Topological Optimization and Optimal Transport

Proposition 4.1. For every ρ ∈ C^∞(Ω) with ρ′(a) = ρ′(b) = 0, and for every η ∈ the limit curve u_* satisfies

where the highly nonlinear term N from (1.18) is given by

The proof of this statement will be treated in two essential steps:

(1) Show the validity of a discrete weak formulation for {u_Δ}_τ, using a discrete flow interchange estimate.

(2) Passing to the limit using Proposition 3.7.

For definiteness, fix a spatial test function ρ ∈ C^∞(Ω) with ...