Existence of a transport map is not the same as regularity. Even smooth positive densities can produce a discontinuous map when the domains or the cost have the wrong geometry. This page separates the Euclidean quadratic theory from the structural conditions needed for general costs.
Twist and nondegeneracy
Definition. A \(C^2\) cost \(c:\Omega\times\Lambda\to\mathbb R\) satisfies the twist condition if, for every \(x\), the map \[ y\longmapsto D_xc(x,y) \] is injective. It is nondegenerate if \(\det D^2_{xy}c(x,y)\ne0\) throughout the domain. Under twist, define the \(c\)-exponential \(Y(x,p)\) by \[ D_xc(x,Y(x,p))=p. \]
Map induced by a differentiable potential. Let \((\varphi,\psi)\) be optimal dual potentials and suppose \(\varphi\) is differentiable at \(x\). If \((x,y)\) belongs to the contact set, then \[ D\varphi(x)=D_xc(x,y). \] Under twist, \(y=Y(x,D\varphi(x))\) is unique.
Proof. The function \(z\mapsto c(z,y)-\psi(y)\) lies above \(\varphi(z)\) and touches it at \(z=x\). Equality of derivatives at a differentiability point gives \(D\varphi(x)=D_xc(x,y)\). Injectivity in \(y\) gives uniqueness. Thus an absolutely continuous source again turns the optimal plan into a map whenever \(c\)-convex potentials are almost everywhere differentiable.
The generated Jacobian equation
Differentiate \(D\varphi(x)=D_xc(x,T(x))\): \[ D^2\varphi(x) =D^2_{xx}c(x,T(x))+D^2_{xy}c(x,T(x))DT(x). \] Nondegeneracy therefore gives \[ DT=(D^2_{xy}c)^{-1}\{D^2\varphi-D^2_{xx}c\}. \] Combining its determinant with \(f(x)=g(T(x))|\det DT(x)|\) yields a generated Jacobian equation. Quadratic cost has \(D^2_{xy}c=-I\) and reduces, after the usual change of potential, to Monge--Ampère.
Interior regularity for quadratic cost
Caffarelli interior regularity. Let \(\Omega,\Lambda\subset\mathbb R^d\) be bounded, with \(\Lambda\) convex. Suppose \[ 0<\lambda\le f,g\le\Lambda_0<\infty \] on their supports, and let \(u\) be the Brenier potential. Then \(u\) is strictly convex and \(u\in C^{1,\alpha}_{\mathrm{loc}}(\Omega)\) for some \(\alpha>0\) depending only on the dimension and density bounds, away from localization issues at the boundary. If \(f,g\in C^{k,\alpha}\) and are bounded away from zero, then \(u\in C^{k+2,\alpha}_{\mathrm{loc}}\).
Proof architecture. For \(h>0\), a section of \(u\) is \[ S_u(x_0,p,h)=\{x:u(x)<u(x_0)+p\cdot(x-x_0)+h\}. \] Alexandrov's maximum principle relates the height \(h\), the volume of a section, and the Monge--Ampère mass. John's ellipsoid theorem normalizes each section by an affine transformation. Bounds on \(f/g\) then give scale-independent volume control. The engulfing property of nested sections implies quantitative strict convexity and Hölder continuity of \(\nabla u\). Once \(D^2u\) is controlled in an appropriate weak class, the linearized equation and classical elliptic estimates bootstrap regularity according to the smoothness of \(f,g\).
Global boundary regularity
Global regularity. If \(\Omega,\Lambda\) are uniformly convex domains with sufficiently smooth boundaries, \(f,g\in C^{k,\alpha}\) are positive and bounded away from zero, and their masses agree, then the Brenier potential satisfies \[ u\in C^{k+2,\alpha}(\overline\Omega) \] up to the standard normalization by an additive constant, and \(\nabla u:\overline\Omega\to\overline\Lambda\) is a diffeomorphism.
The proof requires oblique boundary estimates for the second boundary condition \(\nabla u(\Omega)=\Lambda\). Ordinary Dirichlet theory is not applicable because the boundary values of \(u\) are not prescribed.
What can go wrong
Nonconvex target. If the target has separated components, nearby source points may be sent to different components. The resulting optimal map can jump across an interface even when both densities are smooth. Convexity of the target prevents this geometric tearing in the quadratic theory.
Vanishing density. If \(g\) approaches zero, the equation \(\det D^2u=f/g(\nabla u)\) can force unbounded Hessian eigenvalues. Positivity and two-sided density bounds are therefore genuine ellipticity assumptions.
The Ma--Trudinger--Wang curvature
For a general cost, twist and nondegeneracy produce a map and an equation but do not guarantee continuity. Define \[ A_{ij}(x,p)=D^2_{x_ix_j}c(x,Y(x,p)). \] With the sign convention used here, the Ma--Trudinger--Wang tensor is \[ \operatorname{MTW}_{(x,p)}(\xi,\eta) =-\frac{\partial^2 A_{ij}}{\partial p_k\partial p_l} \xi_i\xi_j\eta_k\eta_l, \qquad \xi\cdot\eta=0. \] The weak MTW condition requires this quantity to be nonnegative; the strong condition requires a positive lower bound proportional to \(|\xi|^2|\eta|^2\).
Regularity principle for general costs. Under smoothness, twist, nondegeneracy, suitable \(c\)-convexity of the domains, positive smooth densities, and an MTW condition, optimal maps are locally Hölder or smoother according to the strength of the assumptions. Failure of the weak MTW condition permits smooth positive data for which the optimal map is discontinuous.
The tensor measures how \(c\)-support functions bend relative to one another. Quadratic Euclidean cost has zero MTW curvature. On a Riemannian manifold, the squared geodesic distance connects this tensor to sectional curvature, while the cut locus creates an additional obstruction to smoothness.
Transport on manifolds
For \(c(x,y)=\tfrac12d_M(x,y)^2\), the optimal map takes the form \[ T(x)=\exp_x(\nabla\phi(x)) \] with a sign depending on the potential convention. Regularity requires that optimal pairs avoid the cut locus, because the squared distance ceases to be smooth there. Curvature, injectivity radius, and \(c\)-convexity of the target jointly replace Euclidean convexity.
A practical hierarchy of conclusions
The assumptions should be read in layers. Lower semicontinuity gives an optimal plan. Twist plus differentiability gives an optimal map. Positive density bounds and convexity yield continuity and interior regularity for quadratic cost. Smooth uniformly convex boundaries give global regularity. For general costs, MTW-type curvature is the additional geometric condition controlling continuity of the map.