Related papers: The Non-Archimedean Monge-Ampere Equation
We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an…
We survey commutative and non-commutative analogs of uniform algebras in the Archimedean settings and also offer some non-Archimedean examples. Constraints on the development of non-complex uniform algebras are also discussed.
In this paper, we consider the Monge-Amp\`{e}re type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence…
We show that the metric defined by the solution to the tropical Monge-Amp\`ere equation, as defined by Hultgren, Mazzon, and the first two authors, on the boundary of the 3-simplex is asymptotic to the Gross-Wilson metric on $S^2$ near each…
The existence, multiplicity and nonexistence of nontrivial radial convex solutions of a system of two weakly coupled Monge-Ampere equations are established with asymptotic assumptions for an appropriately chosen parameter. The proof of the…
The general solution to the Complex Monge-Amp\`ere equation in a two dimensional space is constructed.
In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the…
The main result asserts the existence of continuous solutions of the complex Monge-Amp\`ere equation with the right hand side in $L^p, p>1$, on compact Hermitian manifolds.
We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct…
We describe some natural relations connecting contact geometry, classical Monge-Ampere equations and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of Monge-Ampere equations and sheds new light on…
This paper studies the numerical approximation of solution of the Dirichlet problem for the fully nonlinear Monge-Ampere equation. In this approach, we take the advantage of reformulation the Monge-Ampere problem as an optimization problem,…
We define a general class of elliptic equations for 2-forms on 4-manifolds, of which the complex Monge-Ampere equation is a prototype. We obtain some regularity results and discuss various connections (some speculative) with modern…
We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere…
We improve our previous gradient estimate for the Monge-Amp\`ere equation on a compact Hermitian manifold and give a estimates for the non-mixed second order derivatives. These estimates are required to apply either the Evans-Krylov…
The existence and multiplicity and nonexistence of nontrivial radial convex solutions of systems of Monge-Amp\`ere equations are established with superlinearity or sublinearity assumptions for an appropriately chosen parameter. The proof of…
We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove…
We show the existence and uniqueness of bounded solutions to the degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds. We also study the asymptotics of these solutions. As applications, we give partial answers to…
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.
In this lecture delivered at the Integrable and Quantum Field Theory at Peyresq sixth meeting, we review the Lychagin's Monge-Ampere operators theory and exhibit the link it establishes between the classical problem of local equivalence for…
In this paper, we study the non-pluripolar complex Monge-Amp\`ere measure on bounded domains in \( \mathbb{C}^n \). We establish a general existence theorem for a non-pluripolar complex Monge-Amp\`ere type equation with prescribed…