Monge-Ampère Equation: Theory & Applications

• Monge-Ampère equation is a fully nonlinear PDE defined by the determinant of a function's Hessian, underpinning theories in convex and differential geometry.
• It embraces diverse solution concepts including classical, Aleksandrov, and viscosity solutions, each with unique regularity and existence criteria.
• Recent advances in numerical and variational methods offer robust computational frameworks, significantly impacting geometric analysis and applied mathematics.

The Monge-Ampère equation is a prototypical fully nonlinear partial differential equation of determinant type, arising fundamentally in convex geometry, differential geometry, optimal transport, and complex geometry. Its general form involves the determinant of the Hessian matrix of a function and a prescribed right-hand side, and its classical, weak, and geometric interpretations underlie a vast array of analytic and geometric theories.

1. General Formulation and Solution Notions

Given a domain $\Omega \subset \mathbb{R}^n$, the real Monge-Ampère equation seeks a convex function $u : \Omega \to \mathbb{R}$ satisfying

$\det D^2 u(x) = f(x), \quad x \in \Omega$

with either Dirichlet ($u = g$ on $\partial\Omega$) or more general boundary conditions. In its classical form, $D^2 u$ is the (real symmetric) Hessian, and $f(x)$ is positive and smooth. For applications—including optimal transport, Kähler geometry, and prescribed Gaussian curvature—variants with additional structure or degeneracy on $f$ or on the domain appear.

Notions of Solution

• Classical solutions: $u \in C^2(\Omega)$ and satisfies the equation everywhere. For smooth $f$ and uniformly convex $\Omega$, classical solutions exist and are unique.
• Aleksandrov (generalized) solutions: For convex, continuous $u$, the Monge-Ampère measure $\mu_u(E) := |\partial u(E)|$ (the Lebesgue measure of the subdifferential image) matches $\int_E f(x) \, dx$ on each Borel set $E$. Existence and uniqueness in this sense hold under $f \in L^1(\Omega)$, $f \geq 0$, and continuous Dirichlet data.
• Viscosity solutions: The comparison principle and stability theory for fully nonlinear PDEs apply to the Monge-Ampère equation regarded as $F(D^2 u, x) = \det D^2 u - f(x) = 0$, yielding unique continuous (viscosity) solutions under appropriate boundary and positivity conditions.

2. Existence, Uniqueness, and Regularity

Classical existence and regularity theory for $\det D^2 u = f(x)$ on smooth, uniformly convex domains $\Omega$ with smooth positive $f$ and boundary data $g$ is based on the seminal results of Caffarelli, Nirenberg, Spruck, and others. Uniqueness derives from the comparison principle for convex solutions and the structure of the Monge-Ampère measure. Regularity theory distinguishes between interior $C^{2,\alpha}$ regularity (from Calabi, Pogorelov, and Caffarelli) and global $C^{2,\alpha}$ up to $\partial\Omega$ (requiring $C^3$-smooth, strictly convex domains and data).

Notable Estimates

• Calabi interior $C^3$ estimate: For $\det D^2 u = 1$, Calabi derived a second-order differential inequality for the scalar curvature of the affine metric, providing uniform $C^3$ bounds in terms of $C^2$ data.
• Pogorelov $C^2$ estimate: Pogorelov's maximum principle controls $|D^2 u|$ inside sublevel sets, leading to sharp boundary blow-up rates for solutions to $\det D^2 u = 1$ with zero boundary data.
• Caffarelli-Nirenberg-Spruck boundary regularity: On $C^3$-regular, uniformly convex domains, the $C^{2,\alpha}$ regularity of solutions up to the boundary holds for $C^3$ boundary data, but is generally false for less regular data.

3. Weak and Degenerate Problems

The Monge-Ampère equation admits a robust weak theory. Aleksandrov and viscosity formulations are fundamental for singular, degenerate, or measure data, as well as in the presence of degeneracies in $f$. Mooney constructed Alexandrov solutions with Cantor-type singular Hessians, showing that strict convexity and higher integrability of $f$ cannot be dispensed with if $W^{2,1}$ or $C^2$ regularity is desired.

Partial regularity is a major area of current research, especially when $f$ vanishes or diverges, with Hausdorff-dimension estimates for the singular set and extensions to isometric embedding and geometric optics.

