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The Monge-Ampère equation: various forms and numerical solution. (English) Zbl 1194.65141

Summary: We present three novel forms of the Monge-Ampère equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampère problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampère equation is a sum of such distributions.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J96 Monge-Ampère equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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