Global $W^{2,p}$ regularity for optimal transport maps with costs far from quadratic, and the structure of the singular set in the partial $W^{2,p}$ regularity theory

Chen and Figalli (J. Funct. Anal. 272 (2017), 4588-4605) proved that, for the optimal transport problem with a cost satisfying standard smoothness and twist conditions and positive continuous densities, the potential function belongs to $W^{2,p}_{loc}$ for every $p ≥ 1$ outside a closed singular set Σ of Lebesgue measure zero that can be chosen independently of $p$; and that global, up-to-the-boundary $W^{2,p}$ estimates hold when the domains are $C^2$ uniformly convex and the cost is a $C^2$-small perturbation of the quadratic cost $-x·y$ (a conclusion new even for the exactly quadratic cost with merely continuous densities). The paper leaves undetermined whether analogous global $W^{2,p}$ estimates hold for costs that are not small perturbations of the quadratic cost and/or under relaxed domain convexity, and leaves the geometric and measure-theoretic structure of Σ (e.g. Hausdorff dimension, rectifiability) open.

Background

Optimal transport, introduced by Monge, asks for the cheapest way to move a mass distribution with density $f$ on a bounded domain $X ⊂ ℝ^n$ onto a density $g$ on a bounded domain $Y$, when the cost of moving a particle from $x$ to $y$ is a prescribed function $c(x,y)$. For the quadratic cost $c(x,y) = |x-y|^2/2$ (equivalently $-x·y$, which differs from it by terms irrelevant for minimization), Brenier's theorem identifies the unique optimal map as the gradient $T = ∇u$ of a convex potential $u$, and $u$ solves the second boundary value problem for the Monge-Ampère equation (the equation prescribing the determinant of the Hessian): $\det D^2 u(x) = f(x)/g(∇u(x))$ with $∇u(X) = Y$. For a general cost the potential is instead $c$-convex, meaning $u(x) = \sup_{y∈Y}\,(-c(x,y) + λ_y)$ for some constants $λ_y$ (the analogue, for costs, of convexity), and the optimal map is $T(x) = c\text{-exp}_x(∇u(x))$, where the $c$-exponential is defined by $c\text{-exp}_x(p) = y$ if and only if $-D_x c(x,y) = p$. The potential then solves a Monge-Ampère-type equation in which the mixed Hessian $D^2_{xy}c$ (the matrix of mixed second derivatives of $c$) plays the role played by the identity for the quadratic cost. Standard structural assumptions on the cost are: $c ∈ C^3$ with $\|c\|_{C^3}$ finite ((C0)); the twist conditions, i.e. injectivity of $y ↦ D_x c(x,y)$ for each $x$ and of $x ↦ D_y c(x,y)$ for each $y$ ((C1), (C2)); and nonvanishing of $\det D^2_{xy}c$ ((C3)).

Here $W^{2,p}$ denotes the Sobolev space of functions whose weak second derivatives are $p$-integrable; interior (local, $W^{2,p}_{loc}$) regularity refers to compact subsets of $X$, while global regularity ($u ∈ W^{2,p}(X)$) includes a full neighborhood of the boundary $∂X$. A domain is uniformly convex if the curvatures of its boundary are bounded from below by a positive constant.

The regularity theory of these equations was developed by Caffarelli in the quadratic case: interior $W^{2,p}$ estimates under bounded positive densities, and global $C^{2,α}$ estimates on uniformly convex domains with Hölder densities, together with counterexamples showing that convexity-type assumptions on the domains cannot be dropped altogether. For general costs, Ma, Trudinger, and Wang discovered that a fourth-order sign condition on the cost (the MTW condition, a curvature-type inequality involving derivatives of $c$ up to order four) guarantees smoothness of the potential in the interior for smooth data on uniformly convex domains, and Loeper proved that MTW is essentially necessary for the optimal map to be continuous for all smooth densities; many costs of interest fail MTW. Without MTW, De Philippis and Figalli established partial regularity: for general costs and smooth positive densities the potential is smooth outside a closed singular set of measure zero; Liu, Trudinger, and Wang proved interior $W^{2,p}$ and sharp $C^{1,α}$ estimates under a suitable normalization of the cost.

