Related papers: Some multi-valued solutions to Monge-Ampere equati…
We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.
We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side.
The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…
In this note, we solve the complex Monge-Amp\`ere equation for measures with a pluripolar part in compact K\"ahler manifolds. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss…
The existence and regularity of the classical plurisubharmonic solution for complex Monge-Amp\`ere equations subject to the semilinear oblique boundary condition which is C^1 perturbation of the Neumann boundary condition, are proved in the…
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Amp\`ere equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second…
We generalize and strenghten Ko{\l}odziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Amp\`ere equation.
In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.
We use a multi-scale similarity analysis which gives specific relations between the velocity, amplitude and width of localized solutions of nonlinear differential equations, whose exact solutions are generally difficult to obtain.
We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on…
The multiresolution analysis of Alpert is considered. Explicit formulas for the entries in the matrix coefficients of the refinement equation are given in terms of hypergeometric functions. These entries are shown to solve generalized…
We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each component which lie in the given multi-recurrence unless the…
We prove the long time existence and uniqueness of solution to a parabolic quaternionic Monge-Amp\`{e}re type equation on a compact hyperK\"{a}hler manifold. We also show that after normalization, the solution converges smoothly to the…
We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere…
Given a function f(x, t), its fourth (symmetric) differential is a quartic form in dx, dt. It is well-known that any quartic form in two variables can be represented as a sum of three 4th powers of linear forms. The particular case of two…
In this paper, we consider a version of parabolic complex Monge-Ampere equations, and use a PDE approach similar to Phong et al to establish $L^{\infty}$ and H\"older estimates. We also generalize the $L^{\infty}$ estimates to parabolic…
We develop an alternative approach to Degenerate complex Monge-Amp\`ere equations on compact K\"ahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We…
Given a compact K\"ahler manifold, we survey the study of complex Monge-Amp\`ere type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of…
In this paper, by the method of moving planes, we prove the symmetry result which says that classical solutions of Monge-Ampere system in the whole plane are symmetric about some point. Our system under consideration comes from the…
We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp\`ere equations. This type of equations is precisely what is needed in order to construct K\"ahler-Einstein metrics over…