Brenier's theorem turns quadratic transport into a convex potential \(u\). When the measures have densities, the pushforward condition becomes a nonlinear partial differential equation for \(u\). Its determinant measures how the gradient map changes volume.
Derivation from change of variables
Jacobian equation. Let \(\mu=f(x)\,\mathrm{d}x\) on a domain \(\Omega\), \(\nu=g(y)\,\mathrm{d}y\) on \(\Lambda\), and suppose \(T:\Omega\to\Lambda\) is a \(C^1\)-diffeomorphism with \(T_\#\mu=\nu\). Then \[ f(x)=g(T(x))|\det DT(x)| \] for almost every \(x\). If \(T=\nabla u\) with \(u\) convex and \(D^2u>0\), then \[ \boxed{\det D^2u(x)=\frac{f(x)}{g(\nabla u(x))}}, \qquad \nabla u(\Omega)=\Lambda. \]
Proof. For every bounded continuous \(h\), \[ \int_\Lambda h(y)g(y)\,\mathrm{d}y =\int_\Omega h(T(x))f(x)\,\mathrm{d}x. \] Changing variables \(y=T(x)\) in the left side gives \[ \int_\Omega h(T(x))g(T(x))|\det DT(x)|\,\mathrm{d}x. \] Equality for all \(h\) yields the Jacobian identity. For a convex potential, \(DT=D^2u\) is positive semidefinite, so the absolute value may be removed. The image condition is the second boundary condition, not a pointwise Dirichlet condition.
Ellipticity and convexity
The operator \(u\mapsto\det D^2u\) is elliptic only on convex functions. Indeed, differentiating the determinant at a positive-definite matrix \(A\) in direction \(H\) gives \[ D(\det)_A[H]=\operatorname{cof}(A):H =\det(A)\operatorname{tr}(A^{-1}H), \] whose coefficient matrix \(\operatorname{cof}(A)\) is positive definite. Convexity is therefore both the transport structure and the ellipticity condition.
Linearized Monge--Ampère operator. At a smooth strictly convex solution \(u\), the linearization of \(\det D^2u=F\) is \[ L_uv=\operatorname{cof}(D^2u):D^2v. \] Moreover, each row of \(\operatorname{cof}(D^2u)\) is divergence free, so \[ L_uv=\nabla\cdot(\operatorname{cof}(D^2u)\nabla v). \]
Alexandrov solutions
Optimal potentials need not be twice differentiable. The correct weak formulation uses the geometry of the subgradient rather than distributional determinants.
Definition. For a convex \(u:\Omega\to\mathbb R\), its Monge--Ampère measure is \[ M_u(E)=\mathcal L^d(\partial u(E)), \qquad E\subset\Omega\ \text{Borel}. \] We call \(u\) an Alexandrov solution of \(\det D^2u=\rho\) if \(M_u(E)=\int_E\rho(x)\,\mathrm{d}x\) for every Borel \(E\).
Consistency. If \(u\in C^2(\Omega)\) is strictly convex, then \[ M_u(E)=\int_E\det D^2u(x)\,\mathrm{d}x. \]
Proof. Strict convexity makes \(\partial u(x)=\{\nabla u(x)\}\), and the area formula for the injective gradient map gives \[ \mathcal L^d(\nabla u(E)) =\int_E|\det D(\nabla u)(x)|\,\mathrm{d}x =\int_E\det D^2u(x)\,\mathrm{d}x. \]
Transport formulation of weak solutions
Brenier--Alexandrov correspondence. Let \(\Omega,\Lambda\) be bounded convex domains, let \(f,g\) be positive densities with equal total mass, and let \(u\) be the Brenier potential satisfying \((\nabla u)_\#(f\,\mathrm{d}x)=g\,\mathrm{d}y\). Then \(u\) solves \[ g(\nabla u)\det D^2u=f \] in the Alexandrov sense, together with \(\partial u(\Omega)\subset\overline\Lambda\) and the full-mass image condition. Conversely, an Alexandrov solution satisfying the transport boundary condition induces the optimal map.
The equation and the image condition must be kept together. The determinant alone controls local volume change but does not specify which target domain receives the mass.
One-dimensional and radial examples
One dimension. The equation is \[ u''(x)=\frac{f(x)}{g(u'(x))}. \] Since \(T=u'\), integration gives \(F_\nu(T(x))=F_\mu(x)\), hence \(T=F_\nu^{-1}\circ F_\mu\). Thus the Monge--Ampère equation reduces exactly to monotone rearrangement.
Radial measures. If \(f,g\) are radial and \(T(x)=t(r)x/r\), mass conservation inside balls gives \[ \int_0^r f_0(s)s^{d-1}\,\mathrm{d}s = \int_0^{t(r)}g_0(s)s^{d-1}\,\mathrm{d}s. \] Differentiation yields \[ f_0(r)r^{d-1}=g_0(t(r))t(r)^{d-1}t'(r), \] which equals the determinant formula because the radial and tangential eigenvalues of \(DT\) are \(t'(r)\) and \(t(r)/r\).
Quadratic potentials. For \(u(x)=\tfrac12x^{\mathsf T}Ax+b\cdot x\) with \(A>0\), \(\det D^2u=\det A\). The gradient map sends ellipsoids to ellipsoids and recovers the Gaussian transport matrix from page 8.
Regularity questions
The equation suggests that smooth positive \(f,g\) should produce a smooth potential, but this is false without geometry. Convexity of the domains, bounds away from zero and infinity, and boundary regularity are essential. The next page separates interior regularity, global regularity, and the additional curvature condition required by general costs.