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2017, Zurich lectures in advanced mathematics, ISBN 9783037191705, x, 200 pages

"The Monge-Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry...

Monge-Ampère equations | Differential equations, Partial | Geometry, Differential | Mathematical physics

Monge-Ampère equations | Differential equations, Partial | Geometry, Differential | Mathematical physics

Book

2016, Graduate studies in mathematics, ISBN 9781470426071, Volume 171., viii, 368

Differential equations, Elliptic | Boundary value problems for second-order elliptic equations | Partial differential equations | Differential equations, Nonlinear | Elliptic equations and systems | Quasilinear elliptic equations with mean curvature operator | Elliptic Monge-Ampère equations | Nonlinear elliptic equations

Book

3.
Full Text
Numerical solution of the Optimal Transportation problem using the Monge–Ampère equation

Journal of computational physics, ISSN 0021-9991, 03/2014, Volume 260, Issue 1, pp. 107 - 126

A numerical method for the solution of the elliptic Monge–Ampère Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT...

Fully nonlinear elliptic partial differential equations | Monotone schemes | Numerical methods | Optimal Transportation | Monge Ampère equation | Convexity | Finite difference methods | Viscosity solutions | Physical Sciences | Computer Science, Interdisciplinary Applications | Technology | Physics, Mathematical | Computer Science | Physics | Science & Technology | Transportation industry | Monge-Ampere equation | Computation | Disengaging | Boundary conditions | Mathematical models | Density | Optimization | Finite difference method | Mathematics

Fully nonlinear elliptic partial differential equations | Monotone schemes | Numerical methods | Optimal Transportation | Monge Ampère equation | Convexity | Finite difference methods | Viscosity solutions | Physical Sciences | Computer Science, Interdisciplinary Applications | Technology | Physics, Mathematical | Computer Science | Physics | Science & Technology | Transportation industry | Monge-Ampere equation | Computation | Disengaging | Boundary conditions | Mathematical models | Density | Optimization | Finite difference method | Mathematics

Journal Article

Acta mathematica vietnamica, ISSN 0251-4184, 9/2019, Volume 44, Issue 3, pp. 723 - 749

In this paper, we consider the Dirichlet problem for nonsymmetric Monge-Ampère type equations in which a skew-symmetric matrix is introduced...

Monge-Ampère type equations | Second derivative estimates | The method of continuity | δ -elliptic solutions | Mathematics, general | Mathematics | 35J66 | Mathematical analysis | Dirichlet problem | Matrix methods

Monge-Ampère type equations | Second derivative estimates | The method of continuity | δ -elliptic solutions | Mathematics, general | Mathematics | 35J66 | Mathematical analysis | Dirichlet problem | Matrix methods

Journal Article

Indiana University mathematics journal, ISSN 0022-2518, 1/2011, Volume 60, Issue 5, pp. 1713 - 1722

We prove that any C1,1 solution to the complex Monge-Ampère equation det(uij̄) = f with 0 < f ∈ Cα is in C2,α for α ∈ (0,1).

A priori knowledge | Continuity equations | Mathematical theorems | Elliptic equations | Dirichlet problem | Bleeding time | Monge Ampere equation | College mathematics | Interior estimate | Complex Monge-Ampere equation

A priori knowledge | Continuity equations | Mathematical theorems | Elliptic equations | Dirichlet problem | Bleeding time | Monge Ampere equation | College mathematics | Interior estimate | Complex Monge-Ampere equation

Journal Article

SIAM journal on numerical analysis, ISSN 0036-1429, 1/2011, Volume 49, Issue 3/4, pp. 1692 - 1714

The elliptic Monge—Ampère 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...

Viscosity | Mathematical monotonicity | Eigenvalues | Boundary conditions | Convexity | Monge Ampere equation | Newtons method | Stencils | Jacobians | Fully nonlinear elliptic partial differential equations | Monotone schemes | Convexity constraints | Monge-Ampère equations | Nonlinear finite difference methods | Viscosity solutions | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Viscosity | Mathematical monotonicity | Eigenvalues | Boundary conditions | Convexity | Monge Ampere equation | Newtons method | Stencils | Jacobians | Fully nonlinear elliptic partial differential equations | Monotone schemes | Convexity constraints | Monge-Ampère equations | Nonlinear finite difference methods | Viscosity solutions | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Journal Article

1990, Lecture notes in mathematics, ISBN 0387531033, Volume 1445., xv, 123

...-Ampère equations, a theory largely developed by H. Lewy and E. Heinz which has never been presented in book form...

