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\title{HW 8. Multiple integrals}
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1. Find Euler-Lagrange equations and the natural boundary conditions: 
$$
\min_{u} \int_\Omega \left( (\nabla u_1)^2 + (\nabla u_2)^2 +  a \det(\nabla u_1| \nabla u_2 )- u_1u_2 \right)dx.
$$
Does the solution depend on $a$?

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2. Find the Lagrangian if the Euler-Lagrange equation is
$$
\nabla \cdot ( \sigma(x) \nabla  u ) + \phi(x) =0 ~~ \mbox{in } \Omega, \quad   \frac{\partial u}{\partial n} + \gamma u=0 \mbox{on }\partial  \Omega .
$$

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3. Consider the problem

$$
\min_{u} \int_\Omega \left( a\,u^2+ b (\nabla \times u)^2\right)dx.
$$
1.Derive the Euler-Lagrange equations and natural boundary conditions. 
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2. Rewrite the problem as 
$$
\min_{u} \int_\Omega \left( a\,u^2 + b \,v^2\right)dx \mbox{ subject to } v=\nabla \times u,
$$
use Lagrange multiplier to account to the last constraint, derive the Euler-Lagrange equations and the Euler-Lagrange equations for a dual problem (exclude $u$).

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4. Derive conditions for an optimal boundary component for a problem
$$
\min_{\Omega} \min_{u} \int_\Omega\left( (\nabla u)^2 + \gamma \oint_{\partial \Omega} u^2 \right)ds \quad \|\Omega \|= R
$$

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