Abstract
The proof of the existence of geodesic lines, given by David Hilbert in 1899 (see D. Hilbert, Ueber das Dirichlet’sche Princip, Jber. Deutsch. Math. Verein., 8 (1900), 184–188), was the first success of the direct method, applied to variational problems.
Hilbert’s method works in any metric space, and gives the existence result if the following compactness property is assumed: all closed and bounded subsets of the metric space are compact.
In the first section we expound the method in a general metric space; in the second section we prove that the length of any Lipschitz line can be calculated by a Lebesgue integral.
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© 1996 Birkhäuser Verlag Basel/Switzerland
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Miranda, M. (1996). Geodesic Lines in Metric Spaces. In: Serapioni, R., Tomarelli, F. (eds) Variational Methods for Discontinuous Structures. Progress in Nonlinear Differential Equations and Their Applications, vol 25. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9244-5_11
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DOI: https://doi.org/10.1007/978-3-0348-9244-5_11
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9959-8
Online ISBN: 978-3-0348-9244-5
eBook Packages: Springer Book Archive