Monge–Ampère equation

In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function $u$ of two variables $x$ , $y$ is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of $u$ and in the second-order partial derivatives of $u$ . The independent variables ( $x$ , $y$ ) vary over a given domain $D$ of $\mathbb {R} ^{2}$ . The term also applies to analogous equations with $n$ independent variables. The most complete results so far have been obtained when the equation is elliptic.

Definition

In two dimension

Given two independent variables $x$ and $y$ , and one dependent variable $u$ , the general Monge–Ampère equation is of the form

{\begin{aligned}L[u]=&A\det(\nabla ^{2}u)+B\Delta u+2Cu_{xy}+(D-B)u_{yy}+E\\=&A(u_{xx}u_{yy}-u_{xy}^{2})+Bu_{xx}+2Cu_{xy}+Du_{yy}+E=0,\end{aligned}}

where $A$ , $B$ , $C$ , $D$ , and $E$ are functions depending on the first-order variables $x$ , $y$ , $u$ , $u_{x}$ , and $u_{y}$ only.

In general

Given a domain $\Omega \subset \mathbb {R} ^{n}$ and a real-valued function $u\colon \Omega \to \mathbb {R}$ , a (real) Monge–Ampère equation is any fully nonlinear second-order equation that can be written in the form

F(x,u,Du,\det D^{2}u)=0,

for some function $F$ . More generally, $\Omega$ can be a Riemannian manifold, since $Du,D^{2}u$ are well-defined on a Riemannian manifold.

If $F$ also depends linearly on all principal minors of the Hessian matrix $D^{2}u$ , then it is an equation of Monge–Ampère type.

Classification

As for other second-order fully nonlinear equations, the type of a Monge–Ampère equation is defined by the linearization of the operator at a sufficiently smooth solution. Of these, the most common is the elliptic case. When people say "Monge–Ampère equation" without adjective, they usually mean the elliptic case.

Let $\Omega \subset \mathbb {R} ^{n}$ be an open set, $u\colon \Omega \to \mathbb {R}$ be a $C^{2}$ function, and consider an operator of Monge–Ampère type

L[u]=F(x,u,Du,D^{2}u),

where $F$ is smooth in all variables and depends on $D^{2}u$ only through its principal minors. The linearization of $L$ is of form

\sum _{ij}a^{ij}(x)\,u_{ij}+{\text{lower order terms}},

where

a^{ij}(x)={\frac {\partial F}{\partial u_{ij}}}(x,u(x),Du(x),D^{2}u(x)).

The quadratic form

Q_{x}(\xi )=a^{ij}(x)\,\xi _{i}\xi _{j},\qquad \xi \in \mathbb {R} ^{n},

is the principal symbol of the linearized operator at the point $x$ .

The equation is said to be

elliptic at $x$ if $Q_{x}(\xi )$ if all eigenvalues are of the same sign,
hyperbolic at $x$ if $Q_{x}(\xi )$ takes both positive and negative values (the matrix is indefinite),
parabolic at $x$ if $Q_{x}(\xi )$ is degenerate (the matrix has vanishing determinant),
degenerate elliptic if it is elliptic everywhere,
elliptic if it is degenerate elliptic, and all eigenvalues are a bounded distance away from zero.

As follows from Jacobi's formula for the derivative of a determinant, this equation is elliptic if $f$ is a positive function and solutions satisfy the constraint of being uniformly convex. If $f$ is merely strictly convex, then the equation is degenerate-elliptic.^[1]

Examples

The Monge–Ampère equation in its simplest form is

\det D^{2}u=f(x)

where $f$ is a given function on a domain $\Omega \subset \mathbb {R} ^{n}$ . This is a special case of equation (1) below with $f(x,u,Du)=f(x)$ .

The classical Liouville theorem has an analogy here. If $\det D^{2}u$ is constant, and $u$ is defined on all of $\mathbb {R} ^{n}$ , then $u$ is a quadratic function. This is the Jörgens–Calabi–Pogorelov theorem.^[2]^{: Sec. 4.3}

If $f(x)$ is positive and uniformly convex, and $u$ is a solution to $\det D^{2}u=f(x)$ , then its Legendre transform $u^{*}$ is a solution to $\det D^{2}u^{*}=1/f(x)$ .

