2026/06/17 by Bwo'nyahre Baidi Barthelemy, Kouakep Tchaptchie Yannick, Houpa Danga Duplex Elvis
#math.AP
We study the Monge--Kantorovich optimal transport problem between two signed measures~μ and~ν on convex compact subsets of~ℝd, with a positional penalization function~λ(x, y) that modulates the cost of inter-sign transport. Using four independent positive measures~(π++, π+-, π-+, π--) as decision variables, we prove that the admissible set~A(μ, ν) is weakly-* compact and non-empty if and only if μ+(X) = ν+(Y) and~μ-(X) = ν-(Y). Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~λ at the intersection of inter-sign supports. The penalization~λ 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 σdet(D2yxΛ) e > 0. The classical Brenier equation is recovered in the limit~λ→ 0.