# Correct setup of constrained variational calculus problem

I wish to solve the following maximization problem:

\begin{equation} \max_C \int_0^1 \int_0^1 F\left(\frac{\partial^2 C(u,v)}{\partial u \, \partial v} \right) \, du \, dv \end{equation}

such that $C$ satisfies the integral constraint

\begin{equation} \int_0^1 \int_0^1 G \left( \frac{\partial^2 C(u,v)}{\partial u \, \partial v} \right) \, du \, dv = \sigma \end{equation}

as well as the boundary constraints

\begin{align} C(0, v) &= C(u, 0) = 0\\ C(u, 1) &= u\\ C(1, v) &= v \end{align}

Ideally, I would like to set this up as a differential equation by the usual means of variational calculus. My question is about how to incorporate the integral constraint. If we let $A[C]$ denote the first functional, the unconstrained approach to construct a solution (Euler-Lagrange) would be to solve

\begin{equation} \frac{dA[C + \epsilon H]}{d \epsilon} = 0 \; \Bigg \vert \; \epsilon = 0 \end{equation}

for some suitable perturbation $H$. In light of the theory of Lagrange Multipliers and with $B[C]$ denoting the second functional (integral constraint) I would like to know if the following equation would be adequate to include the integral constraint:

\begin{equation} \frac{dA[C + \epsilon H]}{d \epsilon} = \lambda \frac{dB[C + \epsilon H]}{d \epsilon} \; \Bigg \vert \; \epsilon = 0 \end{equation}

Of course, any recommended resources on this problem setup are greatly appreciated!

• What is $\sigma$ here? Also, you might like to construct a $C$ that satisfies the BCs (such as $C = uv$) and apply it to your question in bold to see what happens. – Mattos Sep 26 '16 at 15:16
• $\sigma$ is a constant of which I know that it admits a solution. Thanks for the hint. I'll give it a try. – R.G. Sep 26 '16 at 15:20