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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
53C22

Retrieve articles in all journals with MSC: 53C22

Additional Information

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

Article copyright:
© Copyright 1991
American Mathematical Society