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

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}[u]= \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}[u]= \int_a^b G(u,\dot{u},t) dt=\xi.
\end{equation*}