Monge-Ampère equations of the form, uxxuyy − u 2 xy = F (u, ux, uy) arise in many areas of fluid and solid mechanics. Here it is shown that in the special case F = u y f(u, ux/uy), where f denotes an arbitrary function, the Monge-Ampère equation can be linearized by using a sequence of Ampère, point, Legendre and rotation transformations. This linearization is a generalization of three examples...

This note presents a detailed and self-contained discussion of the Dirichlet problem of real Monge-Ampère equations in strictly convex domains and complex Monge-Ampère equations in strongly pseudo-convex domains. Sections 1.1 and 1.2 follow [2] and [3] respectively, while Sections 2.1, 2.2 and 2.3 are based on [5], [4] and [1] respectively. This note is written for lectures in the Special Lectu...

A grammar-compressed (GC) string is a string generated by a context-free grammar. This compression model captures many practical applications, and includes LZ78 and LZW compression as a special case. We give an efficient algorithm for threshold approximate matching on a GC-text against a plain pattern. Our algorithm improves on existing algorithms whenever the pattern is sufficiently long. The ...

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampère type equations. An ...

We give a sufficient condition on the Monge-Ampère mass of a plurisubharmonic function u for exp(−2u) to be locally integrable.

A classical result of Aleksandrov allows us to estimate the size of a convex function u at a point x in a bounded domain Ω in terms of the distance from x to the boundary of Ω if ∫ Ω detD 2u dx < ∞. This estimate plays a prominent role in the existence and regularity theory of the Monge-Ampère equation. Jerison proved an extension of Aleksandrov’s result that provides a similar estimate, in som...

[1] Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter W. Shor, and Robert E. Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2:195–208, 1987. [2] M.T. de Berg, O. Schwarzkopf, M.J. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2000. [3] Haim Kaplan, Shay Mozes, Yahav Nussbaum, and Micha Sharir. Submatrix maximu...

In this paper, we study the optimal transportation on the hemisphere, with the cost function c(x, y) = 1 2 d(x, y), where d is the Riemannian distance of the round sphere. The potential function satisfies a Monge-Ampère type equation with natural boundary condition. We obtain the a priori oblique estimate without using any uniform convexity of domains, and in particular for two dimensional case...