4.1. Linear algebra of Legendre transform and Monge-Ampére equations [05CW]

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4.1. Linear algebra of Legendre transform and Monge-Ampére equations

The classical fact, implicitly used in [GW00], is that solving the two-dimensional Laplace equation is equivalent to solving the real two-dimensional Monge-Ampère equation. This can be easily generalized to higher dimensions. Namely, the chain rule and elementary linear algebra for a particular coordinate transformation yields the following.

Lemma 4.1.

If $K(s,t)$ is a local solution to the split Monge-Ampère equation (17), then $\Psi(y)$ is a (local) solution of the classical (real) Monge-Ampère equation

(19)

\det\frac{\partial^{2}\Psi}{\partial y_{i}\partial y_{p}}=1,\quad 1\leq i,j\leq n,

where

(20)

y_{i}=\frac{\partial K}{\partial s_{i}},\ 1\leq i\leq k,\qquad y_{p}=t_{p},\ k+1\leq p\leq n,

and $\Psi(y)$ is the partial Legendre transform of $K(s,t)$ , defined by

(21)

\frac{\partial\Psi}{\partial y_{i}}=s_{i},\ 1\leq i\leq k,\qquad\frac{\partial\Psi}{\partial y_{p}}=-\frac{\partial K}{\partial t_{p}},\ k+1\leq p\leq n.

Proof.

First, we check the very existence of the partial Legendre transform. Consider the following Jacobian and Hessian matrices:

(22)

\frac{\partial(y_{i},y_{p})}{\partial(s,t)}=\left(\begin{array}[]{cc}\frac{\partial^{2}K}{\partial s_{j}\partial s_{i}}&\frac{\partial^{2}K}{\partial t_{q}\partial s_{i}}\\ 0&\mathbbm{1}\end{array}\right)=\left(\begin{array}[]{cc}V&B\\ 0&\mathbbm{1}\end{array}\right),\quad\operatorname{Hess}K(s,t)=\left(\begin{array}[]{cc}V&B\\ {}^{t}B&-W\end{array}\right).

Then

(23)

\frac{\partial(s,t)}{\partial(y_{i},y_{p})}=\left(\begin{array}[]{cc}V^{-1}&-V^{-1}B\\ 0&\mathbbm{1}\end{array}\right),\quad\operatorname{Hess}\Psi(y)=\left(\begin{array}[]{cc}V^{-1}&-V^{-1}B\\ {}^{t}(-V^{-1}B)&W+{{}^{t}B}V^{-1}B\end{array}\right).

This shows that if both $V$ and $W$ are symmetric and positive definite, then $\operatorname{Hess}\Psi$ is also a symmetric positive definite matrix, so there exists (locally) a convex function $\Psi$ – the partial Legendre transform.

On the other hand the inverse of a non-degenerate $2\times 2$ block matrix with $\det A\neq 0$ and $\det D\neq 0$ is:

(24)

\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)^{-1}=\left(\begin{array}[]{cc}(A-BD^{-1}C)^{-1}&(-A+BD^{-1}C)^{-1}BD^{-1}\\ (-D+CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{array}\right),

which implies that

(25)

\det\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)=1\quad\Longleftrightarrow\quad\det A^{-1}=\det(D-CA^{-1}B).

Applied to $\operatorname{Hess}\Psi$ the last observation shows that $\Psi$ is a local Monge-Ampère solution if and only if $\det V=\det W$ . ∎