×

Some regularity properties of solutions of Monge Ampère equation. (English) Zbl 0761.35028

Let \(\text{det} D_{ij}U=d\mu\), where \(U\) is a function defined in a domain \(\Omega\subset R^ N\), \(D_{ij}U\) denotes the Hessian matrix of \(U\), \(\mu\) is a measure satisfying the “doubling-like” geometric property.
The following theorems are proved. Theorem 1. Let \(U\) be a weak, locally Lipschitz, convex solution of the above equation. Then, if \(lx_ 0\) is a supporting linear function to \(U\) at \(x_ 0\) we have: \(\{U=lx_ 0\}=\{X_ 0\}\) or \(\{U=lx_ 0\}\) has no extremal points in the domain of definition of \(U\). Theorem 2. A strictly convex solution \(U\) of the above equation is \(C^{1,\alpha}\), with \(C^{1,\alpha}\) norm depending only on the Lipschitz norm of \(U\), and on its strict convexity.
Reviewer: G.Porru (Cagliari)

MSC:

35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexandrov, Vestnik Leningrad 21 1 (1966)
[2] A.M.S. Transl. (2) 68 pp 120– (1968)
[3] Brenier, Comm. Pure Appl. Math. 44 pp 375– (1991)
[4] Caffarelli, Ann. Math. 131 pp 129– (1990)
[5] Caffarelli, Ann. Math. 131 pp 135– (1990)
[6] Harmonic measure in convex domains, Bull. Amer. Math. Soc., New Series 21, October 1989. · Zbl 0696.30009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.