Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna. (English) Zbl 1231.35280
Building on the results of X.-N. Ma, N.S. Trudinger and X.-J. Wang [Arch. Ration. Mech. Anal. 177, No. 2, 151–183 (2005; Zbl 1072.49035)], and of himself [Acta Math., to appear], the author investigates two problems of optimal transportation on the sphere: the first corresponds to the cost function \(d ^{2}(x, y)\), where \(d(\cdot , \cdot )\) is the Riemannian distance of the round sphere; the second corresponds to the cost function \(- \log |x - y|\), known as the reflector antenna problem. He proves that in both cases, the cost-sectional curvature is uniformly positive, and establishes the geometrical properties so that the results of the author [loc. cit.] and Ma, Trudinger and Wang [loc. cit.] can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.
Reviewer: Ömer Kavaklioglu (Washington)
MSC:
| 35R01 | PDEs on manifolds |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
cost function; reflector antenna; global smooth solutions; Hölder continuous; optimal transportationCitations:
Zbl 1072.49035References:
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