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Sur la régularité des solutions non strictement convexes de l’équation de Monge-Ampère réelle. (On the regularity of non- strictly convex solutions of the real Monge-Ampère equation). (French) Zbl 0702.35050

The author considers the real Monge-Ampère equation in the case when the right-hand side is not strictly positive and thus the solution is not strictly convex. The corresponding linearized operator is then a second order operator with non-negative characteristic form. Using the theory for such linear equations. C. J. Xu has obtained results for the solutions of general nonlinear second order equations. The conditions however depend on the solution and are thus complicated. The author of the present paper gives a simpler and more precise form of similar conditions for regularity and non-regularity in the case of Monge- Ampère equation using a direct calculus of the Poisson brackets.
Reviewer: J.Madjarova

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
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References:

[1] L. Caffarelli - L. Nirenberg - J. Spruck : The Dirichlet problem for nonlinear second order elliptic equation I: Monge-Ampère equation , Comm. on Pure and Applied Mathematics , XXXVII , ( 1984 ), 369 - 402 . MR 739925 | Zbl 0598.35047 · Zbl 0598.35047 · doi:10.1002/cpa.3160370306
[2] Xu Chao Jiang : Régularité des solutions des e.d.p. non-linéaires , C.R. Acad. Sci. Paris , 300 ( 1985 ), 267 - 270 . MR 785066 | Zbl 0587.35034 · Zbl 0587.35034
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