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Variational methods - minimization problem proof

  1. Mar 15, 2008 #1
    Consider the minimization problem
    Inf (u E D) F(u)
    where F(u) = 1/2 integ(0->T) |u (with circle on top)|^2 dt + 1/2 integ(0->T) |u|^2 dt + 1/2 integ(0->T) f(t)u(t) dt, f E L^2 [0,T], and H = {u:[0,T]->R, uEL^2[0,T], u(circle on top) E L^2 [0,T]} is a Hilbert space equipped with the norm
    ||u|| = (integ(0->T) |u(circle on top)|^2 dt + integ(0->T) |u|^2 dt)^1/2 and D={uEH; u(0)=u(T)}
    Prove that there exist u(bar)ED such that Inf (u E D) F(u) = F(u(bar))
    Then show that u(bar) is unique.

    I need some serious help on this question. I can't even decipher the meaning of all that on there and I am not good at proofs at all.
  2. jcsd
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