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Transport Energy

We introduce the \emph{transport energy} functional $\mathcal E$ (a variant of the Bouchitté-Buttazzo-Seppecher shape optimization functional) and we prove that its unique minimizer is the optimal transport density $μ^*$, i.e., the solution of Monge-Kantorovich equations. We study the gradient flow of $\mathcal E$ showing that $μ^*$ is the unique global attractor of the flow. We introduce a two parameter family $\{\mathcal E_{λ,δ}\}_{λ,δ>0}$ of strictly convex functionals approximating $\mathcal E$ and we prove the convergence of the minimizers $μ_{λ,δ}^*$ of $\mathcal E_{λ,δ}$ to $μ^*$ as we let $δ\to 0^+$ and $λ\to 0^+.$ We derive an evolution system of fully non-linear PDEs as gradient flow of $\mathcal E_{λ,δ}$ in $L^2$, showing existence and uniqueness of solutions. All the trajectories of the flow converge in $W^{1,p}_0$ to the unique minimizer $μ_{λ,δ}^*$ of $\mathcal E_{λ,δ}.$ Finally, we characterize $μ_{λ,δ}^*$ by a non-linear system of PDEs which is a perturbation of Monge-Kantorovich equations by means of a p-Laplacian.