We study strictly convex Alexandrov solutions u of the real MongeAmpère equation det(∇2u) = f , where f is measurable, positive, and bounded away from 0 and ∞. Under only these assumptions we prove interior Wregularity of u.

are smooth, given that k is smooth and nonnegative. When u is radial, (1) reduces to a nonlinear ODE on [0, 1) that is singular at the endpoint 0. It is thus easy to prove that u is always smooth away from the origin, even where k vanishes, but smoothness at the origin is more complicated, and determined by the order of vanishing of k there. In fact, Monn [9] proves that if k = k (x) is indepen...

Given an n× n× p cost array C we consider the problem p-P3AP which consists in finding p pairwise disjoint permutations φ1, φ2, . . . , φp of {1, . . . , n} such that ∑p k=1 ∑n i=1 ciφk(i)k is minimized. For the case p = n the planar 3-dimensional assignment problem P3AP results. Our main result concerns the p-P3AP on cost arrays C that are layered Monge arrays. In a layered Monge array all n×n...

The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ainpere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations. 2000 Mathematics Subject Classification: 35J15, 35J20, 35J70. When learning complex analysis, it was a remarkable fact t...

We define the fractional powers Ls=(−aij(x)∂ij)s, 0<s<1, of nondivergence form elliptic operators L=−aij(x)∂ij in bounded domains Ω⊂Rn, under minimal regularity assumptions on coefficients aij(x) and boundary ∂Ω. show that these appear several applications such as Monge–Ampère equations, elasticity, finance. The solution u to nonlocal Poisson problem{(−aij(x)∂ij)su=finΩu=0on∂Ω is characterized ...

We present efficient data structures for submatrix maximum queries in Monge matrices and Monge partial matrices. For n× n Monge matrices, we give a data structure that requires O(n) space and answers submatrix maximum queries in O(logn) time. The best previous data structure [Kaplan et al., SODA‘12] required O(n logn) space and O(log n) query time. We also give an alternative data structure wit...

We study inverse-Monge assignment games, namely cooperative assignment games in which the assignment matrix satisfies the inverse Monge property. For square inverse-Monge assignment games, we describe their cores and we obtain a closed formula for the buyers-optimal and the sellers-optimal core allocations. We also apply the above results to solve the non-square case.

The eigenspace structure for a given n × n concave Monge matrix in a max-plus algebra is described. Based on the description, an O(n) algorithm for computing the eigenspace dimension is formulated, which is faster than the previously known algorithms. Analogous results for convex Monge matrices have been published in [M. Gavalec, J. Plavka, Structure of the eigenspace of a Monge matrix in max-p...

We present the exact solution to the non linear Monge differential equation λ(x, t)λx(x, t) = λt(x, t). It is widely accepted that the Monge equation is equivalent to the ODE .. X= 0 of free motion for particular conditions. Furthermore, the Monge Type equations are connected withX = F ( . X, X; t), which can be integrated with quadratures [1]. Other asymptotic solutions are discussed, see e.g....