2019/09/10 by Enrico Facca, Facca, Enrico, Federico Piazzon +1
Mathematics · #35J20 #49J40 #49J45 #49Q20 #58E50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1909.04417
openalex publication_date 2019/09/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We introduce the 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 Eλ,δ\λ,δ>0 of strictly convex functionals approximating \mathcal E and we prove the convergence of the minimizers μλ,δ^* of \mathcal Eλ,δ to μ^* as we let δ→ 0+ and λ→ 0+. We derive an evolution system of fully non-linear PDEs as gradient flow of \mathcal Eλ,δ in L2, showing existence and uniqueness of solutions. All the trajectories of the flow converge in W1,p0 to the unique minimizer μλ,δ^* of \mathcal Eλ,δ. Finally, we characterize μλ,δ^* by a non-linear system of PDEs which is a perturbation of Monge-Kantorovich equations by means of a p-Laplacian.