Monge Ampere and Calculus I am learning about mass transportation theory and the Monge-Ampere equation, to transport a function $f$ toward $g$ by a change of variable $T$. In particular, in order to solve for :
$$ \min \int c(x, T(x)) f(x) dx $$
under the constraint $ f = g\circ T~(\det(DT))$ where $\det(DT)$ is the jacobian of T (here, the determinant is supposed to be positive to remove absolute values), and $c(x,y)$ is a cost function, I found in a lecture that one introduces Lagrange multipliers $\lambda = \lambda(x)$ and solve for the extended functional:
$$ \min \int \left[ c(., T) f + \lambda~ g\circ T (\det DT) \right] $$
By computing the Euler Lagrange equation of the above equation, one get :
$$ f~c_{yi} = D_i[\lambda g~(cof~DT)^{i,j}] -\lambda~(\det DT) g_{yi} $$
where $cof DT$ stands for the matrix of cofactors for the jacobian matrix $DT$.
I have a few basic questions, that I'd like to be answered assuming very little knowledge on my side :
- Why would the lagrange multipliers depend on $x$ ? Usually, when I have a set of equations to minimize with a set of equations as constraints, I have one $\lambda$ per constraint, and it doesn't depend on $x$
- I don't understand at all how one arrives to this Euler Lagrange equation. I mean, I know that in general, deriving a determinant with respect to a matrix gives a cofactor matrix... but nothing more that I can use here. Could someone add 5-6 steps in between ? what are those $y_i$ and $i,j$ indices ?? Please, treat me as a newbie :)
Thank you very much in advance !
 A: Could you please check your equations to make sure indices weren't dropped?  I think there is a missing index $y$ in the first term on the right.  I assume you meant $g$ to be $g_{yj}$.
But, for now, recognize that the Euler-Lagrange equation comes from varying with respect to $T$ the action you've written.  So, do you see how to the the left hand side by varying the first term in the action?  If so, are you familiar with varying the determinant of a matrix with respect to that matrix?  Check out this answer from Physics.SE and see if that clarifies things.  
https://physics.stackexchange.com/questions/3873/how-do-i-calculate-the-perturbations-to-the-metric-determinant
See if that clarifies.  If not, let me know where you're stuck.
Rewrite your equation as
$$
(f~c_{yi} + \lambda~(\det DT) g_{yi}) - D_i[\lambda g_{yj}~(cof~DT)^{i,j}] = 0
$$
and recognize that
$$
\frac{\partial L}{\partial T} = (f~c_{yi} + \lambda~(\det DT) g_{yi})\,,
$$
and
$$
\frac{\partial L}{\partial D_iT} = \lambda g_{yj}~(cof~DT)^{i,j}\,,
$$
where
$$
L = c(., T) f + \lambda~ g\circ T (\det DT)\,.
$$
I assume the floating indices refer to the location and direction (?) of the variation.
