Prescribed Jacobian Equations (PJE)

• Prescribed Jacobian Equations (PJE) are nonlinear PDEs that constrain the determinant of mappings, generalizing the Monge–Ampère framework.
• They are applied in optimal transport, elasticity, and grid generation, and rely on variational principles and convexity conditions.
• Ongoing research tackles challenges in regularity, uniqueness, and operator surjectivity, revealing deep analytical and topological obstructions.

Prescribed Jacobian Equations (PJE) are nonlinear partial differential equations that constrain the Jacobian determinant of a mapping between domains. They generalize classical Monge–Ampère equations and encompass problems in optimal transport, geometric optics, elasticity, grid generation, and compensated compactness analysis. Modern theory for PJE incorporates regularity, existence, structural convexity, uniqueness, and non-regularity mechanisms, with significant connections to measure theory and Banach space geometry.

1. Formulation and Generalizations

Standard PJE demands a mapping $\pi: \mathbb{R}^d \rightarrow \mathbb{R}^d$ or a scalar potential $u:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}$ such that

$$\det D\pi(x) = \rho(x) \qquad \text{or equivalently} \qquad \det D_p Y(x, u(x), Du(x)) = \psi(x, u(x), Du(x))$$

where $D\pi$ denotes the derivative matrix (Fréchet differential), $\rho$ (or $\psi$) is a measurable $L^\infty$ function or prescribed density, and $Y$ is a $C^1$ or $C^4$ generating function, often depending on auxiliary parameters.

The generating function formalism, elaborated by Trudinger (Trudinger, 2012), extends the classical cost function in optimal transport by introducing extra variables and recasting PJE as a fully nonlinear Monge–Ampère-type equation:

$$\det \left[D^2u(x) - A(x, u(x), Du(x))\right] = B(x, u(x), Du(x))$$

where $A$ and $B$ are determined algorithmically from the generating function $Y$ and its derivatives.

2. Regularity, Existence, and Convexity Theory

Existence of solutions heavily depends on domain geometry, convexity conditions of the generating function, and the monotonicity, integrability, and positivity of the prescribed density. Critical convexity hypotheses are encoded in Ma–Trudinger–Wang (MTW)-type conditions. Weak MTW (A3w) suffices to guarantee existence of generalized and regular solutions under mild domain convexity, while monotonicity in the potential ($u$) is not structurally necessary (Trudinger, 2020):

• Generalized solution existence is established when generating functions satisfy injectivity, nondegeneracy, and gradient bounds (A1, A2, A1*, G1–G5).
• Regularity: Classical $C^{3,\alpha}$ regularity holds for strictly g-convex solutions under strong or weak convexity (Trudinger, 2012, Rankin, 2022).
• Uniqueness: Aleksandrov solutions of generated PJE coincide almost everywhere in gradient if they agree at a differentiable point, as shown by Awanou–Tsogtgerel (Awanou et al., 2021).

Hypothesis	Regularity Implication	Reference
A3w (weak MTW)	Strict g-convexity, $C^1$	(Trudinger, 2020, Rankin, 2022)
A3 (strict MTW)	$C^{3,\alpha}$ regularity	(Trudinger, 2012)
No monotonicity (A4w)	Convexity, regularity	(Trudinger, 2020)

The notion of "g-convexity" generalizes convexity for potentials, ensuring the subgradient map is a bijection between source and target domains and that the push-forward density matches prescribed constraints.

3. Non-Regularity and Limitations

Recent work (Takáč, 16 Jun 2025) demonstrates profound non-regularity near $L^\infty$ for the prescribed Jacobian equation in top-dimensional settings ($d\geq 2$). The key result is that the convex hull

$$\operatorname{conv} \left( \{\det D\pi : \operatorname{Lip}(\pi) \le L \} \right) \subset L^\infty$$

is Baire-meager (a countable union of weak*-nowhere-dense sets), indicating severe topological smallness and failure of surjectivity for the Jacobian operator under uniform Lipschitz bounds. This yields strong non-regularity: for any fixed $L$, there exist smooth target functions $\rho$ for which no Lipschitz solution $\pi$ (even as formal convex combinations) can exist within the prescribed constraints. This topological obstruction directly correlates to the failure of the Flat Chain Conjecture (Lang's version), showing that not every metric current is represented by a flat chain via the prescribed Jacobian mechanism.

