Recently, following a new approach, the symmetry of the solutions to overdetermined problems has been established for a class of Hessian type operators. In this paper we prove that the radial solution of the overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.

S. T. Yau has done extremely deep and powerful work in differential geometry and partial differential equations. His resolution of the Calabi conjecture on the existence of KählerEinstein metrics, by solving a complex Monge-Ampère equation on Kähler manifolds, is of fundamental importance in both mathematics and physics. We would like to recall in this article the contributions of S. Y. Cheng a...

1. Background 5 2. Plurisubharmonic functions 8 3. The complex Monge–Ampère operator 10 3.1. Bedford’s and Taylor’s definition of the complex Monge–Ampère operator 11 3.2. Cegrell’s definition of the complex Monge–Ampère operator 12 4. The Dirichlet problem for the complex Monge–Ampère operator 14 4.1. Boundary blow-up problems for the complex Monge–Ampère operator 17 4.2. Comparison principles...

A complete description of the eigenspace structure for a given n × n Monge matrix in a max-plus algebra is presented. Based on the description, an O(n2) algorithm for computing the eigenspace dimension is formulated, which is faster than the previously known algorithms. © 2007 Elsevier B.V. All rights reserved. MSC: primary 04A72; secondary 05C50;15A33

An M x n matrix C is called Mange matrix if c,, + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j < s d n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to o...

This thesis develops a body of versatile algorithmic techniques. We demonstrate the power and generality of these techniques by applying them to a wide variety of problems. These problems are drawn from such diverse areas of study as computational geometry, VLSI theory, operations research, and molecular biology. The algorithmic techniques described in this thesis are centered around a family o...

We introduce a convergent finite difference method for solving the optimal transportation problem on sphere. The applies to both traditional squared geodesic cost (arising in mesh generation) and logarithmic reflector antenna design problem). At each point sphere, we replace surface PDE with Generated Jacobian equation posed local tangent plane using normal coordinates. discretization is inspir...

In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez’s interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upp...