Fully Nonlinear Equations and Monge–Ampère

A fully nonlinear equation is nonlinear in the highest derivatives:

The most important example is the Monge–Ampère equation $\det D^2u = f$, the equation of prescribed Gauss curvature, of optimal transport and, in its complex form, of Kähler–Einstein metrics. Others come from stochastic control (the Bellman equation $\inf_\alpha L_\alpha u = 0$ of 6D.8 Stochastic Control and the HJB Equation, an infimum of linear operators), from geometry (the equations of special Lagrangian submanifolds and of prescribed curvature functions), and from game theory.

None of the earlier methods apply directly. There is no divergence form, so energy methods and De Giorgi–Nash–Moser are out. There is no variational structure in general. And the linearised operator at a solution depends on $D^2u$ itself, so even the Schauder theory needs a bound on second derivatives before it can start. The theory that overcame this, built between 1979 and 1983 by Nicolai Krylov, Mikhail Safonov and Lawrence Evans, rests on two pillars: a Hölder estimate for non-divergence equations with merely measurable coefficients (Krylov–Safonov, the non-divergence analogue of De Giorgi–Nash–Moser, proved with the Alexandrov–Bakelman–Pucci estimate of 6B.8 Lᵖ Theory and Strong Solutions), and the observation that when $F$ is concave, second derivatives of solutions are subsolutions of a linear equation (Evans–Krylov). This chapter explains both, then turns to Monge–Ampère: its geometry, its weak solutions, optimal transport, and the complex Monge–Ampère equation behind the Calabi conjecture and the Kähler–Ricci flow.

By the end of this chapter you will be able to:

• define uniform ellipticity and concavity for $F(D^2u) = 0$, and check them for the Pucci, Bellman and Monge–Ampère operators;
• prove that first derivatives of solutions solve linear non-divergence equations, and that for concave $F$ second derivatives are subsolutions, and explain how this leads to the Krylov–Safonov and Evans–Krylov theorems;
• prove the comparison principle for the Monge–Ampère equation on convex functions, compute radial solutions and the Monge–Ampère measure of a cone;
• derive the Monge–Ampère equation of optimal transport from Brenier's theorem, and compute optimal maps in one dimension;
• reduce the Calabi conjecture and the Kähler–Ricci flow to scalar complex Monge–Ampère equations.

10.1Moving earth

10.2Ellipticity and concavity

Consider first $F(D^2u) = 0$, with $F$ a function on the space $\mathcal S^n$ of symmetric $n\times n$ matrices. Write $N \geq 0$ for positive semidefinite $N$ and $\|N\|$ for the operator norm.

For linear $F(M) = \operatorname{tr}(AM)$ with $\lambda I \leq A \leq \Lambda I$ this is the usual uniform ellipticity, with $\Lambda$ replaced by $n\Lambda$. If $F$ is $C^1$, uniform ellipticity says the derivative $F_{ij}(M) = \frac{\partial F}{\partial M_{ij}}$ satisfies $\lambda I \leq (F_{ij}) \leq \Lambda I$ (up to a factor $n$ in the constants, since $\|N\| \leq \operatorname{tr}N \leq n\|N\|$ for $N \geq 0$), and concavity says the second derivative is non-positive: $\sum F_{ij,kl}(M)N_{ij}N_{kl} \leq 0$.

Examples. (1) The Pucci operators $\mathcal M^+(M) = \Lambda\sum_{e_i > 0}e_i + \lambda\sum_{e_i < 0}e_i$ and $\mathcal M^-(M) = \lambda\sum_{e_i > 0}e_i + \Lambda\sum_{e_i < 0}e_i$, with $e_i$ the eigenvalues of $M$, are the extremal uniformly elliptic operators: $\mathcal M^-(M) = \inf\{\operatorname{tr}(AM) : \lambda I \leq A \leq \Lambda I\}$ is concave, $\mathcal M^+$ is convex, and every uniformly elliptic $F$ satisfies $\mathcal M^-(N) \leq F(M + N) - F(M) \leq \mathcal M^+(N)$ for all $N$, with suitably adjusted constants (Caffarelli and Cabré, chapter 2). (2) The Bellman operators $F(M) = \inf_\alpha\operatorname{tr}(A_\alpha M)$ over a family with $\lambda I \leq A_\alpha \leq \Lambda I$ are concave and uniformly elliptic: the HJB equations of stochastic control (6D.8 Stochastic Control and the HJB Equation). (3) The Monge–Ampère operator $F(M) = \log\det M$ is defined on positive definite matrices, elliptic there with $F_{ij} = (M^{-1})_{ij}$, and concave (Exercise 10.11). It is not uniformly elliptic: the derivative degenerates as eigenvalues of $M$ grow or shrink, and controlling them is the whole difficulty.