Geometry

Monge–Ampère equations arise naturally in several problems in Riemannian geometry, conformal geometry, affine geometry, and CR geometry.

Given a twice-differentiable real-valued function $f:\Omega \to \mathbb {R}$ defined over a domain $\Omega \subset \mathbb {R} ^{n}$ , its graph is a manifold of n dimensions. At any $x\in \Omega$ , the Gaussian curvature of the manifold at $f(x)$ is ${\frac {\det D^{2}u}{(1+|Du|^{2})^{(n+2)/2}}}$ . Thus, if we want to find a manifold whose Gaussian curvature is an arbitrary function we pick ourselves, then we need to solve the following Monge–Ampère equation:

\det D^{2}u-K(x)(1+|Du|^{2})^{(n+2)/2}=0

where $K$ is the Gaussian curvature we want. Given such a function K, it is nontrivial to find a solution, if any. The problem of finding a solution is the Minkowski problem, or the prescribed Gaussian curvature problem.^[1]

For example, the rigidity of the 2-sphere manifests as the fact that if we require $n=3,K=1,u(0)=0,Du=0$ , then there are just two unique solutions, which is the unit 2-sphere and its reflection.

The affine spheres can be characterized by a Monge–Ampère equation.

Optimal transport

Consider the problem of optimal transport with quadratic cost (this is also called the 2-Wasserstein metric problem) on $\mathbb {R} ^{n}$ . That is, suppose $\mu ,\nu$ are distributions on $\mathbb {R} ^{n}$ with probability density functions $\rho _{\mu },\rho _{\nu }$ . In this case, a map $T:\operatorname {supp} (\mu )\to \operatorname {supp} (\nu )$ is a transport map iff it satisfies $\int h(T(x))\rho _{\mu }(x)dx=\int h(y)\rho _{\nu }(y)dy$ for any integrable test function $h\in L^{1}(\operatorname {supp} (\nu ))$ . The problem is to find the $T$ that minimizes the following quadratic cost function: $\min _{T}\int \|x-T(x)\|^{2}f(x)dx$ By a theorem of Brenier, the optimal transport map exists, and is the gradient of a convex function $\psi :\operatorname {supp} (\mu )\to \mathbb {R} ^{n}$ , with $T=D\psi$ . The convex function satisfies a Monge–Ampère equation:^[3]^: 282^[4]^[5] ${\begin{cases}\det(D^{2}\psi )={\frac {\rho _{\mu }}{\rho _{\nu }\circ D\psi }}\\D\psi (\partial \operatorname {supp} (\mu ))=\partial \operatorname {supp} (\nu )\end{cases}}$ The boundary condition simply states that the optimal transport maps the boundary of the source to the boundary of the target. Furthermore, the solution $\psi$ is almost everywhere unique.

The function $\psi$ is called the potential function of the problem in this case.

Conversely, some Monge–Ampère equations can be interpreted optimal transport. Weak-solutions of a Monge–Ampère equations obtained by optimal transport are often called Brenier solutions in the literature. Brenier solutions satisfy their corresponding Monge–Ampère equations almost everywhere.^[3]^: 323

Rellich's theorem

Let $\Omega$ be a bounded domain in $\mathbb {R} ^{3}$ , and suppose that on $\Omega$ the coefficients $A$ , $B$ , $C$ , $D$ , and $E$ are continuous functions of $x$ and $y$ only. Consider the Dirichlet problem to find $u$ so that

L[u]=0,\quad {\text{on}}\ \Omega

u|_{\partial \Omega }=g.

If

BD-C^{2}-AE>0,

then the Dirichlet problem has at most two solutions.^[6]

Ellipticity results

Suppose now that $\mathbf {x}$ is a variable with values in a domain in $\mathbb {R} ^{n}$ , and that $f(\mathbf {x} ,u,Du)$ is a positive function. Then the Monge–Ampère equation

L[u]=\det D^{2}u-f(\mathbf {x} ,u,Du)=0\qquad \qquad (1)

is a nonlinear elliptic partial differential equation (in the sense that its linearization is elliptic), provided one confines attention to convex solutions.

