Regularity of the positional penalization function in inter-sign optimal transport on real measures

Mathematics > Analysis of PDEs [Submitted on 17 Jun 2026] Title:Regularity of the positional penalization function in inter-sign optimal transport on real measures View PDF HTML (experimental)Abstract:We study the Monge--Kantorovich optimal transport problem between two signed measures~$\mu$ and~$\nu$ on convex compact subsets of~$\mathbb{R}^d$, with a positional penalization function~$\lambda(x, y)$ that modulates the cost of inter-sign transport. Using four independent positive measures~$(\pi^{++}, \pi^{+-}, \pi^{-+}, \pi^{--})$ as decision variables, we prove that the admissible set~$\mathcal{A}(\mu, \nu)$ is weakly-$*$ compact and non-empty if and only if $\mu^+(X) = \nu^+(Y)$ and~$\mu^-(X) = \nu^-(Y)$. Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~$\lambda$ at the intersection of inter-sign supports. The penalization~$\lambda$ is shown to be Lipschitz and to admit Alexandrov second derivatives almost everywhere. Modified Monge--Ampère equations governing inter-sign transport maps are derived in the Alexandrov sense, with well-posedness characterized by $\sigma \det(D^2_{yx}\Lambda) e > 0$. The classical Brenier equation is recovered in the limit~$\lambda \to 0$. Submission history From: Yannick Tchaptchie Kouakep Dr [view email][v1] Wed, 17 Jun 2026 21:59:08 UTC (7 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.