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Monge-Ampère equation

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

It is named after Gaspard Monge who introduced descriptive geometry and the first form of the partial differential equation in 1784,[1] and after André-Marie Ampère who introduced the nonlinear partial differential equation in 1820[2] when studying the geometry of surfaces.[3]

Luis Caffarelli earned the 2023 Abel Prize for his work on this equation.[4]

Definition

In two dimension

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

L [ u ] = A det ( ∇ 2 u ) + B Δ u + 2 C u x y + ( D − B ) u y y + E = A ( u x x u y y − u x y 2 ) + B u x x + 2 C u x y + D u y y + E = 0 , {\displaystyle {\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}}} {\displaystyle {\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 {\displaystyle A} {\displaystyle A}, B {\displaystyle B} {\displaystyle B}, C {\displaystyle C} {\displaystyle C}, D {\displaystyle D} {\displaystyle D}, and E {\displaystyle E} {\displaystyle E} are functions depending on the first-order variables x {\displaystyle x} {\displaystyle x}, y {\displaystyle y} {\displaystyle y}, u {\displaystyle u} {\displaystyle u}, u x {\displaystyle u_{x}} {\displaystyle u_{x}}, and u y {\displaystyle u_{y}} {\displaystyle u_{y}} only.

In general

Given a domain Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} {\displaystyle \Omega \subset \mathbb {R} ^{n}} and a real-valued function u : Ω → R {\displaystyle u\colon \Omega \to \mathbb {R} } {\displaystyle 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 , D u , det D 2 u ) = 0 , {\displaystyle F(x,u,Du,\det D^{2}u)=0,} {\displaystyle F(x,u,Du,\det D^{2}u)=0,}

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

If F {\displaystyle F} {\displaystyle F} also depends linearly on all principal minors of the Hessian matrix D 2 u {\displaystyle D^{2}u} {\displaystyle 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 Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} {\displaystyle \Omega \subset \mathbb {R} ^{n}} be an open set, u : Ω → R {\displaystyle u\colon \Omega \to \mathbb {R} } {\displaystyle u\colon \Omega \to \mathbb {R} } be a C 2 {\displaystyle C^{2}} {\displaystyle C^{2}} function, and consider an operator of Monge–Ampère type

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

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

∑ i j a i j ( x ) u i j + lower order terms , {\displaystyle \sum _{ij}a^{ij}(x)\,u_{ij}+{\text{lower order terms}},} {\displaystyle \sum _{ij}a^{ij}(x)\,u_{ij}+{\text{lower order terms}},}

where

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

The quadratic form

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

The equation is said to be

  • elliptic at x {\displaystyle x} {\displaystyle x} if Q x ( ξ ) {\displaystyle Q_{x}(\xi )} {\displaystyle Q_{x}(\xi )} if all eigenvalues are of the same sign,
  • hyperbolic at x {\displaystyle x} {\displaystyle x} if Q x ( ξ ) {\displaystyle Q_{x}(\xi )} {\displaystyle Q_{x}(\xi )} takes both positive and negative values (the matrix is indefinite),
  • parabolic at x {\displaystyle x} {\displaystyle x} if Q x ( ξ ) {\displaystyle Q_{x}(\xi )} {\displaystyle 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 {\displaystyle f} {\displaystyle f} is a positive function and solutions satisfy the constraint of being uniformly convex. If f {\displaystyle f} {\displaystyle f} is merely strictly convex, then the equation is degenerate-elliptic.[5]

Examples

The Monge–Ampère equation in its simplest form is

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

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

The classical Liouville theorem has an analogy here. If det D 2 u {\displaystyle \det D^{2}u} {\displaystyle \det D^{2}u} is constant, and u {\displaystyle u} {\displaystyle u} is defined on all of R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}}, then u {\displaystyle u} {\displaystyle u} is a quadratic function. This is the Jörgens–Calabi–Pogorelov theorem.[6]: Sec. 4.3 

If f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} is positive and uniformly convex, and u {\displaystyle u} {\displaystyle u} is a solution to det D 2 u = f ( x ) {\displaystyle \det D^{2}u=f(x)} {\displaystyle \det D^{2}u=f(x)}, then its Legendre transform u ∗ {\displaystyle u^{*}} {\displaystyle u^{*}} is a solution to det D 2 u ∗ = 1 / f ( x ) {\displaystyle \det D^{2}u^{*}=1/f(x)} {\displaystyle \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 : Ω → R {\displaystyle f:\Omega \to \mathbb {R} } {\displaystyle f:\Omega \to \mathbb {R} } defined over a domain Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} {\displaystyle \Omega \subset \mathbb {R} ^{n}}, its graph is a manifold of n dimensions. At any x ∈ Ω {\displaystyle x\in \Omega } {\displaystyle x\in \Omega }, the Gaussian curvature of the manifold at f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} is det D 2 u ( 1 + | D u | 2 ) ( n + 2 ) / 2 {\displaystyle {\frac {\det D^{2}u}{(1+|Du|^{2})^{(n+2)/2}}}} {\displaystyle {\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 + | D u | 2 ) ( n + 2 ) / 2 = 0 {\displaystyle \det D^{2}u-K(x)(1+|Du|^{2})^{(n+2)/2}=0} {\displaystyle \det D^{2}u-K(x)(1+|Du|^{2})^{(n+2)/2}=0}

where K {\displaystyle K} {\displaystyle 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.[5]

