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Geodesics in Euclidean space with analytic obstacle


Authors: Felix Albrecht and I. D. Berg
Journal: Proc. Amer. Math. Soc. 113 (1991), 201-207
MSC: Primary 53C22
DOI: https://doi.org/10.1090/S0002-9939-1991-1077783-8
MathSciNet review: 1077783
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Abstract: In this note we are concerned with the behavior of geodesies in Euclidean $ n$-space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form $ {x_n} = f({x_1}, \ldots ,{x_{n - 1}})$ for a real analytic function $ f$, then a geodesic can have, in any segment of finite arc length, only a finite number of distinct switch points, points on the boundary that bound a segment not touching the boundary.

This result is certainly false that for a $ {C^\infty }$ boundary. Indeed, even in $ {E^2}$, where our result is obvious for analytic boundaries, we can construct a $ {C^\infty }$ boundary so that the closure of the set of switch points is of positive measure.


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  • [AA] R. Alexander and S. B. Alexander, Geodesics in Riemannian manifolds--with--boundary, Indiana Univ. Math J. 30 (1981), 481-488. MR 620261 (82j:58040)
  • [ABB1] S. B. Alexander, I. D. Berg, and R. L. Bishop, The Riemannian obstacle problem, Illinois J. Math. 31 (1987), 167-184. MR 869484 (88a:53038)
  • [ABB2] -, Cauchy uniqueness in the Riemannian obstacle problem, Lecture Notes in Math., vol. 1209, Diff. Geom. Peñiscola, 1985, Springer-Verlag, New York, pp. 1-7. MR 863742 (88e:53064)
  • [Alv] A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, Trudy Mat. Inst. Steklov 38 (1951), 5-23. MR 0049584 (14:198a)
  • [Ar] V. I. Arnol'd, Singularities in the calculus of variations, J. Soviet Math. 27 (1984), 2679-2712. MR 735439 (85m:58031)
  • [G] M. Gromov, Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics, Proc. of the 1978 Stony Brook Conf., Princeton Univ. Press, Princeton, NJ, 1980, pp. 183-213. MR 624814 (82m:53035)
  • [K] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. MR 567696 (81g:49013)

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DOI: https://doi.org/10.1090/S0002-9939-1991-1077783-8
Article copyright: © Copyright 1991 American Mathematical Society

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