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Existence of minimizers to isoperimetric problem

  1. Jul 2, 2012 #1
    Does anybody know where can I find theorem ensuring the existence of minimizers for isoperimetric problems? I also need the proof.
     
  2. jcsd
  3. Jul 2, 2012 #2

    chiro

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    Science Advisor

    Hey Tatianaoo and welcome to the forums.

    For the benefit of the other members, can you give a description of the problem (or if its on a wiki page, point to the specific definition)?
     
  4. Jul 3, 2012 #3
    Yes, a description of the problem would be very helpful, as the isoperimetric problem has a number of settings and generalizations.

    Frank Morgan's Introduction to Geometric Measure Theory is a good start, specifically chapter 5 which gives an outline of the compactness theorem. If you're unaware of the compactness theorem, this book probably isn't what you're looking for.
     
  5. Jul 9, 2012 #4
    Thank you very much for your response. I was thinking about the following problem: we look for the minimizer of the following variational functional
    \begin{equation*}
    \mathcal{J}= \int_a^b F(u,\dot{u},t) dt ,
    \end{equation*}
    subject to the boundary conditions
    \begin{equation*}
    u(a)=u_a, u(b)=u_b
    \end{equation*}
    and an isoperimetric constraint
    \begin{equation*}
    \mathcal{I}= \int_a^b G(u,\dot{u},t) dt=\xi.
    \end{equation*}
     
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