Monge-Ampère equations | Global analysis (Mathematics) | Differential Geometry | Global differential geometry | Analysis

Monge-Ampère equations | Global analysis (Mathematics) | Differential Geometry | Global differential geometry | Analysis

Book

1994, ISBN 3540136207, xxi, 510

Book

Journal of computational physics, ISSN 0021-9991, 2011, Volume 230, Issue 3, pp. 818 - 834

The elliptic Monge–Ampère equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing...

Monotone schemes | Monge–Ampère equations | Fully nonlinear elliptic Partial Differential Equations | Convexity constraints | Nonlinear finite difference methods | Viscosity solutions | Monge-Ampère equations | Physical Sciences | Computer Science, Interdisciplinary Applications | Technology | Physics, Mathematical | Computer Science | Physics | Science & Technology | Image processing | Dynamic meteorology | Equipment and supplies | Nonlinear dynamics | Construction | Monge-Ampere equation | Discretization | Computation | Mathematical analysis | Solvers | Mathematical models

Monotone schemes | Monge–Ampère equations | Fully nonlinear elliptic Partial Differential Equations | Convexity constraints | Nonlinear finite difference methods | Viscosity solutions | Monge-Ampère equations | Physical Sciences | Computer Science, Interdisciplinary Applications | Technology | Physics, Mathematical | Computer Science | Physics | Science & Technology | Image processing | Dynamic meteorology | Equipment and supplies | Nonlinear dynamics | Construction | Monge-Ampere equation | Discretization | Computation | Mathematical analysis | Solvers | Mathematical models

Journal Article

Journal of scientific computing, ISSN 1573-7691, 03/2018, Volume 76, Issue 3, pp. 1839 - 1867

In this paper, we propose a monotone mixed finite difference scheme for solving the two-dimensional Monge–Ampère equation...

Mixed schemes | Computational Mathematics and Numerical Analysis | Monotone schemes | Algorithms | Monge–Ampère equations | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics | Nonlinear elliptic partial differential equations | Hamilton–Jacobi–Bellman equations | Viscosity solutions | Finite difference methods | Analysis | Differential equations | Viscosity | Monge-Ampere equation | Discretization | Mathematical analysis | Optimization | Convergence | Finite difference method

Mixed schemes | Computational Mathematics and Numerical Analysis | Monotone schemes | Algorithms | Monge–Ampère equations | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics | Nonlinear elliptic partial differential equations | Hamilton–Jacobi–Bellman equations | Viscosity solutions | Finite difference methods | Analysis | Differential equations | Viscosity | Monge-Ampere equation | Discretization | Mathematical analysis | Optimization | Convergence | Finite difference method

Journal Article

Mathematical models & methods in applied sciences, ISSN 0218-2025, 05/2015, Volume 25, Issue 5, pp. 803 - 837

.... This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge-Ampere equation...

Inverse reflector problem | elliptic Monge-Ampère equation | B-spline collocation method | Picard-type iteration | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Inverse reflector problem | elliptic Monge-Ampère equation | B-spline collocation method | Picard-type iteration | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Journal Article

Journal of mathematical sciences (New York, N.Y.), ISSN 1072-3374, 07/2020, Volume 248, Issue 3, pp. 303 - 337

.... We present a proof of the Jörgens–Calabi–Pogorelov theorem about improper convex affine spheres based on the study of complete convex solutions of the simplest Monge–Ampère equation...

Nonlinear equations | Polynomials | Elliptic functions | Mathematical analysis

Nonlinear equations | Polynomials | Elliptic functions | Mathematical analysis

Journal Article

Computer methods in applied mechanics and engineering, ISSN 0045-7825, 02/2006, Volume 195, Issue 13-16, pp. 1344 - 1386

... functional involves the determinant of the gradient of vector-valued functions. To solve the Monge–Ampère equation we consider two methods...

Monge–Ampere equations | Dirichlet problem | Elliptic equations | Monge-Ampere equations | Mathematics, Interdisciplinary Applications | Engineering | Physical Sciences | Technology | Engineering, Multidisciplinary | Mechanics | Mathematics | Science & Technology | Methods | Algorithms | Numerical Analysis

Monge–Ampere equations | Dirichlet problem | Elliptic equations | Monge-Ampere equations | Mathematics, Interdisciplinary Applications | Engineering | Physical Sciences | Technology | Engineering, Multidisciplinary | Mechanics | Mathematics | Science & Technology | Methods | Algorithms | Numerical Analysis

Journal Article

Acta mathematica scientia, ISSN 0252-9602, 07/2018, Volume 38, Issue 4, pp. 1285 - 1295

...-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem...

