Related papers: Robust and fast iterative method for the elliptic …
We prove a convergence result for a mixed finite element method for the Monge-Ampere equation to its weak solution in the sense of Aleksandrov. The unknowns in the formulation are the scalar variable and the Hessian matrix.
In this paper we propose a finite element method for solving elliptic equations with the observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier. We show…
We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Amp\`ere operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is…
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.
We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Amp\`ere Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.
We prove the convergence of a hybrid discretization to the viscosity solution of the elliptic Monge-Ampere equation. The hybrid discretization uses a standard finite difference discretization in parts of the computational domain where the…
An equation of Monge-Amp\`ere type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport…
In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtain the existence of the solutions to the Dirichlet problem for such equations in strictly pesudoconvex domains in quaternionic space. The stability and…
The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In…
This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Amp\`ere equation $\det(D^2u^0)=f$ based on the vanishing moment method which was…
We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral…
We consider a Monge-Amp\`ere functional and its corresponding second boundary value problem, a nonlinear fourth order PDE with two Dirichlet boundary conditions. This problem was solved by Trudinger-Wang and Le under the assumption that the…
This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Amp\`ere problem in the $n$-dimensional radial symmetric case. A detailed study of the fourth order problem is presented. In…
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 propose weighted and unweighted enrichment strategies to enhance the accuracy of the linear lagrangian finite element for solving the Poisson problem with Dirichlet boundary conditions. We first recall key examples of…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
This paper is devoted to study the following degenerate Monge-Amp\`ere equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=\Lambda_q (-u)^q \quad \text{in}\quad \Omega,\\ u=0 \quad\text{on}\quad \partial\Omega \end{cases}…
We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each…
We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized…
We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Amp\`{e}re type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for…