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 \(\mu^*\), i.e., the solution of Monge-Kantorovich equations. We study the gradient...

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Bibliographic Details
Published in:arXiv.org 2020-05
Main Authors: Facca, Enrico, Piazzon, Federico
Format: Article
Language:English
Subjects:
Gradient flow
Nonlinear systems
Shape optimization
Online Access:Get full text
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Summary: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 \(\mu^*\), i.e., the solution of Monge-Kantorovich equations. We study the gradient flow of \(\mathcal E\) showing that \(\mu^*\) is the unique global attractor of the flow. We introduce a two parameter family \(\{\mathcal E_{\lambda,\delta}\}_{\lambda,\delta>0}\) of strictly convex functionals approximating \(\mathcal E\) and we prove the convergence of the minimizers \(\mu_{\lambda,\delta}^*\) of \(\mathcal E_{\lambda,\delta}\) to \(\mu^*\) as we let \(\delta\to 0^+\) and \(\lambda\to 0^+.\) We derive an evolution system of fully non-linear PDEs as gradient flow of \(\mathcal E_{\lambda,\delta}\) 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 \(\mu_{\lambda,\delta}^*\) of \(\mathcal E_{\lambda,\delta}.\) Finally, we characterize \(\mu_{\lambda,\delta}^*\) by a non-linear system of PDEs which is a perturbation of Monge-Kantorovich equations by means of a p-Laplacian.
ISSN:2331-8422