Geodesics in Euclidean space with analytic obstacle
Authors:
Felix Albrecht and I. D. Berg
Journal:
Proc. Amer. Math. Soc. 113 (1991), 201207
MSC:
Primary 53C22
MathSciNet review:
1077783
Fulltext 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]
Ralph
Alexander and S.
Alexander, Geodesics in Riemannian manifoldswithboundary,
Indiana Univ. Math. J. 30 (1981), no. 4,
481–488. MR
620261 (82j:58040), http://dx.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
(88a:53038)
 [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
(88e:53064), http://dx.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
(14,198a)
 [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
(85m:58031)
 [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
(82m:53035)
 [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 YorkLondon, 1980.
MR 567696
(81g:49013)
 [AA]
 R. Alexander and S. B. Alexander, Geodesics in Riemannian manifoldswithboundary, Indiana Univ. Math J. 30 (1981), 481488. MR 620261 (82j:58040)
 [ABB1]
 S. B. Alexander, I. D. Berg, and R. L. Bishop, The Riemannian obstacle problem, Illinois J. Math. 31 (1987), 167184. MR 869484 (88a:53038)
 [ABB2]
 , Cauchy uniqueness in the Riemannian obstacle problem, Lecture Notes in Math., vol. 1209, Diff. Geom. Peñiscola, 1985, SpringerVerlag, New York, pp. 17. 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), 523. MR 0049584 (14:198a)
 [Ar]
 V. I. Arnol'd, Singularities in the calculus of variations, J. Soviet Math. 27 (1984), 26792712. 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. 183213. 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
53C22
Retrieve articles in all journals
with MSC:
53C22
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110777838
PII:
S 00029939(1991)10777838
Article copyright:
© Copyright 1991
American Mathematical Society
