Functional Analysis – Lagrange Multipliers for Function Spaces

Question

For some constant $A > 1$ I am trying to solve the constrained minimization problem
minimize $F(u)$ in $C$
subject to $H(u) = 0$.

Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx – A$, where our integral is from 0 to 1.

I have two main questions:

1. What is the geometric interpretation of this problem?

2. If I assume that $u$ is a smooth minimizer I am to use the Lagrange Multiplier
   Theorem to compute the Euler–Lagrange equations. Derive a differential equation for $u$∗.

I have heard that this is called the isoparametric problem. Is it fair to say that the geometric interpretation here is that of minimizing area given a certain perimeter? Is using the Lagrange Multiplier Theorem similar to using Euler-Lagrange to minimize an unconstrained functional? Thank you!

Answer 1

Have a look at my answer to a different question. The same procedure should work for your problem:

1. Set up the Lagrangian with multiplier $\lambda$: $$L(u, \lambda) = F(u) + \lambda \, H(u)$$
2. Solve the following system for $u$ and $\lambda$: \begin{align*} \partial_u L(u, \lambda) &= 0 \\ \partial_\lambda L(u, \lambda) = H(u) &= 0\end{align*} Typically, this can be done by using the Euler-Lagrange-equations for the first equation. This gives you $u$ in dependence of $\lambda$. Then, choose $\lambda$, such that the corresponding $u$ solves the second equation.