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\(C^{1,1}\) estimates for solutions of a problem of Alexandrov. (English) Zbl 0879.53047

More than fifty years ago, A. D. Alexandrov gave a necessary and sufficient condition for a nonnegative, completely additive function \(\mu\) on the Borel sets of \(\mathbb{S}^n\) to be the integral Gaussian curvature of some finite convex hypersurface in Euclidean space \(\mathbb{R}^{n+1}\). It follows from regularity results of Monge-Ampère equations that the solution to the Alexandrov problem is smooth if the density of \(\mu\) is a smooth positive function on \(\mathbb{S}^n\). It is natural to ask that when the density of \(\mu\) is smooth, but only nonnegative, are the solutions to the Alexandrov problem always smooth? Here we encounter certain degenerate Monge-Ampère equations.
We show in this paper that when the density of \(\mu\) is smooth and nonnegative, solutions of the Alexandrov problem are at least \(C^{1,1}\) in dimension \(n=2,3\). For higher dimensions, we have the same conclusion under some further hypothesis on the density function of \(\mu\). We also produce some \(\mu\) with smooth nonnegative density function, but nevertheless the solution is merely \(C^{2,2/n} (\mathbb{S}^n)\). In particular, it is not \(C^3\).

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
58J05 Elliptic equations on manifolds, general theory
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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