The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge–Ampère operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge–Ampère measures \((\triangle u_j)^n\rightarrow (\triangle u)^n\). We also obtain an equivalent condition of “convergence in \(C_{n-1}\)-capacity” by using methods from Xing (Proc Am Math Soc 124(2):457–467, 1996). As an application, the range of the quaternionic Monge–Ampère operator is discussed.     

Global and local definition of the Monge–Ampère operator on compact Kähler manifolds

The aim of this Note is to give a sufficient condition in order for a function in the global domain of definition of the Monge–Ampère operator not to belong to the local domain of the former in the sense of Cegrell, when one looks at the n-dimensional complex projective space. Using this result, we show that the subsolution theorem is false for functions in the local domain of definition of the Monge–Ampère operator on such a projective space.     

Dirichlet problem of quaternionic Monge–Ampère equations

In this paper, the author studies quaternionic Monge–Ampère equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper aims to answer the question proposed by Semyon Alesker in [3]. It also extends relevant results in [8] to the quaternionic vector space.     

Monge–Ampère measures on pluripolar sets

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kołodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.     

Maximum and Comparison Principles for Convex Functions on the Heisenberg Group

Abstract

We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge–Ampère type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge–Ampère measures for convex functions in the Heisenberg group.     

On certain degenerate and singular elliptic PDEs II: Divergence-form operators,Harnack inequalities,and applications

By means of the Monge–Ampère real-analysis and PDE techniques associated to certain convex functions, an approach towards Harnack inequalities is developed that simultaneously extends the one for uniformly elliptic operators from the De Giorgi–Nash–Moser theory and the one for the linearized Monge–Ampère operator from the Caffarelli–Gutiérrez theory. Applications include regularity properties for solutions to divergence-form elliptic equations with power-like singularities and $C^2$-estimates for solutions to the Monge–Ampère equation.     

The Wolff potential estimate for solutions to elliptic equations with signed data

In this note, we obtain sharp bounds for the Green’s function of the linearized Monge–Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge–Ampère measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman–Stampacchia–Weinberger for uniformly elliptic operators in divergence form. We also obtain the L ^p integrability for the gradient of the Green’s function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge–Ampère equation.     

Global W^{2,p} estimates for solutions to the linearized Monge–Ampère equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge–Ampère equations under natural assumptions on the domain, Monge–Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global $W^{2,p}$ estimates for the Monge–Ampère equations.     

Radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations

In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations. By applying a well‐known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge‐Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

An extension of Jörgens–Calabi–Pogorelov theorem to parabolic Monge–Ampère equation

We extend a theorem of Jörgens, Calabi and Pogorelov on entire solutions of elliptic Monge–Ampère equation to parabolic Monge–Ampère equation, and obtain delicate asymptotic behavior of solutions at infinity. For the dimension \(n\ge 3\), the work of Gutiérrez and Huang in Indiana Univ. Math. J. 47, 1459–1480 (1998) is an easy consequence of our result. And along the line of approach in this paper, we can treat other parabolic Monge–Ampère equations.     

Local indicators for plurisubharmonic functions

The notion of index, classical in number theory and its calculation by P. Lelong (1997) for plurisubharmonic functions, allows to define an indicator which is applied to the study of the Monge–Ampère operator and a pluricomplex Green function.     

Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>: the case of compact Kähler manifolds

We prove that on compact Kähler manifolds solutions to the complex Monge–Ampère equation, with the right-hand side in L ^p , p > 1, are Hölder continuous.     

Stability of isoperimetric type inequalities for some Monge–Ampère functionals

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampère functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampère eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\) -torsional rigidity, a natural extension of the usual torsional rigidity.     

Quasi-Neutral Limit of the Euler–Poisson and Euler–Monge–Ampère Systems

ABSTRACT

This paper studies the pressureless Euler–Poisson system and its fully nonlinear counterpart, the Euler–Monge–Ampère system, where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.     

On Yau’s theorem for effective orbifolds

In 1978 Yau (Yau, 1978) confirmed a conjecture due to Calabi (1954) stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature (Joyce, 2000; Boyer and Galicki, 2008; Demailly and Kollár, 2001). In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau’s original continuity method to the setting of effective orbifolds in order to solve a Monge–Ampère equation. We then outline how to obtain Kähler–Einstein metrics on orbifolds with negative first Chern class by solving a slightly different Monge–Ampère equation. We conclude by listing some explicit examples of Calabi–Yau orbifolds, which consequently admit Ricci flat metrics by Yau’s theorem for effective orbifolds.     

Monge–Ampère Operator on Four Dimensional Almost Complex Manifolds

We define the Monge–Ampère operator \({(i\partial {\bar{\partial }}u)^{2}}\) for continuous J-plurisubharmonic functions on four dimensional almost complex manifolds.     

A note on the weighted log canonical thresholds of plurisubharmonic functions

In this note, we give a characterization for the weighted log canonical thresholds of plurisubharmonic functions. As an application, we prove an inequality for weighted log canonical thresholds and Monge–Ampère masses.     

Partial Regularity for Singular Solutions to the Monge‐Ampère Equation

We prove that solutions to the Monge‐Ampère inequality in ?ⁿ are strictly convex away from a singular set of Hausdorff (n‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D²u = 1 with singular set of Hausdorff dimension as close as we like to n‐1. As a consequence we obtain W^2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc.     

Light reflection is nonlinear optimization

In this paper, we show that the near field reflector problem is a nonlinear optimization problem. From the corresponding functional and constraint function, we derive the Monge–Ampère type equation for such a problem.