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

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-$\ast$ 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 $\sigma \det (D_{yx}^2\mathrm{\Lambda })e>0$. The classical Brenier equation is recovered in the limit~$\lambda \to 0$.