Existence of minimizers to isoperimetric problem

[Tatianaoo]

Does anybody know where can I find theorem ensuring the existence of minimizers for isoperimetric problems? I also need the proof.

 

[chiro]

Does anybody know where can I find theorem ensuring the existence of minimizers for isoperimetric problems? I also need the proof.

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)?

 

[theorem4.5.9]

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.

 

[Tatianaoo]

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*}