For example, the rigidity of the 2-sphere manifests as the fact that if we require n = 3 , K = 1 , u ( 0 ) = 0 , D u = 0 {\displaystyle n=3,K=1,u(0)=0,Du=0} {\displaystyle 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 R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}}. That is, suppose μ , ν {\displaystyle \mu ,\nu } {\displaystyle \mu ,\nu } are distributions on R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} with probability density functions ρ μ , ρ ν {\displaystyle \rho _{\mu },\rho _{\nu }} {\displaystyle \rho _{\mu },\rho _{\nu }}. In this case, a map T : supp ⁡ ( μ ) → supp ⁡ ( ν ) {\displaystyle T:\operatorname {supp} (\mu )\to \operatorname {supp} (\nu )} {\displaystyle T:\operatorname {supp} (\mu )\to \operatorname {supp} (\nu )} is a transport map iff it satisfies ∫ h ( T ( x ) ) ρ μ ( x ) d x = ∫ h ( y ) ρ ν ( y ) d y {\displaystyle \int h(T(x))\rho _{\mu }(x)dx=\int h(y)\rho _{\nu }(y)dy} {\displaystyle \int h(T(x))\rho _{\mu }(x)dx=\int h(y)\rho _{\nu }(y)dy}for any integrable test function h ∈ L 1 ( supp ⁡ ( ν ) ) {\displaystyle h\in L^{1}(\operatorname {supp} (\nu ))} {\displaystyle h\in L^{1}(\operatorname {supp} (\nu ))}. The problem is to find the T {\displaystyle T} {\displaystyle T} that minimizes the following quadratic cost function: min T ∫ ‖ x − T ( x ) ‖ 2 f ( x ) d x {\displaystyle \min _{T}\int \|x-T(x)\|^{2}f(x)dx} {\displaystyle \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 ψ : supp ⁡ ( μ ) → R n {\displaystyle \psi :\operatorname {supp} (\mu )\to \mathbb {R} ^{n}} {\displaystyle \psi :\operatorname {supp} (\mu )\to \mathbb {R} ^{n}}, with T = D ψ {\displaystyle T=D\psi } {\displaystyle T=D\psi }. The convex function satisfies a Monge–Ampère equation:[7]: 282 [8][9] { det ( D 2 ψ ) = ρ μ ρ ν ∘ D ψ D ψ ( ∂ supp ⁡ ( μ ) ) = ∂ supp ⁡ ( ν ) {\displaystyle {\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}}} {\displaystyle {\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 ψ {\displaystyle \psi } {\displaystyle \psi } is almost everywhere unique.

The function ψ {\displaystyle \psi } {\displaystyle \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.[7]: 323 

Rellich's theorem

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

L [ u ] = 0 , on   Ω {\displaystyle L[u]=0,\quad {\text{on}}\ \Omega } {\displaystyle L[u]=0,\quad {\text{on}}\ \Omega }
u | ∂ Ω = g . {\displaystyle u|_{\partial \Omega }=g.} {\displaystyle u|_{\partial \Omega }=g.}

If

B D − C 2 − A E > 0 , {\displaystyle BD-C^{2}-AE>0,} {\displaystyle BD-C^{2}-AE>0,}

then the Dirichlet problem has at most two solutions.[10]

Ellipticity results

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

L [ u ] = det D 2 u − f ( x , u , D u ) = 0 ( 1 ) {\displaystyle L[u]=\det D^{2}u-f(\mathbf {x} ,u,Du)=0\qquad \qquad (1)} {\displaystyle 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 {\displaystyle L} {\displaystyle L} satisfies versions of the maximum principle, and in particular solutions to the Dirichlet problem are unique, provided they exist.

See also

References

  1. Monge, Gaspard (1784). "Mémoire sur le calcul intégral des équations aux différences partielles". Mémoires de l'Académie des Sciences (in French).
  2. Ampère, André-Marie (1819). Mémoire contenant l'application de la théorie exposée dans le XVII. e Cahier du Journal de l'École polytechnique, à l'intégration des équations aux différentielles partielles du premier et du second ordre (in French). Paris, France: Imprimerie royale, 1819.
  3. Arendt, Wolfgang; Urban, Karsten (2023-01-01). Partial Differential Equations: An Introduction to Analytical and Numerical Methods. Springer Nature. ISBN 978-3-031-13379-4.
  4. Vázquez, Juan Luis (2024-06-19). "Luis Caffarelli, winner of the Abel Prize 2023". European Mathematical Society Magazine (132): 14–22. doi:10.4171/mag/192. ISSN 2747-7894.
  5. Gilbarg & Trudinger 2001.
  6. 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.
  7. Villani 2009.
  8. 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. Bibcode:2015SJSC...37B.937P. doi:10.1137/140986414. ISSN 1064-8275.
  9. Villani 2003; Villani 2009.
  10. Courant & Hilbert 1962, p. 324.

Additional references