On the Rate of Convergence of the Finite-Difference Approximations for Parabolic Bellman Equations with Constant Coefficients

The error bounds of order for two types of finite-difference approximation schemes of parabolic Bellman equations with constant coefficients are obtained, where h is x-mesh size and τ is t-mesh size. The key methods employed are the maximum principles for the Bellman equation and the approximation schemes.     

Counterexamples to the Local Solvability of Monge-Ampère Equations in the Plane

In this paper, we present C ^∞ examples of degenerate hyperbolic and mixed type Monge-Ampère equations in the plane, which do not admit a local C ³ solution.     

A Class of Singular Coupled Systems of Superlinear Monge-Ampère Equations

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by...     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

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.     

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).     

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.     

On the Dirichlet problem for Monge-Ampère type equations

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampére type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in Ann. Scoula Norm. Sup. Pisa Cl. Sci. VIII, 143–174 2009 in their study of global regularity in optimal transportation as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.     

The Third Initial-Boundary Value Problem for a Class of Parabolic Monge-Ampère Equations

For the more general parabolic Monge-Ampère equations defined by the operator $F(D^2u + σ(x))$, 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.     

The Third Initial-boundary Value Problem for a Class of Parabolic Monge-Ampère Equations

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 e...     

A variational approach to complex Monge-Ampère equations

We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.     

Regularity Properties of the Degenerate Monge-Ampère Equations on Compact Khler Manifolds

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampère equations on compact Khler manifolds.     

On the action of the complex Monge-Ampère operator on piecewise linear functions

The action of the mixed complex Monge-Ampère operator (h ₁, …, h _k ) ? dd ^c h ₁ ∧ … ∧ dd ^c h _k on piecewise linear functions h _i is considered. The language of Monge-Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge-Ampère operator depends only on the product of the functions h _i .     

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.