The proposition shows both the opportunity and the problem. The derivatives of a solution satisfy linear equations, so linear theory can be applied to them, but with coefficients $a^{ij}(x) = F_{ij}(D^2u(x))$ about which nothing is known beyond ellipticity: they are bounded and measurable, no more. And the equation is in non-divergence form, so De Giorgi–Nash–Moser does not apply. What is needed is a regularity theory for non-divergence equations with rough coefficients.

10.3Krylov–Safonov and Evans–Krylov

Nicolai Krylov and Mikhail Safonov proved this in 1979–80. The heart of the proof is a measure estimate: if a non-negative supersolution is at least $1$ on a definite fraction of a ball, it is bounded below by a positive constant on a smaller ball. That estimate comes from the Alexandrov–Bakelman–Pucci inequality of 6B.8 Lᵖ Theory and Strong Solutions, applied to a cleverly chosen auxiliary function, and a covering argument (the "growing ink-spots" lemma) propagates it across scales, much as the De Giorgi iteration does in the divergence case (6B.9 De Giorgi–Nash–Moser). Luis Caffarelli showed in 1989 that the whole theory, and a perturbative $C^{2,\alpha}$ theory, works for viscosity solutions of fully nonlinear equations.

Applied to $w = u_e$ (by Proposition 10.2), Krylov–Safonov gives a $C^{1,\alpha}$ estimate for solutions of every uniformly elliptic $F(D^2u) = 0$: $\|u\|_{C^{1,\alpha}(B_{1/2})} \leq C\|u\|_{L^\infty(B_1)}$. To go further, to $C^{2,\alpha}$, where Schauder theory takes over, concavity is needed.

Lawrence Evans (1982) and Krylov (1982–83) found this independently. The idea: by Proposition 10.2, every pure second derivative $u_{ee}$ is a subsolution, so it cannot dip much below its supremum on large sets; on the other hand, since $F(D^2u) = 0$ and $F$ is uniformly elliptic, the second derivatives are tied together, and a sum of them is a supersolution. Combining a weak Harnack inequality for the supersolutions with the subsolution property for each $u_{ee}$, over finitely many directions $e$ chosen to span the space of matrices, the oscillation of $D^2u$ decreases by a fixed factor from one ball to the half ball, which is Hölder continuity.

Concavity is necessary. Nikolai Nadirashvili and Serge Vlăduţ constructed in 2007 uniformly elliptic (but neither concave nor convex) equations $F(D^2u) = 0$ in dimension 12 with viscosity solutions that are not $C^2$, and later examples went down to dimension 5. In dimension 2, every uniformly elliptic $F(D^2u) = 0$ has $C^{2,\alpha}$ solutions (Nirenberg, 1953), because in two dimensions solutions of linear non-divergence equations with measurable coefficients are already $C^{1,\alpha}$ (a theorem going back to Bernstein and Morrey), and the first derivatives $u_e$ solve such equations.

The continuity method again. With the Evans–Krylov estimate, the Dirichlet problem for concave uniformly elliptic equations is solved exactly as in 6B.7 Schauder Theory and the Continuity Method and 6C.6 Quasilinear Elliptic Equations and Minimal Surfaces: embed in a family, prove a priori bounds in $C^{2,\alpha}$ (here Evans–Krylov in the interior plus boundary estimates of Krylov), and run the continuity method. The pattern is always the same: existence reduces to a priori estimates, and the estimates reduce to the regularity theory of the linearised equation.

10.4The real Monge–Ampère equation

with $f > 0$. Three facts make it special.

It is elliptic only on convex functions. The linearisation at $u$ is $v \mapsto \operatorname{tr}\big(\operatorname{cof}(D^2u)D^2v\big) = \det(D^2u)\operatorname{tr}\big((D^2u)^{-1}D^2v\big)$, where $\operatorname{cof}$ is the cofactor matrix. This is elliptic exactly when $D^2u$ is positive definite. Without convexity the equation can change type: in two dimensions $u = x_1x_2$ solves $\det D^2u = -1$, a hyperbolic equation, and in even dimensions the concave function $-u$ solves the same equation as $u$, with negative definite linearisation. So one always imposes convexity.

