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Topological Optimization and Optimal Transport by Filippo Santambrogio, Thierry Champion, Guillaume Carlier, Martin Rumpf, Édouard Oudet, Maïtine Bergounioux

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Proposition 4.1. For every ρC(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 {uΔ}τ, using a discrete flow interchange estimate.

(2) Passing to the limit using Proposition 3.7.

For definiteness, fix a spatial test function ρC(Ω) with ...

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