Related papers: Cegrell classes and a variational approach for the…
We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral…
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity…
Given a domain $\Omega\subset \mathbf C^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal G(\Omega)$ and Monge-Amp\`ere operators $u\mapsto [dd^c u]^p$, $p\leq n$, on $\mathcal G(\Omega)$ that extend the…
We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural…
The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…
In this paper we prove an existence and uniqueness result for a Monge-Amp\`{e}re type equation in the Cegrell class $\mathcal{E}_{\chi}$.
The complex Monge-Amp\`ere operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded…
We provide the sharp $C^0$ estimate for the quaternionic Monge-Ampere equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends…
In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal…
We prove several approximation theorems of the complex Monge-Ampere operator on a compact Kahler manifold. As an application we give a new proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex…
Let $1 \leq m \leq n$ be two integers and $\Omega \Subset \C^n$ a bounded $m$-hyperconvex domain in $\C^n$. Using a variational approach, we prove the existence of the first eigenvalue and an associated eigenfunction which is…
This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish that solutions possessing comparable singularities to the complex Monge-Amp\`ere equation…
We introduce the class of semiweak Cullen-regular quaternionic functions by interpreting Cullen-regular functions as solutions to an inhomogeneous PDE in terms of the Fueter operator.
We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This…
In this paper, we study interior estimates for solutions to linearized Monge-Amp\`ere equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear…
We give a formula for the complex Monge-Ampere operator applied to the maximum of a finite number of functions.
In this paper we firstly introduce the concepts of capacity and Cegrell's classes associated to any $m$-positive closed current $T$. Next, after investigating the most imporant related properties, we study the definition and the continuity…
In this work, we study Monge-Ampere equations over closed K\"ahler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.
We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge-Amp\`ere type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu…
Quaternionic analysis relies heavily on results on functions defined on domains in $\mathbb R^4$ (or $\mathbb R^3$) with values in $\mathbb H$. This theory is centered around the concept of $\psi-$hyperholomorphic functions i.e.,…