It is geometric. The Gauss curvature of the graph of $u$ is $K = \frac{\det D^2u}{(1 + |\nabla u|^2)^{(n+2)/2}}$, so prescribing the Gauss curvature of a convex hypersurface is a Monge–Ampère equation (the Minkowski and Weyl problems). And $\det D^2u$ is the Jacobian of the gradient map $\nabla u$: for a smooth convex $u$ and a Borel set $E$,

(the area formula, 6B.8 Lᵖ Theory and Strong Solutions; $\nabla u$ is injective when $u$ is strictly convex).

It is invariant under affine maps. If $A$ is linear with $\det A = 1$, then $u(Ax)$ solves the same equation as $u$ (with $f(Ax)$). So solutions can be stretched enormously in one direction and compressed in another without changing the equation: the estimates cannot be uniform unless something controls the shape of level sets. This is why Caffarelli's regularity theory is built on sections $\{u < \ell + h\}$ ($\ell$ affine), normalised by John's ellipsoid lemma.

In particular, the Dirichlet problem for (MA) has at most one convex classical solution. For existence, the analogue of the four estimates of 6C.6 Quasilinear Elliptic Equations and Minimal Surfaces works, with convexity providing the $C^0$ and $C^1$ bounds and the hard steps being the second-derivative bounds up to the boundary; the theorem is due to Luis Caffarelli, Louis Nirenberg and Joel Spruck (1984), and to Krylov.

Weak solutions. Formula (1) suggests a definition that needs no derivatives. For a convex $u$, the subdifferential at $x$ is $\partial u(x) = \{p : u(y) \geq u(x) + p\cdot(y - x) \text{ for all } y\}$, the set of slopes of supporting planes, and the Monge–Ampère measure is $\mu_u(E) = |\partial u(E)|$. Aleksandr Alexandrov showed that this is a Borel measure, and $u$ is an Alexandrov solution of $\det D^2u = f$ if $\mu_u = f\,dx$. The cone $u = |x|$ has $\mu_u = |B_1|\delta_0$ (Exercise 10.14): all its curvature is at the vertex. Alexandrov solutions exist and are unique for the Dirichlet problem under mild conditions, and they are stable under uniform convergence, playing the role that viscosity solutions play for general fully nonlinear equations.

Regularity is subtle. Aleksei Pogorelov found Alexandrov solutions of $\det D^2u = f$ with $f$ smooth and positive that are not $C^2$, in dimensions $n \geq 3$: functions of the form $u(x', x_n) = |x'|^{2 - 2/n}h(x_n)$ that contain a line segment in their graph. Caffarelli proved in 1990 that this is the only obstruction: a strictly convex Alexandrov solution with $0 < c \leq f \leq C$ is $C^{1,\alpha}$, and if $f \in C^\alpha$ it is $C^{2,\alpha}$. In dimension two, Alexandrov solutions with positive bounded $f$ are automatically strictly convex.

10.5Optimal transport

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with finite second moments. A map $T$ pushes $\mu$ forward to $\nu$, $T_\#\mu = \nu$, if $\mu(T^{-1}(B)) = \nu(B)$ for all Borel $B$; equivalently $\int\varphi(T(x))\,d\mu(x) = \int\varphi(y)\,d\nu(y)$ for bounded continuous $\varphi$. Monge's problem with quadratic cost is to minimise $\int|x - T(x)|^2\,d\mu(x)$ over such maps.

The proof (Villani, chapter 2) goes through Kantorovich's relaxation and duality. The reason gradients of convex functions appear is cyclical monotonicity: an optimal plan cannot be improved by permuting destinations among finitely many points, and in the quadratic case this says the support of the plan lies in the subdifferential of a convex function (a theorem of Rockafellar). In one dimension, this is the familiar fact that the optimal way to match two sorted lists is in order: the optimal map is the monotone rearrangement $T = G^{-1}\circ F$, where $F$ and $G$ are the distribution functions (Exercise 10.15).

This is a Monge–Ampère equation with a second boundary condition: instead of prescribing $u$ on $\partial U$, one prescribes that $\nabla u$ maps the support of $\mu$ onto that of $\nu$. Caffarelli proved in 1992 that if the target is convex and the densities are bounded above and below, the Brenier map is $C^{0,\alpha}$, and smooth if the densities are; without convexity of the target, the map can be discontinuous.