4. Geometric and Complex Extensions

Complex and geometric Monge-Ampère equations arise on compact Kähler manifolds, in optimal transport, and in the theory of special Lagrangian submanifolds. Given a compact Kähler manifold $(X,\omega)$, the generalized (complex) Monge-Ampère equation reads

$$a_0 (\omega + dd^c\phi)^n + a_1 (\omega + dd^c\phi)^{n-1} + \cdots + a_{n-1} (\omega + dd^c\phi) = \eta$$

where $dd^c = \frac{i}{2\pi} \partial \bar{\partial}$, $a_i$ are closed $(i, i)$ forms, and $\eta$ is a prescribed volume form. The method of continuity, together with a priori $C^0$ and $C^2$ estimates, yields existence and uniqueness under cohomological and positivity constraints, conditional in higher dimensions on Hessian lower bounds (Pingali, 2012).

Complex Monge-Ampère reductions and twistor-theoretic constructions further link the subject to special Lagrangian geometry, self-dual gravity (Plebanski equations), and integrable systems (Banos, 2011).

5. Numerical Methods

Modern numerical analysis addresses the Monge-Ampère equation using monotone wide-stencil finite differences (Neilan et al., 2019), two-scale approaches, variational (mixed and penalty) finite element methods (Awanou et al., 2015), meshfree collocation (Böhmer et al., 2017), and power-diagram geometric methods. With degenerate ellipticity and non-variational structure, monotonicity and stability are key for convergence to viscosity/Aleksandrov solutions, validated using the Barles–Souganidis framework.

Recent advances:

• Quadrature-based monotone finite difference schemes (Brusca et al., 2022) use an angular integral representation of $\det D^2u$ to achieve arbitrarily high-order angular accuracy without sacrificing monotonicity, drastically reducing required stencil width and improving convergence even in degenerate problems.
• Hybrid Newton/finite-difference solvers switch between centered, high-accuracy (non-monotone) and monotone discretizations based on local regularity to combine accuracy and robustness (Froese et al., 2010).
• Bellman-type algorithms reformulate $\det D^2u$ as an infimum over linear elliptic operators, iteratively updating a minimizer $B$ and solving a resulting linear PDE per step, providing orders of magnitude speedup over fixed-point or Gauss-Seidel algorithms, especially for degenerate or mildly singular problems (Le et al., 7 May 2025).
• Variational convex programming: Alexander-type solutions as minimizers of strictly convex functionals under convexity and nonlinear constraints, with discretized convex programs solved by modern conic solvers (Awanou et al., 2015).

6. Generalized and Geometric Structures

The Monge-Ampère equation is naturally interpreted in the context of exterior differential systems (EDS). On jet spaces, the PDE is encoded by a differential ideal generated by the contact 1-form and a determinantal $n$-form (Kawamata et al., 2020). This framework generalizes to higher-order and systems, with a precise bijection between generalized Monge-Ampère PDEs and certain EDSs, and encompasses classical equations such as the KdV and Cauchy-Riemann equations.

In integrable systems, generalized (e.g., third-order) Monge-Ampère equations can admit bi-Hamiltonian, symplectic, and recursion operator structures, complete infinite hierarchies of commuting flows and conservation laws, and connections to topological field theory (notably, the WDVV/associativity equations) (Kersten et al., 2011).

7. Boundary Phenomena and Asymptotics

Boundary singularities and regularity failures are intrinsic to the Monge-Ampère equation's geometry. For polygonal domains with Guillemin boundary conditions (singular logarithmic terms), full asymptotic expansions (in powers of the boundary distance and logs) characterize solutions away from corners (Rubin, 2014). On exterior domains, solutions asymptotic to quadratic polynomials (plus possible logarithmic terms in two dimensions) are sharply classified under decay rates for $f(x) - 1$ at infinity, with sharp threshold $\beta > 2$ (Bao et al., 2013).

In degenerate or mixed-type problems, Nash–Moser iteration (energy-based, tame estimates) can yield local regular solutions even when the PDE degenerates or changes type (as in prescribed Gaussian curvature or local isometric embedding) (Khuri, 2010).

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The Monge-Ampère equation thus forms a central nexus between nonlinear PDE theory, convex geometry, complex and symplectic geometry, geometric analysis, regularity theory, and computational mathematics. Continuing advances in both analytic structure and algorithmic solution have far-reaching consequences for geometric analysis, optimal transportation, and applied mathematics.