$J[y]=\int_{0}^{1}[(y')^{2}-(y')^{4}]dx$ ,subject to boundary conditions $y(0)=0,y(1)=0.$
A broken extremal is a continuous extremal whose derivative has jump discontinuities at a finite number of points. Then which of the following is /are true?
- 1).There are no broken extremals and $y=0$ is an extremal.
- 2).There is a unique broken extremal.
- 3).There exist more then one and finitely many broken extremals.
- 4).There exist infinitely many broken extremals.
Here the smooth extremal satisfying the boundary conditions is clearly $y=0$, but how can I investigate the broken extremals here?
I tried using the Weierstrass-Erdmann Corner conditions and it seems there are no broken extremals at all. So I want to go with the first option. But the answer key says option 4.
EDIT
Utlilizing the vital comments below, I actually got broken extremals. But I am still stuck with the number of extremals that should exist here. Here's a rough sketch of my understanding.
But I am missing out on something.
Please help.
Thank you.