相关论文: The Monge-Amp\`ere equation in cones and polygonal…
The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\`ere equation. While recent…
We present the results from our earlier paper (arXiv:math/0602484) on the affine normal flow on noncompact convex hypersurfaces in affine space from a more PDE point of view, emphasizing the estimates involved. Our results concern the…
In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is $C^{2,\alpha}$…
In this paper, we investigate the interior H\"older regularity of solutions to the linearized Monge-Amp\`ere equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic…
In this paper, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge--Amp\`ere equation. Among the results we obtain through these connections are: $(i)$~a theorem on the continuous extension…
We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Amp\`ere equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This…
We determine the global behavior of every C^2-solution to the two-dimensional degenerate Monge-Ampere equation, u_{xx}u_{yy}-u_{xy}^2=0, over the finitely punctured plane. With this, we classify every solution in the once or twice punctured…
In this paper, we study the global regularity for regular Monge-Amp\`ere type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of…
In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on a strongly pseudoconvex domain by the viscosity method. We extend the results in [EGZ15b] on the existence of solution and the…
In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of…
This paper introduces a novel neural network-based approach to solving the Monge-Amp\`ere equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks…
In this survey article we discuss the interior and boundary regularity of Alexandrov solutions to $\det D^2u = 1$. We include some topics which it seems were not recently revisited in similar articles, including Calabi's interior $C^3$…
We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Amp\'ere equation. Mainly we use the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to…
We study a Monge-Amp\`ere type equation that interpolates the classical {\sigma_2} -Yamabe equation in conformal geometry and the 2-Hessian equation in dimension 4.
In this paper, we solve the Dirichlet problem for Monge-Amp\`ere type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.
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…
In this paper, we introduce a family of real Monge-Amp\`ere functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated…
The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image…
The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge--Amp\`ere type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature.
In this article, we implement the algorithm based on the convex integration result proved in [Lewicka-Pakzad Analysis and PDE (2017)] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approx- imating the anomalous…