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 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 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].