Monge–Ampère equation

Nonlinear second-order partial differential equation of special kind

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 R2. 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.

Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784[1] and later by André-Marie Ampère in 1820.[2] Important results in the theory of Monge–Ampère equations have been obtained by Sergei Bernstein, Aleksei Pogorelov, Charles Fefferman, and Louis Nirenberg. More recently, Alessio Figalli and Luis Caffarelli were recognized for their work on the regularity of the Monge–Ampère equation, with the former winning the Fields Medal in 2018 and the latter the Abel Prize in 2023.[3][4]

Description

Given two independent variables x and y, and one dependent variable 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 L[u]=A\,{\text{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,}

where A, B, C, D, and E are functions depending on the first-order variables x, y, u, ux, and uy only.

Rellich's theorem

Let Ω be a bounded domain in R3, and suppose that on Ω 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 , on   Ω {\displaystyle L[u]=0,\quad {\text{on}}\ \Omega }
u | ∂ Ω = g . {\displaystyle u|_{\partial \Omega }=g.}

If

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

then the Dirichlet problem has at most two solutions.[5]

Ellipticity results

Suppose now that x is a variable with values in a domain in Rn, and that f(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)}

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]

Applications

Monge–Ampère equations arise naturally in several problems in Riemannian geometry, conformal geometry, and CR geometry. One of the simplest of these applications is to the problem of prescribed Gauss curvature.[6] Suppose that a real-valued function K is specified on a domain Ω in Rn, the problem of prescribed Gauss curvature seeks to identify a hypersurface of Rn+1 as a graph z = u(x) over x ∈ Ω so that at each point of the surface the Gauss curvature is given by K(x). The resulting partial differential equation is

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

The Monge–Ampère equations are related to the Monge–Kantorovich optimal mass transportation problem, when the "cost functional" therein is given by the Euclidean distance.[7]

See also

References

  1. ^ Monge, Gaspard (1784). "Mémoire sur le calcul intégral des équations aux différences partielles". Mémoires de l'Académie des Sciences. Paris, France: Imprimerie Royale. pp. 118–192.
  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. Paris: De l'Imprimerie royale. Retrieved 2017-06-29.
  3. ^ "Figalli long citation" (PDF). Fields Medals 2018. International Mathematical Union.
  4. ^ De Ambrosio, Martín. "A nivel de los grandes del siglo: Luis Caffarelli, el Messi de la matemática que ganó el equivalente al Nobel de la disciplina". LA NACION. LA NACION. Retrieved 22 March 2023.
  5. ^ Courant & Hilbert 1962, p. 324.
  6. ^ Gilbarg & Trudinger 2001.
  7. ^ Villani 2003; Villani 2009.

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.
  • 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.

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