Kantorovich-Rubinstein duality theory for the Hessian

TL;DR

This work develops a Hessian-constrained Kantorovich–Rubinstein duality, showing that maximizing over $u\in C^{1,1}$ with ${\rm lip}(\nabla u)\le1$ is equivalent to a three-marginal OT problem with third marginal in convex order, and to a second-order Beckmann problem with $\mathrm{div}^2\sigma=f$. A central three-point equality links the dual Hessian problem to a convex-order framework, enabling a decomposition of optimal tensor measures into rank-one pieces supported on a graph. The results yield explicit solutions in several examples (Gaussian, two-point, and ordered measures) and, crucially, provide existence and structural results for optimal grillages in 2D, connecting OT, convex order, and structural optimization. The framework thus bridges Monge–Kantorovich theory, second-order PDE constraints, and engineering design (grillages/trusses), with practical implications for finite-support designs and potential extensions to domain confinement and more general loads.

Abstract

The classical Kantorovich-Rubinstein duality theorem establishes a significant connection between Monge optimal transport and maximization of a linear form on the set of 1-Lipschitz functions. This result has been widely used in various research areas. In particular, it unlocks the optimal transport methods in some of the optimal design problems. This paper puts forth a similar theory when the linear form is maximized over $C^{1,1}$ functions whose Hessian lies between minus and plus identity matrix. The problem will be identified as the dual of a specific optimal transport formulation that involves three-point plans. The first two marginals are fixed, while the third must dominate the other two in the sense of convex order. The existence of optimal plans allows to express solutions of the underlying Beckmann problem as a combination of rank-one tensor measures supported on a graph. In the context of two-dimensional mechanics, this graph encodes the optimal configuration of a grillage that transfers a given load system.

Figures (6)

• Figure 1: The tensor measure $\sigma^{x,y,z}$; the density with respect to $\mathcal{H}^1 \mathbin{}([x,z] \cup [z,y])$ is illustrated. Blue and red indicate the positive and the negative part, respectively.
• Figure 2: Various optimal $\sigma$ (blue) for the data $\mu = \delta_0$ (gray) and $\nu = \sum_{i=1}^4 \frac{1}{4} \delta_{y_i}$ (black). Only the density of the 1D measure $\sigma$ is displayed.
• Figure 3: Data $\mu$ (gray) and $\nu$ (black), optimal $\rho$ (magenta), and optimal $\sigma$ (blue and red for the positive and negative part). (a) generic data in the case (A); (b,c,d) solutions for various data in the case (A); (e) solution for the limit case; (f) solution for data in the case (B).
• Figure 4: (a) Michell structure for a finitely supported system of forces $F$; (b) optimal grillage for a finitely supported torque $f = - {\mathrm{div}}\, F$.
• Figure 5: Numerical solution of the optimal grillage problem: (a) finitely supported data $\mu, \nu$; (b) optimal grillage $\sigma_{\overline\pi}$ where blue and red indicate, respectively, the positive and the negative part; (c) solution $\overline\rho$ of the optimal dominance problem ${\mathcal{V}}(\mu,\nu)$.

Theorems & Definitions (35)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Definition 2.2
• Theorem 2.3: Strassen
• Proposition 3.1
• Remark 3.2
• proof