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Global $W^{2,1+ε}$ regularity for potentials of optimal transport of non-convex planar domains

This paper establishes a global $W^{2,1+\epsilon}$ regularity estimate for optimal transport potentials when the source domain is a non-convex polygon in $\mathbb{R}^2$ , extending the result to a broader class of domains.

Original authors: Shengnan Hu, Yuanyuan Li

Published 2026-01-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shengnan Hu, Yuanyuan Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Moving a Mountain of Sand

Imagine you have a pile of sand (the source) sitting on a strangely shaped, jagged piece of land. You want to move this sand to a new location (the target) which is a perfectly smooth, convex shape (like a circle or a square).

The goal of Optimal Transport is to move every grain of sand to the new spot in the most efficient way possible, using the least amount of total energy. In math, the "map" that tells you exactly where each grain of sand should go is called the optimal transport map.

Usually, if the starting land is a nice, smooth shape, we know exactly how the sand will flow. But what if the starting land is a non-convex polygon? Think of a shape like a "Pac-Man" or a star with sharp, inward-pointing corners (concave vertices). These sharp corners create "traps" or confusing spots where the math gets very messy.

The Problem: The "Sharp Corner" Glitch

The authors, Shenan Hu and Yuanyuan Li, are tackling a specific problem: How smooth is the map when the starting shape has these sharp, inward-pointing corners?

In the world of math, "smoothness" is measured by how well-behaved the derivatives (slopes and curvatures) of the map are.

If the map is $C^1$ , it means the map is continuous and has no sudden jumps in direction (like a smooth road).
If the map is $W^{2,1+\epsilon}$ , it means the map is even better: its curvature is well-behaved enough that we can integrate it and get a finite result, even if it's not perfectly smooth everywhere.

The authors prove that even with those tricky, sharp inward corners, the map is still "good enough" (specifically, it has $W^{2,1+\epsilon}$ regularity). This is a big deal because, in these sharp corners, the usual rules of geometry often break down.

The Strategy: Zooming In and "Doubling"

To prove this, the authors use a clever strategy involving "sections."

1. The "Section" (The Zoom Lens)
Imagine placing a magnifying glass over a specific point on your jagged land. A "section" is the area you see through that lens. The authors define these sections based on the "height" of the sand pile relative to a flat plane.

The Challenge: Near a sharp, inward corner, the shape of the land changes drastically. A small magnifying glass might show a nice curve, but a slightly bigger one might suddenly include a huge chunk of empty space outside the land.
The Solution: The authors prove a "Doubling Property." Think of this like a rule for a rubber band. If you stretch a rubber band to cover a certain area of the sand, and then you stretch it to cover twice that area, the amount of sand inside the new, larger rubber band doesn't explode unpredictably. It grows in a controlled, predictable way. They prove this holds true even near the sharp corners, provided the "height" of the zoom is small enough.

2. The "Localization" (Keeping it Small)
They show that if you look at a very small area (a small height), the section of the land you are looking at is "well-localized." It doesn't stretch out weirdly across the whole map. It stays contained near the point you are studying. This allows them to treat the messy corner as if it were a manageable, small puzzle.

3. The "Engulfing" (The Nesting Dolls)
Near the sharp corners, they had to prove a property called "engulfing." Imagine you have a set of Russian nesting dolls. If you have a small doll (a small section) inside a slightly larger one, and you move the center of the small doll, the larger doll should still be able to "swallow" or contain the new position of the small one.
The authors had to prove this works even when the dolls are sitting right next to a jagged cliff edge (the concave vertex). They did this by splitting the area near the corner into two halves and proving the rule works for each half separately.

The Result: Why It Matters

By proving these geometric rules (Doubling and Engulfing) work even on jagged, non-convex shapes, the authors can apply advanced mathematical tools (specifically methods developed by De Philippis, Figalli, and others) to show that the curvature of the transport map is controlled.

In plain English: Even though the starting shape is jagged and ugly, the instructions for moving the sand are still mathematically "smooth" and predictable.

The Practical Hook (From the Paper)

The paper mentions one specific real-world reason this matters: Computer Mesh Generation.

When engineers use computers to simulate things (like airflow over a wing or stress on a bridge), they break the object into a grid of tiny shapes (a mesh).
If the object is a complex polygon, creating a high-quality mesh is hard.
Optimal transport can be used as an algorithm to distribute mass evenly over these polygonal domains, creating better, higher-quality grids.
The authors' result guarantees that this mathematical tool works reliably even when the shapes have sharp, inward corners, ensuring the computer simulations will be accurate and efficient.

Summary

The paper is a mathematical proof that says: "Don't worry about the sharp, inward corners of your shape. Even there, the optimal way to move mass is well-behaved and predictable." They achieved this by inventing a way to zoom in on those corners and prove that the geometry, while tricky, follows strict, controllable rules.

Technical Summary: Global $W^{2,1+\epsilon}$ Regularity for Potentials of Optimal Transport of Non-Convex Planar Domains

Problem Statement
This paper investigates the regularity of optimal transport maps in two dimensions ( $\mathbb{R}^2$ ) where the source domain $\Omega$ is a bounded non-convex polygon, and the target domain $\Omega^*$ is a bounded convex domain. The problem involves finding a globally Lipschitz convex potential function $u$ such that the gradient map $\nabla u$ pushes a source probability density $f$ (supported on $\Omega$ ) to a target density $g$ (supported on $\Omega^*$ ), satisfying $(\nabla u)_{\sharp} f = g$ . The densities are assumed to be bounded between positive constants ( $1/\lambda \leq f, g \leq \lambda$ ).