10.6The complex Monge–Ampère equation and the Calabi conjecture

On a compact Kähler manifold $(M, \omega)$ of complex dimension $n$, with $\omega = i\,g_{j\bar k}\,dz^j\wedge d\bar z^k$, the Ricci form is locally

and it represents $2\pi c_1(M)$, the first Chern class, in cohomology. Eugenio Calabi conjectured in 1954–57 that every closed real $(1, 1)$-form $\rho$ representing $2\pi c_1(M)$ is the Ricci form of a unique Kähler metric in the cohomology class of $\omega$. Every Kähler metric in that class has the form $\omega_\varphi = \omega + i\partial\bar\partial\varphi$ for a function $\varphi$ (the $\partial\bar\partial$-lemma), and $\rho = \operatorname{Ric}(\omega) - i\partial\bar\partial F$ for some function $F$, normalised by $\int_Me^F\omega^n = \int_M\omega^n$. Since $\operatorname{Ric}(\omega_\varphi) - \operatorname{Ric}(\omega) = -i\partial\bar\partial\log\frac{\omega_\varphi^n}{\omega^n}$, the conjecture becomes the complex Monge–Ampère equation

which in local coordinates reads $\det(g_{j\bar k} + \varphi_{j\bar k}) = e^F\det(g_{j\bar k})$.

Shing-Tung Yau proved this in 1976 (announced 1977, published 1978). The proof is the continuity method of 6B.7 Schauder Theory and the Continuity Method applied to $(\omega + i\partial\bar\partial\varphi)^n = e^{tF + c_t}\omega^n$, $t \in [0, 1]$, and openness is the invertibility of the linearised operator, the Laplacian of $\omega_\varphi$. Closedness is a sequence of a priori estimates, which is where the work lies: a $C^0$ bound on $\varphi$ (Yau's Moser iteration, the hardest step), a bound on the Laplacian of $\varphi$ from a maximum principle computation in the spirit of 6C.7 Harmonic Maps and Their Heat Flow (Aubin and Yau), a third-order bound (Calabi's estimate, or Evans–Krylov, since $\log\det$ is concave), and Schauder bootstrapping. Thierry Aubin and Yau independently solved the case $c_1(M) < 0$, where $\operatorname{Ric}(\omega_\varphi) = -\omega_\varphi$ leads to $(\omega + i\partial\bar\partial\varphi)^n = e^{F + \varphi}\omega^n$, whose extra $e^\varphi$ gives the $C^0$ bound by the maximum principle. Ricci-flat Kähler manifolds with $c_1 = 0$ are the Calabi–Yau manifolds of string theory, where six of the ten dimensions of spacetime are curled up into one. Yau received the Fields Medal in 1982.

10.7The Kähler–Ricci flow

On a Kähler manifold, the Ricci flow $\partial_tg = -\operatorname{Ric}(g)$ (with this normalisation, Hamilton's flow at half speed) preserves the Kähler condition, and it becomes $\partial_t\omega = -\operatorname{Ric}(\omega)$. Because the Ricci form is $-i\partial\bar\partial$ of a function in the Kähler setting, the whole flow reduces to a scalar parabolic equation for a potential.

The equation is parabolic because $\log\det$ is elliptic, and concave, so the Evans–Krylov theory applies. Huai-Dong Cao proved in 1985 that on compact Kähler manifolds with $c_1(M) = 0$ or $c_1(M) < 0$ (with the appropriate normalisation) the flow exists for all time and converges to the Kähler–Einstein metric, giving a parabolic proof of the theorems of Yau and Aubin. The Fano case ($c_1(M) > 0$) is much harder, because Kähler–Einstein metrics need not exist there. Perelman proved unpublished estimates for the Kähler–Ricci flow on Fano manifolds (a uniform bound on scalar curvature and diameter, written up by Natasa Sesum and Gang Tian in 2008) using his $\mathcal W$-entropy (12A.3 The 𝓦-Entropy); the eventual solution of the Yau–Tian–Donaldson conjecture (Chen, Donaldson and Sun, 2015) characterised existence by algebraic stability. The analytic minimal model program of Jian Song and Gang Tian studies the Kähler–Ricci flow on general projective varieties, where it performs algebraic surgeries (contractions and flips) at singular times, the complex analogue of Perelman's surgery.

10.8History

Monge posed his transport problem in 1781, and André-Marie Ampère studied the equation that bears both names around 1820. Hermann Minkowski posed the problem of convex bodies with prescribed curvature around 1900. Aleksandr Alexandrov developed the weak theory in the 1940s and 1950s, and Pogorelov and, independently, Shiu-Yuen Cheng and Yau solved the Minkowski problem in the 1970s. Calabi's conjecture dates from 1954–57 and Yau's solution from 1976–78. Krylov and Safonov proved their Harnack inequality in 1979–80; Evans and Krylov the $C^{2,\alpha}$ estimate in 1982–83; Caffarelli, Nirenberg and Spruck the Dirichlet problem in 1984; Caffarelli the regularity theory for viscosity solutions in 1989 and for Monge–Ampère in 1990–92. Kantorovich's relaxation dates from 1942 and Brenier's theorem from 1987–91.

