Monge-Ampère equation

A Monge-Ampère equation , or Monge-Ampère differential equation , is a special nonlinear second order partial differential equation in variables. ${\displaystyle n}$

It was introduced by Gaspard Monge at the beginning of the 19th century in order to solve a mass transport problem ( "Problem du remblai- déblai") for military purposes. Despite its fairly simple shape, it is generally difficult to solve. The equation is also named after André-Marie Ampère , who dealt with it around 1820.

Mathematical formulation

In general, a Monge-Ampère equation over an open area has the form ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$

${\displaystyle \det \,D^{2}u=f}$

where , where is the unknown function, a given function , and ${\displaystyle u\colon \Omega \to \mathbb {R} }$ ${\displaystyle u=u(x_{1},\ldots ,x_{n})}$ ${\displaystyle f\colon \Omega \times \mathbb {R} ^{n+1}\to \mathbb {R} }$ ${\displaystyle f=f(x_{1},\ldots ,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})}$

${\displaystyle D^{2}u={\begin{pmatrix}u_{x_{1}x_{1}}&\cdots &u_{x_{1}x_{n}}\\\vdots &\ddots &\vdots \\u_{x_{n}x_{1}}&\cdots &u_{x_{n}x_{n}}\end{pmatrix}}\qquad {\mbox{mit }}u_{x_{i}x_{j}}={\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}.}$

the Hessian matrix of . The simple shape results especially for the two-dimensional case ${\displaystyle u}$ ${\displaystyle n=2}$

${\displaystyle u_{xx}u_{yy}-u_{xy}^{2}=f}$

with and the functions and . In the case of n = 2, however, the following representation is often referred to as a general Monge-Ampère equation: ${\displaystyle (x,y)\in \Omega \subset \mathbb {R} ^{2}}$ ${\displaystyle u(x,y)}$ ${\displaystyle f(x,y,u,u_{x},u_{y})}$

${\displaystyle Ar+2Bs+Ct+(rt-s^{2})=E,\qquad {\mbox{mit }}r=u_{xx},\ s=u_{xy},\ t=u_{yy},\ p=u_{x},\ q=u_{y},}$

where and are functions of ( ). One immediately recognizes that with and the above simpler shape results. ${\displaystyle A,B,C}$ ${\displaystyle E}$ ${\displaystyle x,y,u,p,q}$ ${\displaystyle A=B=C=0}$ ${\displaystyle E=f}$

Concrete example

Be and . Then there is a solution of the Monge-Ampère differential equation, because and therefore ${\displaystyle n=2}$ ${\displaystyle f(x,y)=4(1-y^{2})(1-x^{2})-16x^{2}y^{2}}$ ${\displaystyle u(x,y)=(1-x^{2})(1-y^{2})}$ ${\displaystyle u_{xx}=-2(1-y^{2}),}$ ${\displaystyle u_{yy}=-2(1-x^{2}),}$ ${\displaystyle u_{xy}=u_{yx}=-4xy,}$ ${\displaystyle \det \,D^{2}u=\det {\begin{pmatrix}-2(1-y^{2})&-4xy\\-4xy&-2(1-x^{2})\end{pmatrix}}=f(x,y).}$

Classification as a partial differential equation

A Monge-Ampère equation is a fully nonlinear partial differential equation of the second order in variables. Explanations: ${\displaystyle n}$

• "Partial differential equation", because we are looking for a function that depends on several variables and whose partial derivatives must obey the given equation. ${\displaystyle u}$

• "fully non-linear", since all terms appear with second (i.e. the highest) derivatives of quadratically. ${\displaystyle u}$

An important class are the elliptical Monge-Ampère equations, which satisfy the conditions and , or in the simpler form simple . ${\displaystyle n=2}$ ${\displaystyle AC-B^{2}+E>0}$ ${\displaystyle t+A>0}$ ${\displaystyle f>0}$

Applications

Most of the applications of the Monge-Ampère equation are of an intramathematical nature, especially in differential geometry. In the Minkowski problem, for example, a strictly convex hypersurface with a given Gaussian curvature is sought, which leads to a Monge-Ampère equation. The problem was solved by Nirenberg in 1953 .

An unexpected application in the field of string theory resulted from a result published by Yau in 1978 , which proved a conjecture by Calabi about the curvature of certain Kähler manifolds by solving a complex Monge-Ampère equation ( Yau's theorem ). Accordingly, one speaks today of Calabi-Yau manifolds .

Significant contributions to Monge-Ampère's equations in the course of the 20th century came from Hermann Weyl , Franz Rellich , Erhard Heinz , Louis Nirenberg , Shing-Tung Yau , Luis Caffarelli , Alexei Wassiljewitsch Pogorelow , Thierry Aubin , Sébastien Boucksom , Alessio Figalli and Guido de Philippis .

Web links

• Plots of solutions of Monge-Ampèrescher equations ( Memento of February 5, 2005 in the Internet Archive )
• Entry in MathWorld (Engl.)
• Pogorelov Monge-Ampère equation, Encyclopedia of Mathematics, Springer