Singular Solutions to Monge-Ampère Equation

We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side.     

Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD²u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C^∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.     

Monge-Ampère Equation with Bounded Periodic Data

We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ 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.     

Second Order Derivative Estimates of the Solutions of a Class of Monge-Ampère Equations

In this paper, we consider a class of Monge-Amp`ere equations in relative differential geometry. Given these equations with zero boundary values in a smooth strictly convex bounded domain, we obtain second order derivative estimates of the convex solutions.     

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Let (W,,H) be an abstract Wiener space assume two _i ,i=1,2 probabilities on (W,(W)). We give some conditions for the Wasserstein distance between ₁ and ₂ with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures on W×W whose first and second marginals are ₁ and ₂. In this case we prove the existence of a unique (cyclically monotone) map T=I _W +, with :WH, such that T maps ₁ to ₂. Moreover, if ₂, then T is stochastically invertible, i.e., there exists S:WW such that ST=I _{W 1} a.s. and TS=I _{W 2} a.s. If, in addition, ₁=, then there exists a 1-convex function in the Gaussian Sobolev space such that =. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W. Mathematics Subject Classification (2000)60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65,35A30, 46N10, 49Q20, 58E12, 26A16, 28C20cf. Theorem 6.1 for the precise hypothesis about ₁ and ₂.In fact this hypothesis is too strong, cf. Theorem 6.1. AcknowledgementThe authors are grateful to Françoise Combelles for all the bibliographical help that she has supplied for the realization of this research. We thank also the anonymous referee for his particular attention and valuable remarks.     

Some Remarks on One Type of Parabolic Monge-Ampère Equation

Inthepresentpaper,weinvestigatethefollowinginitial--boundaryvalueproblemforatypeofparabolicMonge-AmpereequationwhereQ=ax(0,TI,nisaboundedconvexdomaininR,aamistheHessianofthefunctionu,ovQdenotestheparabolicboundaryofQ.Recently,IvochkinaandLadyzhenskaya[1]consideredtheaboveproblem,andundersomestructureconditions,theyobtainedtheuniquesolvabilityofitinclassicalsense.Afterthat,WangRouhuaiandWangGuanglieimprovedtheresultsof[ifundersomeweakerconditionsandgetthatproblem(1),(2)hasauniquesolution…     

Lie Symmetry Analysis and Exact Solutions for?the?Extended mKdV Equation

By using Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation, the all exact solutions based on the Lie group method are given. Especially, the bifurcations and traveling wave solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlinear science.     

Isolated singularities of Monge-Ampère equations

Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s ²=E in the punctured disk 0<(x–x ₀)²+(y–y ₀)²<². Assume thatq is continuous at (x₀, y₀). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x ₀,y ₀) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344).     

Symplectic Reduction and the Homogeneous Complex Monge–Ampère Equation

A Riemannian manifold ( ⁿ , g) is said to be the center of thecomplex manifold ⁿ if is the zero set of a smooth strictly plurisubharmonic exhaustion function ² on such that is plurisubharmonic and solves theMonge–Ampère equation ( ) ⁿ = 0 off , and g is induced by the canonical Kähler metric withfundamental two-form ². Insisting that be unbounded puts severe restrictions on as acomplex manifold as well as on ( , g). It is an open problemto determine the class Riemannian manifolds that are centers of complexmanifolds with unbounded . Before the present work, the list of knownexamples of manifolds in that class was small. In the main result of thispaper we show, by means of the moment map corresponding to isometric actionsand the associated bundle construction, that such class is larger than originally thought and contains many metrically and diffeomorphically`exotic' examples.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

Stability of radial symmetry for a Monge-Ampère overdetermined problem

Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.      

Global regularity for a class of Monge-Ampère type equations

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.     

Curvature Estimates for the Level Sets of Solutions to the Monge-Ampère Equation det D~2u = 1

For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.     

Integrable structures for a generalized Monge-Ampère equation

We consider a third-order generalized Monge-Ampère equation u_yyy ? u _xxy ² + u_xxxu_xyy = 0, which is closely related to the associativity equation in two-dimensional topological field theory. We describe all integrable structures related to it: Hamiltonian, symplectic, and also recursion operators. We construct infinite hierarchies of symmetries and conservation laws.     

Exact Solutions and Integrability of the Duffing – Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.     

Scalar differential invariants of symplectic Monge-Ampère equations

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an example we study equations of the form u _xy + f(x, y, u _x , u _y ) = 0 and in particular find a simple linearization criterion.     

Exact Vacuum Solutions to the Einstein Equation

In this paper,the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation.This procedure of resolution is based on a canonical form of the metric.According to this procedure,the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution.     

On Harnack’s Inequality for the Linearized Parabolic Monge-Ampère Equation

It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampère quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampère equation under minimal geometric assumptions.