Probability measures with either finite Monge-Ampere energy or entropy have played a central role in recent developments Kahler geometry. In this note we make systematic study of quasi-plurisubharmonic potentials whose entropy. We show that these belong to the class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes complex dimension, and provide examples showing critical exponent is sharp. Our ...

We introduce a notion of fractional convexity that extends naturally the usual in Euclidean space to setting. With this convexity, we study convex envelope inside domain an exterior datum (the largest possible function is below outside) and show characterized as viscosity solution non-local equation given by infimum among all directions 1-dimensional laplacian. For prove existence, uniqueness c...

This paper studies the complex Monge–Ampère equations for F-plurisubharmonic functions in bounded F-hyperconvex domains. We give sufficient conditions this equation to solve measures with a singular part.

In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich's weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence ...

One of the most efficient methods for the study of boundary and periodic problems for nonlinear differential equations and inclusions, consists in the operator treatment of these problems in suitable functional spaces. However, for a number of problems of this sort, the maps constructed in functional spaces do not possess “nice” properties on the whole domain, but only on some open neighborhood...

In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich’s weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence ...

On any compact Kähler manifold, Mabuchi [16], Semmes [17], and Donaldson [5] introduced a Weil-Peterson type metric in the space of Kähler metrics and proved that it is a formally non-positively curved symmetric space of “noncompact” type. According to [17], the geodesic equation can be transformed into a homogenous complex Monge-Ampere equation. In [5], Donaldson proposed an ambitious program ...