L-1 Transport Energy

We introduce the transport energy functional epsilon (a variant of the Bouchitte-Buttazzo-Seppecher shape optimization functional) and prove that its unique minimizer is the optimal transport density mu*, i.e., the solution of Monge-Kantorovich partial differential equations. We study the gradient flow of epsilon and show that mu* is the unique global attractor of the flow. Next, we introduce a two parameter family {epsilon(lambda,delta))(lambda,delta>0) of strictly convex regularized functionals approximating epsilon and prove the convergence of the minimizers mu(lambda,delta)* of epsilon(lambda,delta) to mu* as we let delta -> 0(+) and lambda -> 0(+). We derive an evolution system of fully non-linear PDEs as the gradient flow of epsilon(lambda,delta) in L-2, showing existence and uniqueness of the solution for all times. We are able to prove that the trajectories of the flow converge in W-0(1,p) to the unique minimizer mu(lambda,delta)* of epsilon(lambda,delta). This allows us to characterize mu(lambda,delta)* by a non-linear system of PDEs that turns out to be a perturbation of the Monge-Kantorovich equations by a p-Laplacian.