and 61720106005)。
文摘Optimal transportation plays a fundamental role in many fi elds in engineering and medicine,including surface parameterization in graphics,registration in computer vision,and generative models in deep learning.For quadratic distance cost,optimal transportation map is the gradient of the Brenier potential,which can be obtained by solving the Monge-Ampère equation.Furthermore,it is induced to a geometric convex optimization problem.The Monge-Ampère equation is highly non-linear,and during the solving process,the intermediate solutions have to be strictly convex.Specifi cally,the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure.In this work,we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions.Experimental results demonstrate the efficiency and efficacy of our method.
基金Yau Mathematical Sciences Center for the support。
文摘In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampère type equations.By employing the concept of(a,η) type domain,we emphasize that the boundary regularity depends on the convexity of the domain in nature.The key idea of our proof is to provide more effective global H?lder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.
