Our aim is to give a version of the Moser-Trudinger inequality in setting complex geometry. As very particular case, our result already gives new for functions Sobolev space $W^{1,2}$ domain $R^2$. We also deduce necessary condition Monge-Ampere equation given measure on compact Kahler manifold admit Holder continuous solution.

We present a method for designing freeform optical surfaces illumination optics. By the laws of reflection, refraction and conservation energy, fully nonlinear PDE, Monge-Ampere equation, is derived surface. edge ray principle transport boundary condition obtained. solve hyperbolic variant PDE using least-squares method, resulting in saddle parallel source far-field target.

This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of our condition A3, under which we proved interior regularity in a recent paper with Xi-nan Ma. Consequently they include the quadratic cost func...

In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities new affine invariant entropy obtain on functional surface area lower upper bounds Kullback-Leibler divergence in terms area. The lead to L_p-affine areas convex bodies.

The Dirichlet problem for a Monge-Ampère equation corresponding to a nonnegative, possible degenerate cohomology class on a Kähler manifold with boundary is studied. C1,α estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C1,α geodesic rays in the space of Kähler potentials are constructed for each test configurat...

We study the Monge-Ampere operator $u \mapsto (\text{dd}^c u)^n$ and compare it with Dinh-Sibony's intersection product defined via density currents. show that if $u$ is a plurisubharmonic function belonging to Blocki-Cegrell class, then Dinh-Sibony $n$-fold self-product of $\text{dd}^c u$ exists coincides $(\text{dd}^c u)^n$.