Accordingly, the operator $L$ satisfies versions of the maximum principle, and in particular solutions to the Dirichlet problem are unique, provided they exist.^{[citation needed]}

References

^ ^a ^b Gilbarg & Trudinger 2001.
^ Figalli, Alessio (2017). The Monge-Ampère equation and its applications. Zurich lectures in advanced mathematics. Zürich: European Mathematical Society. ISBN 978-3-03719-170-5.
^ ^a ^b Villani 2009.
^ Prins, C. R.; Beltman, R.; ten Thije Boonkkamp, J. H. M.; IJzerman, W. L.; Tukker, T. W. (January 2015). "A Least-Squares Method for Optimal Transport Using the Monge--Ampère Equation". SIAM Journal on Scientific Computing. 37 (6): B937–B961. doi:10.1137/140986414. ISSN 1064-8275.
^ Villani 2003; Villani 2009.
^ Courant & Hilbert 1962, p. 324.

Additional references

Aubin, Thierry (1998). Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Berlin: Springer-Verlag. doi:10.1007/978-3-662-13006-3. ISBN 3-540-60752-8. MR 1636569. Zbl 0896.53003.
Courant, R.; Hilbert, D. (1962). Methods of mathematical physics. Volume II: Partial differential equations. New York–London: Interscience Publishers. doi:10.1002/9783527617234. ISBN 9780471504399. MR 0140802. Zbl 0099.29504. {{cite book}}: ISBN / Date incompatibility (help)
Gilbarg, David; Trudinger, Neil S. (2001). Elliptic partial differential equations of second order. Classics in Mathematics (Reprint of the 1998 ed.). Berlin: Springer-Verlag. doi:10.1007/978-3-642-61798-0. ISBN 3-540-41160-7. MR 1814364. Zbl 1042.35002.
Spivak, Michael (1999). A comprehensive introduction to differential geometry: volume five (Third edition of 1975 original ed.). Publish or Perish, Inc. ISBN 0-914098-74-8. MR 0532834. Zbl 1213.53001.
Villani, Cédric (2003). Topics in optimal transportation. Graduate Studies in Mathematics. Vol. 58. Providence, RI: American Mathematical Society. doi:10.1090/gsm/058. ISBN 0-8218-3312-X. MR 1964483. Zbl 1106.90001.
Villani, Cédric (2009). Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften. Vol. 338. Berlin: Springer-Verlag. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3. MR 2459454. Zbl 1156.53003.
Figalli, Alessio (2017). The Monge-Ampère equation and its applications. Zurich lectures in advanced mathematics. Zürich: European Mathematical Society. ISBN 978-3-03719-170-5.
Gutiérrez, Cristian E. (2016). The Monge-Ampère Equation. Progress in Nonlinear Differential Equations and Their Applications (2nd ed. 2016 ed.). Cham: Birkhäuser. ISBN 978-3-319-43372-1.
Lê, Nam Q. (2024). Analysis of Monge-Ampère equations. Graduate studies in mathematics. Providence, Rhode Island: American Mathematical Society. ISBN 978-1-4704-7420-1.

External links

"Monge–Ampère equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Monge-Ampère Differential Equation". MathWorld.

[rdp-we-cite_note-FOOTNOTEGilbargTrudinger2001-1] Gilbarg & Trudinger 2001.

[rdp-we-cite_note-2] Figalli, Alessio (2017). The Monge-Ampère equation and its applications. Zurich lectures in advanced mathematics. Zürich: European Mathematical Society. ISBN 978-3-03719-170-5.

[rdp-we-cite_note-FOOTNOTEVillani2009-3] Villani 2009.

[rdp-we-cite_note-4] Prins, C. R.; Beltman, R.; ten Thije Boonkkamp, J. H. M.; IJzerman, W. L.; Tukker, T. W. (January 2015). "A Least-Squares Method for Optimal Transport Using the Monge--Ampère Equation". SIAM Journal on Scientific Computing. 37 (6): B937–B961. doi:10.1137/140986414. ISSN 1064-8275.

[rdp-we-cite_note-FOOTNOTEVillani2003Villani2009-5] Villani 2003; Villani 2009.

[rdp-we-cite_note-FOOTNOTECourantHilbert1962324-6] Courant & Hilbert 1962, p. 324.

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