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\title{M}
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\section{HW1 }

\begin{enumerate}
 \item Consider 
$$
J_1=\min_{u(x): u(0)=1} \int_0^1 \left(a (u')^2 + 2 b \,u\, u' + c \,u^2 \right) dx, ~~ a\geq 0, ~c \geq 0 
$$
Find Euler equation and natual boundary condition at $x=1$. Solve. Discuss dependence on $b$. What happens when $a=c=0$? 

 \item Approximate $y=\mbox{sign } x, ~ x\in [-1, 1] $ by a function $u(x)$ using quadratic penalty, that is minimize 
$$
J_2=\min_{u(x)} \frac{1}{2} \int_{-1}^1 \left[ (u-y)^2 + \varepsilon^2 (u')^2  \right] dx. 
$$
Here
$$ \mbox{sign } x= \left\{\begin{array}{lll} 
-1   & \mbox{if } & x<0 \\
 1   & \mbox{if } & x>0 \\
 0   & \mbox{if } & x=0 \end{array}
 \right.
$$
 \item Approximate $y=\sin x, ~ x\in [0, 2 \pi] $ by a function $u(x)$ using TV penalty, that is minimize
$$J_3= \min_{u(x)}  \int_0^{2 \pi} \left[\frac{1}{2} (y-u)^2 + \varepsilon | y'| \right] dx
$$
\end{enumerate}

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