Chen-Figalli (2017) strengthened partial regularity to second order: for general costs satisfying (C0)-(C3) and merely continuous positive densities, there exists a closed singular set $Σ ⊂ X$ of Lebesgue measure zero, independent of $p$, such that $u ∈ W^{2,p}_{loc}(X \setminus Σ)$ for every $p ≥ 1$ (Theorem 2.1). Combining these estimates with Savin's global $W^{2,p}$ estimates for the Dirichlet problem for Monge-Ampère, the authors also proved the first global up-to-the-boundary $W^{2,p}$ result for the second boundary value problem with continuous densities: if $X, Y$ are $C^2$ uniformly convex and $\|c + x·y\|_{C^2(X×Y)} ≤ δ$ with $δ$ smaller than a constant $\hat{δ}$ depending only on the dimension, the moduli of continuity of $f, g$, and the uniform convexity and $C^2$-smoothness of $X$ and $Y$, then $u ∈ W^{2,p}(X)$ for every $p ≥ 1$ (Theorem 2.3). The paper establishes nothing, however, for costs that are not $C^2$-close to the quadratic cost, nor about the structure of $Σ$ beyond closedness and measure zero.

Problem Statement

Let $X, Y ⊂ ℝ^n$ be bounded domains, let $c : X × Y → ℝ$ be a cost satisfying (C0)-(C3) — $c ∈ C^3$; the twist conditions, i.e. injectivity of $y ↦ D_x c(x, y)$ for each fixed $x$ and of $x ↦ D_y c(x, y)$ for each fixed $y$; and $\det D^2_{xy}c \ne 0$, where $D^2_{xy}c$ is the matrix of mixed second derivatives of $c$ — and let $f, g$ be positive continuous densities on $X, Y$. Let $u$ be the $c$-convex potential of the optimal transport map from $(X, f)$ to $(Y, g)$, i.e. $u(x) = \sup_{y∈Y}\,(-c(x,y) + λ_y)$ for constants $λ_y$. Chen-Figalli (2017) proved that there is a closed singular set $Σ ⊂ X$ of Lebesgue measure zero, independent of $p$, with $u ∈ W^{2,p}_{loc}(X \setminus Σ)$ for all $p ≥ 1$ (here $W^{2,p}$ denotes the Sobolev space of functions whose weak second derivatives are $p$-integrable, and $W^{2,p}_{loc}$ its local version on compact subsets), and that $u ∈ W^{2,p}(X)$ for all $p ≥ 1$ when $X, Y$ are $C^2$ uniformly convex and $\|c + x·y\|_{C^2(X×Y)}$ is sufficiently small. Determine what happens beyond this near-quadratic, uniformly convex regime.

(1) Global estimates for non-near-quadratic costs. For which costs that are not $C^2$-small perturbations of the quadratic cost $-x·y$ — for instance costs satisfying the MTW condition (the Ma-Trudinger-Wang condition, a fourth-order sign condition on the cost, i.e. a curvature-type inequality involving derivatives of $c$ up to order four, that guarantees interior smoothness), or general twisted costs failing MTW — does the potential satisfy global, up-to-the-boundary $W^{2,p}$ estimates on $C^2$ uniformly convex (or more generally convex) domains with positive continuous densities? Establish such estimates for a class of costs quantitatively bounded away from the quadratic cost — that is, with $\|c + x·y\|_{C^2(X×Y)} ≥ ε$ for a fixed $ε > 0$ independent of $f, g, X, Y$ — or exhibit a cost in this class, with positive continuous (or smooth) densities and uniformly convex domains, for which $u \notin W^{2,p}(X)$ for some $p ≥ 1$. (For the exactly quadratic cost this is now known to hold on arbitrary convex domains, and costs affinely equivalent to it — of the form $c(x,y) = -A x·y + b(x) + d(y)$ with $A$ invertible linear — are treated as that settled quadratic case and are excluded from the class in question; the open content concerns the remaining non-quadratic costs, with or without relaxed domain convexity.)

(2) Structure of the singular set. What is the geometric and measure-theoretic structure of the singular set $Σ$ of the general-cost theory: is its Hausdorff dimension bounded (for example by $n-1$), is $Σ$ rectifiable (contained in a countable union of sufficiently regular hypersurfaces), or can it be essentially arbitrary among closed measure-zero sets? Provide general bounds or explicit constructions.

The two items share one scientific objective: to complete the second-order regularity picture left open by Chen-Figalli (2017) — item (1) is the global counterpart of their Theorem 2.3 with the cost-closeness assumption removed, and item (2) is the structural refinement of their Theorem 2.1.

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