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$J[y]=\int_{0}^{1}[(y')^{2}-(y')^{4}]dx$ ,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. enter image description here But I am missing out on something. Please help.

Thank you.

S.S
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    Could you expand how you have used the WE conditions? They will actually show the existence of broken extremals with the slope $\pm\frac{1}{\sqrt{2}}$, so the mistake is somewhere in your calculations/interpretations. – A.Γ. Jul 27 '20 at 11:21
  • @A.Γ. I got $F_{y'}=2y'-4y'^3$ and $F-y'F_{y'}=3y'^4-y'^2$, where $F$ is the given functional I tried solving them simultaneously for equality on the right and left hand limits at the corner point. Kindly point out where I am going wrong. Thanks. – S.S Jul 27 '20 at 12:05
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    It is a right approach. Solve the system $\omega_1-2\omega_1^3=\omega_2-2\omega_2^3$, $3\omega_1^4-\omega_1^2=3\omega_2^4-\omega_2^2$ for $\omega_1\ne\omega_2$. – A.Γ. Jul 27 '20 at 14:21
  • @A.Γ. I got it! It was a computational error. I got precisely two broken extremals now. So will it it be wise to rule out option 4 and go for option 3? – S.S Jul 27 '20 at 14:30
  • No. You've got two possible slopes at a corner. Now try to figure out how to make arbitrarily many corners if you know that between corners it is straight lines (from the E-L equation). Think like wwww. – A.Γ. Jul 27 '20 at 15:26
  • @A.Γ. But should I not confine myself to the interval [0,1] due to the given boundary conditions? – S.S Jul 27 '20 at 15:56
  • Yes, you should, however it is still possible. Think the shape of extremals like wwwwww – A.Γ. Jul 27 '20 at 16:30
  • @A.Γ. Is that possible within the interval [0,1] itself? If yes, then I am unable to perceive it. :( :( Please can you elaborate a bit on the figure? – S.S Jul 27 '20 at 16:36
  • Given the slope, one possibility is to go up to x=0.5 then down. Next possibility: go up to x=0.25 then down up to x=0.5, then up again and then down. Can you continue? – A.Γ. Jul 27 '20 at 17:04
  • Let us continue this discussion in chat. – S.S Jul 27 '20 at 17:32

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