We give the Lax representations for for the elliptic, hyperbolic and homogeneous second order Monge-Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Mong...

Multivariate probability distributions with given marginals are considered, along with linear functionals, to be minimized or maximized, acting on them. The functionals are supposed to satisfy the Monge or inverse Monge or some higher order convexity property and they may be only partially known. Existing results in connection with Monge arrays are reformulated and extended in terms of LP dual ...

When restricted to cost arrays possessing the sum Monge property, many combinatorial optimization problems with sum objective functions become signi cantly easier to solve. The more general algebraic assignment and transportation problems are similarly easier to solve given cost arrays possessing the corresponding algebraic Monge property. We show that Monge-array results for two sum-of-edge-co...

Geodesics on the infinite dimensional symmetric space H of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in X×A, where A ⊂ C is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces GC/G. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Ampère geodesics c...

We find normal forms for parabolic Monge-Ampère equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exhaust the different classes of parabolic Monge-Ampère equations with symmetry p...

We present two numerical methods for the fully nonlinear elliptic MongeAmpère equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods are proved to converge for smooth solutions. We give numerical evidence that they are also able to capture the viscosity solution of the Monge-Ampère equation. Even in the case of the degener...

In this paper, we establish global W 2,p estimates for solutions to the linearizedMonge–Ampère equations under natural assumptions on the domain, Monge– Ampère measures and boundary data. Our estimates are affine invariant analogues of the global W 2,p estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global W 2,p estimates for th...

Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature of the method. The Euler-Monge equations may be interpreted as a boundary theory arising from a linearized bulk system such that all boundary solutions follow...

We propose a model to describe the optimal distributions of residents and services in a prescribed urban area. The cost functional takes into account the transportation costs (according to a Monge–Kantorovich-type criterion) and two additional terms which penalize concentration of residents and dispersion of services. The tools we use are the Monge–Kantorovich mass transportation theory and the...