A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations |
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Authors: | Cristian Rios |
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Institution: | a McMaster University, Hamilton, Ontario, Canada b Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA |
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Abstract: | In dimension n?3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge-Ampère equations in dimension n?3. A corollary is that if k?0 vanishes only at nondegenerate critical points, then a C2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be when not smooth. |
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Keywords: | 35B65 35B45 |
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