Generated Jacobian Equations, part 3/n

 

(26 Feb 2019)

The exponential map

Example 5 (the standard gradient map):

Example 7 (the exponential in Riemannian geometry)

The Jacobian equation

Example 8 (Far field reflector)

Example 9 (Near field reflector)

The (real) Monge-Ampère equation

The most basic Generated Jacobian Equation, and also one of the most important elliptic equations, is the real Monge-Ampère equation

\[\det(D^2u) = f(x)\]

where one only considers convex solutions $u$, and $f\geq 0$.

If we denote the $d$ eigenvalues of $D^2u(x)$ by $0\leq \lambda_1(x)\leq \ldots \leq \lambda_d(x)$ then the equation reads

\[\prod \limits_{i=1}^d \lambda_i(x) = f(x)\]

If we knew a priori that our solution satisfied the estimate $\inf \lambda_1(x) \geq c_0>0$, then

\[\lambda_d \leq c_0^{-(d-1)} f(x)\]

In which case $D^2u \in L^p$ if $f \in L^p$.

Since we are looking at $u$ which is convex, it can be represented by

\[u(x) = \max \limits_{y \in \overline{\Omega}} \{ x\cdot y - v(y) \}\]

for some function $v(y)$ such that

\[v(y) = \max \limits_{x \in \Omega} \{ x\cdot y - u(x) \}\]

Suppose, for moment, that for a given $x$ there is exactly one $y$ realizing the maximum, then this $y$ must be $y = \nabla u(x)$.

Add discussion of 1) continuity method, 2) Urbas notes 3) Caffarelli’s estimate.


Optimal transport and the $c$-Monge-Ampére equation

\[J(T) = \int_{X}c(x,T(x))f(x)\;dx\]

A theorem of Brenier and Gangbo-McCann says that

\[Dc(x,T(x)) = Du(x)\] \[T(x) = \text{exp}_x^{c}(Du(x))\] \[u(x) = \max \limits_{y \in \overline{\Omega}} \{c(x,y)-v(y) \}\]