Optimal transport of signed fractal measures with dimensional distortion: a variational characterization

Mathematics > Analysis of PDEs [Submitted on 17 Jun 2026] Title:Optimal transport of signed fractal measures with dimensional distortion: a variational characterization View PDF HTML (experimental)Abstract:We extend the optimal transport theory for signed measures supported on Ahlfors-regular fractal sets (Bwo'Nyahre et al., 2026) to allow a controlled dimensional distortion between source and target. A penalization term $\varepsilon \Phi(d_s(x) - d_t(y))$ -- where $\Phi$ is a fixed smooth strictly convex function and $d_s, d_t$ are the local Hausdorff dimensions of the fractal supports -- is added to the transport cost on inter-sign regions, with~$\varepsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1--H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\varepsilon}$ for every~$\varepsilon > 0$; coupled Monge--Ampère equations with a distortion correction term, generalizing the classical Brenier--Caffarelli equation; a double Legendre--Fenchel characterization of the optimal potentials, giving a complete variational description of the transport in each of the four sign regimes. The double Legendre--Fenchel system (Theorem~4.2) is the central contribution: it shows that the optimal potentials are the unique fixed points of a system of conjugacy equations, one per transport regime, and it provides the foundation for numerical algorithms and asymptotic analysis. Submission history From: Yannick Tchaptchie Kouakep Dr [view email][v1] Wed, 17 Jun 2026 22:20:48 UTC (8 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.