Originally Posted by

**WilliamH** hey fobos, thanks for the reply, much appreciated!

So I don't have any initial conditions. However I know that this crossover will happen exaclty once, that is there exists $\displaystyle y_1$ such that $\displaystyle \forall y\leq y_1 $$\displaystyle -\phi(y_1)\geq y $ and $\displaystyle \forall y\geq y_1$ $\displaystyle -\phi(y_1)\leq y$, with equality only at $\displaystyle y_1$

so just to check whether I am understanding correctly what you're saying:

should I split the integral up in two parts (where the breaking point is the kink in x, say $\displaystyle y_1$) and then apply the Euler-Lagrange equation for both integrals, using, in both cases, the constrait $\displaystyle \phi(y_1)=-y_1$?

or am I misunderstanding you?

EDIT: and then maximize wrt $\displaystyle y_1$