4. Variational and Computational Frameworks

Variational formulations underpin PJE in numerous applications. For grid generation and elasticity, the minimization of Dirichlet-type energies subject to prescribed Jacobian and, in some settings, prescribed curl, is central (Chen et al., 2015, Guerra et al., 2020). The Euler–Lagrange system derived from such energy functionals typically takes the form:

$$E[u] = \frac{1}{2} \int_\Omega [(J(T) - f_0)^2 + \alpha ( \operatorname{curl} T - g_0 )^2 ]\, dA, \qquad T(x) = x + u(x)$$

with stationarity conditions yielding adjoint PDEs that can be discretized and solved efficiently with standard Poisson solvers.

Prescribing only the Jacobian yields deformations with correct area-ratio but potentially undesirable angular distortion; adding curl constraints improves grid orientation and uniqueness (Chen et al., 2015, Zhou et al., 2017). The combination of Dirichlet boundary conditions, positive Jacobian, and prescribed curl ensures local uniqueness in the $H^2$ norm, though global uniqueness remains unresolved.

5. Symmetry, Uniqueness, and Energy-Minimizing Maps

Symmetry and uniqueness in energy-minimizing maps with prescribed Jacobian, under Dirichlet-type energies, arise when radial monotonicity of the prescribed density is enforced (Guerra et al., 2020). The sufficient and necessary condition for unique radial minimizers is:

$f(r) \leq \fint_{B_r} f$

Where this fails, explicit counterexamples and constructions yield uncountably many energy-minimizing, non-symmetric maps, even under fixed boundary conditions. The PJE is thus underdetermined unless the data exhibits strong monotonicity; energy minimization alone does not ensure a unique or symmetric solution.

6. Differential Inclusions and Flexibility in Subcritical Sobolev Spaces

For $p < d$, the Jacobian determinant loses rigidity: convex integration and Young measure theory enable the characterization of all possible oscillation/concentration patterns of gradients under det-constraints (Koumatos et al., 2013). Orientation-preserving and incompressible mappings can be constructed as solutions to first-order differential inclusions in $W^{1,p}$, highlighting the flexibility of PJE in low regularity settings.

No convexity condition in $W^{1,p}$ (even $p$-quasiconvexity) suffices to prevent collapse of the determinant to zero, establishing the limitations of standard lower semicontinuity theorems for energies blowing up as det$\to 0$.

7. Banach Space Geometric Structure and Compensated Compactness

The surjectivity problem for the Jacobian operator $J$ mapping Sobolev spaces into Hardy spaces $\mathcal{H}^1$ is analyzed via Banach space and duality geometry (Lindberg, 2022). The failure of strict convexity and uniqueness of minimum-norm solutions for the planar Jacobian links analytic, geometric, and topological obstructions in PJE to open conjectures about compensated compactness and operator surjectivity. The critical technical barrier is the global a priori estimate in the norm correspondence:

$$\|\omega\|_{L^2}^2 \lesssim \|Q\omega\|_{\mathcal{H}^1}$$

which would imply surjectivity. The framework unifies the operator-theoretic perspective for quadratic Jacobian-type PDEs and characterizes the critical hurdles via James boundary and Krein–Milman-type results.

8. Integrability and Algebraic Systems for Polynomial Jacobian Constraints

For polynomial mappings, the PJE reduces to finite algebraic systems among homogeneous components (Medeiros et al., 2014). The recursive solution structure establishes that specifying the Jacobian as constant imposes rigidity: only certain combinations of the homogeneous parts of the polynomial maps qualify, yielding pure algebraic obstructions and facilitating the explicit integration of the Jacobian equation for polynomial solutions.

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In synthesis, Prescribed Jacobian Equations form a nexus for fully nonlinear PDE theory, regularity and convexity analysis, compensated compactness, and variational grid generation, with pervasive flexibility in low-regularity regimes and strict non-regularity at high regularity. The topological and geometric structure of the solution sets, especially under Lipschitz or Sobolev constraints, is fundamental, with ongoing research addressing both analytic uniqueness, effective algorithms, and deep obstructions inherent to the operator landscape.