While global $C^{1,\alpha}$ estimates for such potentials have been established for source domains with convex holes (Andriyanova and Chen [1]) and sharp global $W^{2,p}$ estimates have been proven for smooth densities and regular holes (Mooney and Rakshit [29]), the case of a general non-convex polygonal source domain presents specific geometric challenges. Specifically, the standard doubling property of the Monge-Ampère measure, which relies on the convexity of the domain, may fail near the concave vertices of a non-convex polygon. The authors aim to establish a global $W^{2,1+\epsilon}$ estimate for the potential $u$ in this setting.

Methodology and Key Technical Tools
The proof strategy relies on a localized analysis of the Monge-Ampère sections associated with the potential $u$ , adapting techniques from Caffarelli's theory of regularity to the non-convex boundary setting.

Localization and Doubling Property:
The authors first establish a doubling property for the Monge-Ampère measure associated with $u$ for sections of small height $h$ . A critical step is proving that sections $S^c_h(x)$ (centered sub-level sets) are "well-localized."
- Lemma 3.2 demonstrates that for sufficiently small $h$ , any section centered in $\Omega$ is contained within a small ball, ensuring the section does not "escape" the domain or span across non-adjacent edges.
- Proposition 1.1 proves the doubling property: $|\nabla u(S^c_h(x))| \leq C |\nabla u(\frac{1}{2}S^c_h(x))|$ . This is achieved by decomposing the domain near concave vertices. For points away from concave vertices, the intersection of the section with $\Omega$ is convex. Near concave vertices, the authors show that the area of the section intersecting $\Omega$ remains a positive proportion of the total section area, allowing the doubling estimate to hold despite the non-convexity.
Geometric Decay and Engulfing Properties:
To handle the singular behavior near concave vertices ( $V_1$ ), the authors introduce a geometric decay property for sections (Lemma 4.1). This lemma asserts that if a point $x_2$ lies deep within a scaled section of $x_1$ , then a sufficiently small section centered at $x_2$ is contained within a slightly smaller scaled section of $x_1$ .
- The proof utilizes a renormalization argument (Blaschke's selection theorem) to derive a contradiction. If the decay property failed, the limit function would satisfy a Monge-Ampère equation with a specific boundary condition on a half-ball that admits no convex solution (referencing Lemma 6.1 in [29]).
- Based on this, an engulfing property (Lemma 4.2) is established for sections near concave vertices. This property is essential for constructing a Vitali-type covering lemma (Lemma 4.3) tailored to the non-convex geometry.
Iterative Integral Estimates:
The core of the $W^{2,1+\epsilon}$ proof follows the framework of De Philippis, Figalli, and Savin [16, 17].
- Proposition 5.1 provides a bound on the integral of the Hessian norm $\|D^2 u\|$ over a section in terms of the measure of the set where the Hessian is comparable to the normalized size of the section.
- Lemma 5.1 establishes a decay estimate for the integrals of $\|D^2 u\|$ over level sets $D_m^+ = \{x : \|D^2 u(x)\| \geq N^m\}$ . By combining the engulfing property with the Vitali covering lemma, the authors show that the integral over $D_{m+1}^+$ is strictly smaller than the integral over $D_m^+$ by a factor $(1-\tau)$ .
- This geometric decay of the integrals implies that the distribution function of $\|D^2 u\|$ decays sufficiently fast to guarantee integrability in $L^{1+\epsilon}$ .

Key Results
The main result is Theorem 1.1, which states:
Let $\Omega$ be a bounded non-convex polygonal domain in $\mathbb{R}^2$ , and $\Omega^*$ be a bounded convex domain. If $u$ is an Alexandrov solution to the optimal transport problem with densities bounded between $1/\lambda$ and $\lambda$ , then $u \in W^{2,1+\epsilon}(\Omega)$ for some $\epsilon > 0$ .

The proof is completed by showing that the $W^{2,1+\epsilon}$ estimate holds both in the interior of the domain (away from concave vertices) and in neighborhoods of the concave vertices, utilizing the specific geometric decomposition and covering arguments developed in the paper.

Significance and Claims
The paper claims to extend the regularity theory of optimal transport potentials to the case of non-convex polygonal source domains.

Theoretical Contribution: It addresses the failure of the standard doubling property in non-convex settings by proving that sections are sufficiently localized and that the "engulfing" property holds even near concave vertices. This allows for the derivation of global $W^{2,1+\epsilon}$ estimates without requiring the source domain to be convex or the holes to be smooth.
Practical Motivation: The authors note that optimal transport between polygonal domains is of significant practical value, particularly in numerical computation and finite element methods. High-quality mesh generation over polygonal domains often relies on optimal transport to distribute mass effectively, thereby enhancing simulation accuracy and efficiency. The established regularity provides a theoretical foundation for the convergence and stability of such mesh generation algorithms.

The authors do not claim global $C^{1,\alpha}$ or $W^{2,p}$ for $p>1+\epsilon$ in this general setting, but rather the specific $W^{2,1+\epsilon}$ regularity, which is a sharp result given the geometric constraints of polygonal domains.