Monge–Ampère equation

A monge - Ampère equation, or monge - Ampère differential equation is a special nonlinear partial differential equation of second order in n variables.

It was introduced by Gaspard Monge in the early 19th century, a mass transport problem ( " problème du remblai - déblai " about " problem of embankment and excavating " ) to solve for military purposes. Despite their relatively simple form, it is generally difficult to solve.

Mathematical formulation

General has a monge - Ampère equation over an open area in the form of

Where, with the unknown function is a given function, and

The Hessian matrix of u specifically for the two-dimensional case n = 2 results in the simple form

And with the functions and. Often n = 2, but also referred to the following representation as a general monge - Ampère equation for the case:

Wherein A, B, C and D functions of ( x, y, u, p, q) are. One immediately recognizes that = f the above results in simpler form with A = B = C = 0 and e.

A concrete example

Let n = 2 and. Then is a solution of monge - Ampère differential equation, and therefore for

Classification as a partial differential equation

A monge - Ampère equation is a fully nonlinear partial differential equation of second order in n variables. Notes:

• " partial differential equation " because it is a depending on several variables function u wanted, its partial derivatives must obey the given equation.

• " fully non-linear", since all the terms with the second (ie the highest ) appear derivatives of u square.

An important class are the elliptic monge - Ampère equations that satisfy the conditions and for n = 2, or simply in the simpler form.

Applications

Most applications of monge - Ampère equation are innermathematical Art In Minkowski problem, for example a strictly convex hypersurface with prescribed Gaußkrümmung is sought, which leads to a monge - Ampère equation. The problem was solved in 1953 by Nirenberg.

An unexpected application in the field of string theory resulted from a 1978 published result of Yau, of a conjecture of Calabi via the curvature of certain Kähler manifolds with the help of the solution of a complex monge - Ampère equation proved ( set of Yau ). We speak today of corresponding Calabi -Yau manifolds.

Significant contributions to monge - Ampère equations in the course of the 20th century came from Hermann Weyl, Franz Rellich, Erhard Heinz, Louis Nirenberg, Shing -Tung Yau, Luis Caffarelli.