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 -space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form for a real analytic function , 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 boundary. Indeed, even in , where our result is obvious for analytic boundaries, we can construct a boundary so that the closure of the set of switch points is of positive measure.

**[AA]**Ralph Alexander and S. Alexander,*Geodesics in Riemannian manifolds-with-boundary*, Indiana Univ. Math. J.**30**(1981), no. 4, 481–488. MR**620261**, https://doi.org/10.1512/iumj.1981.30.30039**[ABB1]**Stephanie B. Alexander, I. David Berg, and Richard L. Bishop,*The Riemannian obstacle problem*, Illinois J. Math.**31**(1987), no. 1, 167–184. MR**869484****[ABB2]**Stephanie B. Alexander, I. David Berg, and Richard L. Bishop,*Cauchy uniqueness in the Riemannian obstacle problem*, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 1–7. MR**863742**, https://doi.org/10.1007/BFb0076617**[Alv]**A. D. Aleksandrov,*A theorem on triangles in a metric space and some of its applications*, Trudy Mat. Inst. Steklov., v 38, Trudy Mat. Inst. Steklov., v 38, Izdat. Akad. Nauk SSSR, Moscow, 1951, pp. 5–23 (Russian). MR**0049584****[Ar]**V. I. Arnol′d,*Singularities in the calculus of variations*, Current problems in mathematics, Vol. 22, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 3–55 (Russian). MR**735439****[G]**M. Gromov,*Hyperbolic manifolds, groups and actions*, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 183–213. MR**624814****[K]**David Kinderlehrer and Guido Stampacchia,*An introduction to variational inequalities and their applications*, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**567696**

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1077783-8

Article copyright:
© Copyright 1991
American Mathematical Society