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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
Book
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
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
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
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
Book
Journal of computational physics, ISSN 0021-9991, 2011, Volume 230, Issue 3, pp. 818 - 834
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
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
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
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
Journal Article
by Guan, Bo and Li, Qun
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
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
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
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
Journal Article