Monge–Ampère equation

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

${\displaystyle L[u]=0,\text{on}\mathrm{\Omega }}$

${\displaystyle u{|}_{\mathrm{\partial }\mathrm{\Omega }}=g.}$

If

${\displaystyle BD-C^2-AE>0,}$

then the Dirichlet problem has at most two solutions.

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

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

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.

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. 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

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

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.

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.