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\title{HW 4 for Calculus of Variations 2012}
\author{Andrej Cherkaev}

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\maketitle

1. Compute convex envelopes of the functions
$$
f_1(u)= \min\left\{ (u-1)^2, (u+1)^2 \right\} 
$$

$$
f_2(u)= - \frac{1}{1+ u^2}
$$

$$
f_3(u_1, u_2)= \left\{ \begin{array}{ll}
 0 & \mbox{ if } u_1^2 + u_2^2=0 \\
1+ u_1^2 + u_2^2 & ~~~\mbox{ otherwise }
\end{array}
\right.
$$

2. Consider variational problems 
$$
\inf \int_0^1 (u^2 + f_i(u'))dx, ~~ u(0)=a, i=1, 2
$$
Find a relaxed formulation, and write the equations for the minimizers. 


3. Solve the equation for eikonal $S=S(x_1, x_2)$
$$ (\nabla S)^2=1, ~~|x_1|+ |x_2|>1, ~~ S=0 ~~\mbox{if } |x_1|+ |x_2|=1
$$
(Guess the solution, assuming that $|x_1|+ |x_2|=1$ is the radiating curve, than check that the solution is correct. )
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