Monge–Ampère equation

E326554

elliptic partial differential equation fully nonlinear equation mathematical concept nonlinear elliptic equation partial differential equation

The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.

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All labels observed (5)

Label	Occurrences
Monge–Ampère equation canonical	2
Monge–Ampère equations	1
Yau’s solution of the Calabi conjecture	1
complex Monge–Ampère flow	1
real Monge–Ampère equation	1

Statements (49)

Predicate	Object
instanceOf	elliptic partial differential equation ⓘ fully nonlinear equation ⓘ mathematical concept ⓘ nonlinear elliptic equation ⓘ partial differential equation ⓘ
classification	fully nonlinear second-order PDE ⓘ
describes	convex geometry problems ⓘ prescribed curvature problems ⓘ
field	differential geometry ⓘ optimal transport ⓘ partial differential equations ⓘ several complex variables ⓘ
hasForm	det(D^2 u(x)) = f(x,u(x),∇u(x)) ⓘ
namedAfter	André-Marie Ampère ⓘ Gaspard Monge ⓘ
relatedTo	Abreu equation ⓘ Kantorovich duality ⓘ Monge problem in optimal transport ⓘ surface form: Monge–Kantorovich optimal transport problem curvature of hypersurfaces ⓘ determinant of the Hessian matrix ⓘ real Hessian equations ⓘ
requires	convexity of solutions in many real formulations ⓘ
solutionType	classical solution ⓘ viscosity solution ⓘ weak solution ⓘ
specialCase	complex Monge–Ampère equation ⓘ degenerate Monge–Ampère equation ⓘ Monge–Ampère equation self-linksurface differs ⓘ surface form: real Monge–Ampère equation
studiedBy	Aleksandr Danilovich Aleksandrov ⓘ Caffarelli ⓘ Louis Nirenberg ⓘ Shing-Tung Yau ⓘ
usedIn	Christoffel–Minkowski problem ⓘ surface form: Aleksandrov problem Calabi conjecture ⓘ Kähler geometry ⓘ Christoffel–Minkowski problem ⓘ surface form: Minkowski problem Yau’s proof of the Calabi conjecture ⓘ affine differential geometry ⓘ complex geometry ⓘ construction of Kähler–Einstein metrics ⓘ geometric optics ⓘ image processing ⓘ mass transport problems ⓘ mesh generation ⓘ meteorology ⓘ optimal transport maps ⓘ prescribed Gauss curvature problems ⓘ prescribed Ricci curvature problems ⓘ theory of convex functions ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Gaspard Monge → knownFor → Monge–Ampère equation ⓘ

Kähler–Ricci flow → relatedTo → Monge–Ampère equation ⓘ

this entity surface form: Yau’s solution of the Calabi conjecture

Kähler–Ricci flow → relatedTo → Monge–Ampère equation ⓘ

this entity surface form: complex Monge–Ampère flow

Monge–Ampère equation → specialCase → Monge–Ampère equation self-linksurface differs ⓘ

this entity surface form: real Monge–Ampère equation

Monge problem in optimal transport → relatedConcept → Monge–Ampère equation ⓘ

Shing-Tung Yau → knownFor → Monge–Ampère equation ⓘ

this entity surface form: Monge–Ampère equations