Related papers: Singular Abreu equations and linearized Monge-Amp\…
We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge-Amp\`ere equationn $$ \left\{\begin{eqnarray} &&-u_t+\det D^2u= \psi(x,t) \quad\quad\ \text{ in } Q_T,\newline…
We give a new proof for the interior regularity of strictly convex solutions of the Monge-Amp\`ere equation. Our approach uses a doubling inequality for the Hessian in terms of the extrinsic distance function on the maximal Lagrangian…
We investigate global H\"older gradient estimates for solutions to the Monge-Amp\`ere equation $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main…
We study generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.
Monge-Amp\`ere equations of the form, $u_{xx}u_{yy}-u_{xy}^2=F(u,u_x,u_y)$ arise in many areas of fluid and solid mechanics. Here it is shown that in the special case $F=u_y^4f(u, u_x/u_y)$, where $f$ denotes an arbitrary function, the…
We establish global $W^{2,\delta}$ estimates, for all $\delta<\frac{1}{n-1}$, for convex solutions to the Monge-Amp\`ere equation with positive $C^{2,\beta}$ right-hand side and zero boundary values on general bounded convex domains in…
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…
The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Amp\`ere equation is known independently of the convexity of the domain or Dirichlet boundary data -- when the Monge-Amp\`ere equation is posed…
We study the complex Monge-Amp\`ere equation $(dd^c u)^n=\mu$ in a strictly pseudoconvex domain $\Omega$ with the boundary condition $u=\varphi$, where $\varphi\in C(\partial\Omega)$. We provide a non-trivial sufficient condition for…
In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that…
We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Amp\`ere measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Amp\`ere…
We study the Dirichlet problem for Monge-Amp\`ere equation in bounded convex polytopes. We give sharp conditions for the existence of global $C^2$ and $C^{2,\alpha}$ convex solutions provided that a global $C^2$, convex subsolution exists.
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem…
The Monge-Amp\`{e}re equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the…
In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Amp\`ere equations \begin{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = \gamma…
We study the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are H\"older…
We give a new probabilistic construction of solutions to real Monge-Amp\`ere equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal…
We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Amp\`ere equation--as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with…
We investigate the Monge-Amp\`ere equation subject to zero boundary value and with a positive right-hand side unction assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are…
The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in H^n. We continue the study of the theory of…