65K10 | gradient conjugate method | finite element Galerkin method | elliptic Monge-Ampère equation | 35J60 | 65N30 | Physical Sciences | Mathematics | Science & Technology

65K10 | gradient conjugate method | finite element Galerkin method | elliptic Monge-Ampère equation | 35J60 | 65N30 | Physical Sciences | Mathematics | Science & Technology

Journal Article

Advances in mathematics (New York. 1965), ISSN 0001-8708, 10/2013, Volume 246, pp. 351 - 367

... torsion and general (non-pseudoconvex) boundary data. We present our methods, which work for more general equations, by considering a specific equation which resembles the complex Monge...

Strict concavity property | Dirichlet problem | Fully nonlinear elliptic equations | A priori estimates | Hermitian manifolds | Donaldson conjecture | Physical Sciences | Mathematics | Science & Technology

Strict concavity property | Dirichlet problem | Fully nonlinear elliptic equations | A priori estimates | Hermitian manifolds | Donaldson conjecture | Physical Sciences | Mathematics | Science & Technology

Journal Article

IMA journal of numerical analysis, ISSN 0272-4979, 2015, Volume 35, Issue 3, pp. 1150 - 1166

We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical solutions of the equation...

Lagrange elements | Time marching | Smooth solutions | Elliptic Monge-Ampère | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Lagrange elements | Time marching | Smooth solutions | Elliptic Monge-Ampère | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology

Journal Article

Journal of scientific computing, ISSN 1573-7691, 09/2018, Volume 79, Issue 1, pp. 1 - 47

We discuss in this article a novel method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation...

Fully nonlinear elliptic partial differential equations | Operator-splitting method | Mixed finite element methods | Computational Mathematics and Numerical Analysis | Algorithms | Monge–Ampère equations | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Finite element approximations | Variational crimes | Tychonoff regularization | Mathematics | Physical Sciences | Mathematics, Applied | Science & Technology | Methods | Differential equations | Boundary value problems | Divergence | Approximation | Monge-Ampere equation | Methodology | Finite element method | Operators (mathematics) | Domains | Splitting | Robustness (mathematics) | Mathematical analysis | Rectangles | Newton methods | Computing time

Fully nonlinear elliptic partial differential equations | Operator-splitting method | Mixed finite element methods | Computational Mathematics and Numerical Analysis | Algorithms | Monge–Ampère equations | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Finite element approximations | Variational crimes | Tychonoff regularization | Mathematics | Physical Sciences | Mathematics, Applied | Science & Technology | Methods | Differential equations | Boundary value problems | Divergence | Approximation | Monge-Ampere equation | Methodology | Finite element method | Operators (mathematics) | Domains | Splitting | Robustness (mathematics) | Mathematical analysis | Rectangles | Newton methods | Computing time

Journal Article

Communications in partial differential equations, ISSN 1532-4133, 12/2019, Volume 45, Issue 6, pp. 1 - 26

We study the obstacle problem for a nonlocal, degenerate elliptic Monge-Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.

Primary: 35J96 | degenerate elliptic nonlinear nonlocal equation | Free boundary problem | regularity | Secondary: 35B65 | fractional Monge-Ampère equation | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology | Regularity | Free boundaries

Primary: 35J96 | degenerate elliptic nonlinear nonlocal equation | Free boundary problem | regularity | Secondary: 35B65 | fractional Monge-Ampère equation | Physical Sciences | Mathematics | Mathematics, Applied | Science & Technology | Regularity | Free boundaries

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 03/2013, Volume 254, Issue 6, pp. 2515 - 2531

.... The central model in this work is given by the following fourth order elliptic equation,Δ2u=det(D2u)+λf,x∈Ω⊂R2,conditions on ∂Ω. The framework to study the problem deeply depends on the boundary conditions.

Higher order elliptic equations | Variational methods | Gaussian curvature | Monge–Ampère type equations | Existence of solutions | Growth problems | Monge-Ampère type equations | Physical Sciences | Mathematics | Science & Technology | Epitaxy | Analysis

Higher order elliptic equations | Variational methods | Gaussian curvature | Monge–Ampère type equations | Existence of solutions | Growth problems | Monge-Ampère type equations | Physical Sciences | Mathematics | Science & Technology | Epitaxy | Analysis

Journal Article