Equation Of Monge-Ampere Type Research Articles

Neumann problem for Monge-Ampere type equations revisited.

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

Noncompact L_p-Minkowski problems

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampere type equations subject to certain boundary conditions depending on the value of $p$. The special case of $p=1$ was previously studied by Pogorelov and Chou-Wang. Here, we give some sufficient conditions for the solvability for general $p\neq1$.

A class of nonlinear optimisation and applications

In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Ampere type equation. We also present some interesting applications in optimal transportation and geometric optics problems.

The Dirichlet Problem on Almost Hermitian Manifolds

We prove second-order a priori estimate on the boundary for the Dirichlet problem of a class of fully nonlinear equations on compact almost Hermitian manifolds with smooth boundary. As applications, we solve the Dirichlet problem of the Monge–Ampere type equation and of the degenerate Monge–Ampere equation.

Global regularity for a class of Monge-Ampère type equations

In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampere type equations. By employing the concept of (a, η) type domain, we emphasize that the boundary regularity depends on the convexity of the domain in nature. The key idea of our proof is to provide more effective global Holder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.

On the Dirichlet problem for degenerate Monge–Ampère type equations

We prove second derivative estimates for the Dirichlet problem of the degenerate Monge–Ampere type equations arising in optimal transportation with a cost function satisfying the optimal condition (A3w). Our proof is based on the behaviour of the equation with respect to affine transformation and on the c-convexity of solutions.

Symmetry of solutions to a class of Monge-Ampère equations

We study the symmetry of solutions to a class of Monge-Ampere type equations from a few geometric problems. We use a new transform to analyze the asymptotic behavior of the solutions near the infinity. By this and a moving plane method, we prove the radially symmetry of the solutions.

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Ampere type equation on unbounded domains. For a few special kinds of unbounded convex domains,we find the explicit formulas of the solutions to the problem. For general unbounded convex domain $\Om$, we prove the existence for solutions to the problem in the space$C^{\infty}(\Om)\cap~C~(\overline{\Om})$. We also obtain the local $C^{\frac{1}{2}}$-estimate up to the $\partial~\Omega$ and the estimate for the lower bound of the solutions.

Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations II. A Priori Estimates and the Dirichlet Problem

In this paper, we consider the Dirichlet problem for nonsymmetric Monge-Ampere type equations in which a skew-symmetric matrix is introduced. We establish uniform with respect to a class of skew-symmetric matrix bounds for δ-elliptic $C^{2, \alpha }(\overline {{\Omega }})$ -solutions to the Dirichlet problem. Then, we prove the classical solvability of the Dirichlet problem, provided those skew-symmetric matrices are sufficiently small in some sense.

Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity

We establish the monotonicity property for the mass of non-pluripolar products on compact Kahler manifolds, and we initiate the study of complex Monge-Ampere type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kahler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for non-pluripolar products with small unbounded locus.

Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature

We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampere type. These surfaces need not be continuous up to the boundary of the domain and the Dirichlet boundary condition must be interpreted in a weak sense. As a consequence, sub-solutions do not always lie below super-solutions, standard comparison principles fail, and existing convergence theorems break down. By relying on a geometric interpretation of weak solutions, we prove a relaxed comparison principle that applies only in the interior of the domain. We provide a general framework for proving existence and stability results for consistent, monotone finite difference approximations and modify the Barles-Souganidis convergence framework to show convergence in the interior of the domain. We describe a convergent scheme for the prescribed Gaussian curvature equation and present several challenging examples to validate these results.

Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations I: the d-Concavity and the Comparison Principle

We introduce the notion of d-concavity, d ≥ 0, and prove that the nonsymmetric Monge-Ampere type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Ampere type equations in the case when they are so-called δ-elliptic with respect to compared functions with 0 ≤ δ < 1.

Logarithmic gradient estimates to a Monge-Ampere Type equation on \(S^n\)

&lt;p&gt;Monge-Ampere type equations arise naturally from many geometric problems. In this paper, we are concerned with one of these Monge-Ampere type equations on dimensional sphere \(S^n\) and obtain logarithmic gradient estimate by using Bernstein technique.&lt;/p&gt;

On Pogorelov estimates in optimal transportation and geometric optics

In this paper, we prove global and interior second derivative estimates of Pogorelov type for certain Monge–Ampere type equations, arising in optimal transportation and geometric optics, under sharp hypotheses. Specifically, for the case of generated prescribed Jacobian equations, as developed recently by the second author, we remove barrier or subsolution hypotheses assumed in previous work by Trudinger and Wang (Arch Ration Mech Anal 192:403–418, 2009), Liu and Trudinger (Discret Contin Dyn Syst Ser A 28:1121–1363, 2010), Jiang et al. (Calc Var Partial Differ Equ 49:1223–1236, 2014), resulting in new applications to optimal transportation and near field geometric optics.

On Asymptotic Behaviour and W 2, p Regularity of Potentials in Optimal Transportation

In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W2, p estimates and sharp C1, α estimates for the potentials, which satisfy a Monge–Ampere type equation. The W2, p estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge–Ampere equation.

Prescribing curvature problems on the Bakry-Emery Ricci tensor of a compact manifold with boundary

The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Ampere type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.

A mathematical theory in the reflector antenna design

In this paper, we discuss a partial differential equation arising in the design of a reflector antenna. It is a fully nonlinear equation of Monge-Ampere type. We first introduce a generalized solution, then consider the existence, uniqueness and regularity of generalized solutions. Finally, we present an approach to its numerical solution.

On the Dirichlet problem for Monge-Ampère type equations

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in Ann. Scoula Norm. Sup. Pisa Cl. Sci. VIII, 143–174 2009 in their study of global regularity in optimal transportation as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.

On a uniform estimate for the quaternionic Calabi problem

We establish a C 0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

On an anisotropic Minkowski problem

In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampere type equation on the anisotropic support function and then prove the existence and uniqueness of the admissible solution to such equation. In conclusion, we give an affirmative answer to the anisotropic Minkowski problem.