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Symmetry Lie group of the Monge-Ampère equation. (English) Zbl 0934.58039

The paper mainly deals with point symmetries of the Monge-Ampère equation \[ \det(\partial^2 u/\partial x^i \partial x^j) = f(x,u),\qquad 1 \leq i,\quad j \leq n. \] From the results: for \(n = 2\) and \(f = 1\), invariant solutions with respect to four optimal subalgebras are found, and a proof is given that among second-order equations the Monge-Ampère equation is uniquely determined by its symmetry algebra.
For related earlier results also on contact symmetries see S. V. Khabirov [Math. USSR, Sb. 71, No. 2, 447-462 (1992); translation from Mat. Sb. 181, No. 12, 1607-1622 (1990; Zbl 0713.76085)].
[Reviewer’s remark: The same paper is reprinted in Appl. Sci. 1, No. 1, 60-73 (1999; Zbl 0934.58040)].
Reviewer: M.Marvan (Opava)

MSC:

58J70 Invariance and symmetry properties for PDEs on manifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
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