Related papers: Symmetry Results for classical solutions of Monge-…
We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side.
We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere…
We prove the existence of C^{\infty} local solutions to a class of mixed type Monge-Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point.…
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…
In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, \theta, z)$, we investigate the shape of solutions whose derivative in $\theta$ vanishes at the…
We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic,…
In this work we prove that the unique 1-convex solution of the Monge problem contructed from the solution of the Monge-Kantorovitch problem between the Wiener measure and a target measure which has a log-concave density w.r.to the Wiener…
We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric…
In this note, we solve the complex Monge-Amp\`ere equation for measures with a pluripolar part in compact K\"ahler manifolds. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss…
We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the H\"older regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$,…
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
This paper develops a rigorous analytic framework for the hyperbolic Monge-Amp\`ere equation on strip-like domains, which model wrinkled patterns in thin elastic sheets. Our work addresses the rigid side of the classical…
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…
In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action…
We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Amp\`ere equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on…
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 propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate…
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
We consider an elliptic polyharmonic problem of any order which takes place in a punctured bounded domain with Navier conditions. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced…
We study an elliptic system coupled by Monge-Amp\`{e}re equations: \begin{center} $\left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^\alpha, & \hbox{in $\Omega,$} det~D^{2}u_{2}={(-u_{1})}^\beta, & \hbox{in $\Omega,$} u_{1}<0, u_{2}<0,&…