Optimal Transportation Networks pp 119-134 | Cite as

# The Landscape of an Optimal Pattern

## 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 *x*_{0} 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 *E*^{α} 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## Preview

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