10.9Exercises

Solution

(a) $\log\det(M + sN) = \log\det M + \log\det(I + sM^{-1}N)$, and $\frac{d}{ds}\log\det(I + sA)|_{s=0} = \operatorname{tr}A$. With $S = M^{-1/2}NM^{-1/2} \geq 0$, $\operatorname{tr}(M^{-1}N) = \operatorname{tr}S > 0$ unless $S = 0$. (b) $\log\det(I + sS) = \sum\log(1 + s\sigma_i)$ over the eigenvalues $\sigma_i$ of $S$, whose second derivative at $0$ is $-\sum\sigma_i^2 = -\operatorname{tr}S^2 = -\operatorname{tr}\big((M^{-1}N)^2\big) \leq 0$.

Solution

(a) Write $M = Q\operatorname{diag}(e_i)Q^T$. For admissible $A$, $\operatorname{tr}(AM) = \sum_ie_i(Q^TAQ)_{ii}$, and $\lambda \leq (Q^TAQ)_{ii} \leq \Lambda$. So $\operatorname{tr}(AM) \geq \sum_{e_i > 0}\lambda e_i + \sum_{e_i < 0}\Lambda e_i = \mathcal M^-(M)$, with equality for $A = Q\operatorname{diag}(a_i)Q^T$, $a_i = \lambda$ if $e_i > 0$ and $\Lambda$ otherwise. (b) An infimum of linear functions of $M$ is concave.

Solution

$D^2u = U''\frac{x\otimes x}{r^2} + \frac{U'}{r}\big(I - \frac{x\otimes x}{r^2}\big)$, with eigenvalue $U''$ in the radial direction and $\frac{U'}{r}$ (multiplicity $n - 1$) in tangential directions. So $\det D^2u = U''(U'/r)^{n-1}$, i.e. $\frac1n\big((U')^n\big)' = r^{n-1}f(r)$. Integrating from $0$: $(U')^n = n\int_0^rs^{n-1}f$. For $f = 1$, $(U')^n = r^n$, $U' = r$, $U = \frac{r^2}{2}$.

Solution

At $0$: $p \in \partial u(0)$ iff $|y| \geq p\cdot y$ for all $y$, iff $|p| \leq 1$. At $x \neq 0$, $u$ is differentiable with gradient $x/|x|$, and for a convex function the subdifferential at a point of differentiability is the gradient alone. So $\partial u(E)$ is $\overline{B_1}$ (plus a null set of unit vectors) if $0 \in E$, and a subset of the unit sphere (measure zero) otherwise.

Solution

(a) $T$ must be increasing and push $N(0, 1)$ to $N(m, \sigma^2)$; $x \mapsto m + \sigma x$ does both, and the monotone map is unique. $W_2^2 = \mathbb{E}\big[(X - m - \sigma X)^2\big] = \mathbb{E}\big[(m + (\sigma - 1)X)^2\big] = m^2 + (\sigma - 1)^2$. (b) $u' = m + \sigma x$, $u'' = \sigma > 0$. With $f(x) = \frac{e^{-x^2/2}}{\sqrt{2\pi}}$ and $g(y) = \frac{e^{-(y - m)^2/(2\sigma^2)}}{\sigma\sqrt{2\pi}}$: $g(m + \sigma x)\sigma = \frac{e^{-x^2/2}}{\sqrt{2\pi}} = f(x)$.

Solution

(a) $\omega = \Omega + i\partial\bar\partial\varphi = \big(1 + \frac12\Delta\varphi\big)\frac i2dz\wedge d\bar z$, so $e^v = 1 + \frac12\Delta\varphi$. The equation $\varphi_t = \log\frac{\omega}{\Omega} = v$. Differentiating $e^v = 1 + \frac12\Delta\varphi$ in $t$: $e^vv_t = \frac12\Delta\varphi_t = \frac12\Delta v$. (b) $\partial_tg = e^vv_t(dx^2 + dy^2) = \frac12\Delta v\,(dx^2 + dy^2) = -Ke^v(dx^2 + dy^2) = -Kg$, and on a surface $\operatorname{Ric} = Kg$.