Optimal (partial) transport to non-convex polygonal domains

Technical Summary: Optimal (Partial) Transport to Non-Convex Polygonal Domains

Problem Formulation
This paper investigates the regularity of optimal transport maps and free boundaries when the target domain is a non-convex polygon in $\mathbb{R}^2$. The study addresses two distinct scenarios:

1. Complete Optimal Transport: Given two bounded domains $\Omega$ and $\Omega^*$ with equal measure, the goal is to find a transport plan minimizing the quadratic cost $\int |x-y|^2 d\gamma$. By Brenier's theorem, this is equivalent to finding a convex potential $u$ such that $(Du)_\# \chi_\Omega = \chi_{\Omega^*}$. The focus is on the structure of the singular set $A = \{x \in \Omega : u \text{ is not differentiable at } x\}$.
2. Optimal Partial Transport: Given a mass $m < \min(|\Omega|, |\Omega^*|)$, the problem seeks to transport a subdomain $U \subset \Omega$ to a subdomain $V \subset \Omega^*$. The boundary of the active transport region, $F = \partial U \cap \Omega$, is termed the "free boundary." The paper analyzes the regularity of $F$ when $\Omega^*$ is a non-convex polygon.

Methodology
The authors employ a combination of convex analysis, geometric measure theory, and regularity theory for Monge-Ampère equations. Key methodological components include:

• Legendre Transform and Dual Potentials: The analysis relies heavily on the dual convex function $v = u^*$ (or $v$ in the partial transport case). The differentiability of $u$ at $x$ corresponds to the differentiability of $v$ at points in the subgradient $\partial^- u(x)$.
• Localization and Sublevel Sets: The authors utilize centered sublevel sets $S^c_h[v](y)$ to localize the behavior of the potential near singularities. By establishing that these sublevel sets remain within small neighborhoods of the target boundary, they invoke quantitative uniform convexity estimates.
• Geometric Decomposition of Singularities: The singular set (or free boundary) is decomposed based on the geometry of the subgradient $\partial^- u(x)$ relative to the target polygon $\Omega^*$. The authors categorize points $x$ based on whether $\partial^- u(x)$ contains a vertex of $\Omega^*$, touches one edge, or touches multiple edges.
• Regularity Bootstrapping: For regions where the target geometry is locally convex (e.g., near the interior of an edge), the authors apply Caffarelli's regularity theory and Schauder estimates for uniformly elliptic equations. They demonstrate that the potential is $C^\infty$ in these regions, implying the singular set is a smooth curve.
• Obliqueness Estimates: In the partial transport case, the proof of smoothness relies on establishing an obliqueness estimate (the angle between the normal to the free boundary and the normal to the target boundary is non-zero), which allows the application of boundary regularity results.

Key Contributions and Results

• Structure of the Singular Set (Complete Transport):
  The paper proves Theorem 1.1, which provides a precise description of the singular set $A$ when $\Omega^*$ is a non-convex polygon. The authors show that $A$ is either a finite set or, away from a finite number of points $\{x_i\}$, is locally a smooth one-dimensional curve.

  • The "exceptional" points $\{x_i\}$ correspond to locations where the subgradient $\partial^- u(x)$ touches a vertex of the polygon or touches three or more edges.
  • Near these exceptional points, the singular set is a union of finitely many Lipschitz curves.
  • The proof involves decomposing $A$ into sets $\Sigma_1$ (touching vertices), $\Sigma'_2$ (touching exactly two edges), and $\Sigma''_2$ (touching $\ge 3$ edges). It is shown that $\Sigma_1$ and $\Sigma''_2$ are finite sets, while $\Sigma'_2$ consists of smooth curves.

• Regularity of the Free Boundary (Partial Transport):
  The paper proves Theorem 1.2, establishing that the free boundary $F$ in the optimal partial transport problem is smooth away from finitely many singular points.

  • Similar to the complete transport case, the singular points on the free boundary arise where the subgradient touches a vertex of $\Omega^*$ or multiple edges.
  • The authors prove that the set of points touching vertices ($F_1$) or multiple edges ($F_2$) is finite.
  • For points touching exactly one edge ($F_3$), the free boundary is shown to be smooth ($C^\infty$) via obliqueness estimates and boundary regularity theory.

• Generalizations and Conjectures:

  • Remark 1.2 notes that these results extend to general positive smooth densities and, with slightly weaker regularity ($C^{1,\alpha}$), to densities bounded above and below.
  • Remark 3.2 indicates the method applies to targets that are unions of two bounded convex domains in general dimensions.
  • Section 5 formulates two conjectures for higher dimensions ($\mathbb{R}^n$). Conjecture 5.1 posits that for complete transport to a non-convex polytope, the singular set is locally a smooth $(n-1)$-dimensional hypersurface away from a set of Hausdorff dimension at most $n-2$. Conjecture 5.2 makes a similar claim for the free boundary in partial transport.

Significance and Claims
The paper claims to provide a more precise description of the singularity structure in optimal transport than previously established for non-convex planar domains. While previous work (e.g., Figalli [16]) established that singular sets are $C^1$ manifolds away from a countable set, this work refines that result for polygonal targets, showing the singular set is smooth away from a finite set of points.

The authors highlight the practical relevance of these results in numerical computations, specifically for mesh generation on polygonal domains using optimal transport algorithms. By characterizing the singularities, the paper aids in understanding the limitations and behavior of such algorithms near non-convex corners.

The paper maintains a modest tone regarding higher dimensions, explicitly stating that the results for $n=2$ are proven, while the extension to $n \ge 3$ remains a conjecture. The authors do not claim to have solved the higher-dimensional case but provide a structural framework and specific conjectures for future investigation.