Abstract
Consider the irrigation of an arbitrary measure on a domain X starting from a single source \( \mu + = \delta _0 \). In this chapter we define following Santambrogio [75] a function z(x) that represents the elevation of the landscape associated with an optimal traffic plan. This function was known in the geophysical community under a very strong discretization [6], [73]. The continuous landscape function will be proven to share all the properties that hold in the discrete case, in particular the fact that, at a point x0 of the irrigation network, z(x) has maximal slope in the direction of the network itself and that this slope is given by a power of the multiplicity, \( \left| x \right|_P^{\alpha - 1} \). Section 11.1 describes the physical discrete model of joint landscape-river network evolution. In Section 11.2 we will prove a general first variation inequality for the energy Eα when \( \mu ^ + \) varies. Section 11.3 proves that the definition of z(x), which will be defined as a path integral along the network from the source to x. does not depend on the path but only on x. Section 11.4 is devoted to a general semicontinuity property of z(x) and Section 11.5 to its Hölder continuity when the irrigated measure dominates Lebesgue on X. The whole chapter follows closely the seminal Santamabrogio paper [75].
Keywords
- Variation Inequality
- River Network
- Maximal Slope
- Irrigation Network
- Landscape Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2009). The Landscape of an Optimal Pattern. In: Optimal Transportation Networks. Lecture Notes in Mathematics, vol 1955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69315-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-69315-4_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69314-7
Online ISBN: 978-3-540-69315-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
