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\noindent{\bf Math 273b}: Calculus of Variations 

\noindent{\bf Homework \#2, due on Monday, May 2nd } (late homework accepted)

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\noindent{\bf [1]} Assume that $V$ and $V^*$ are normed vector spaces in duality. Let 
$F:V\rightarrow \overline{\rit}$ and let $F^*:V^*\rightarrow \overline{\rit}$ be the polar or conjugate of $F$. We define $F^*$ by 
$$F^*(u^*)=\sup_{u\in V}\Big\{\langle u^*,u\rangle-F(u)\Big\}.$$

Show 

(i) $F^*(0)=-inf _{u\in V} F(u)$. 

(ii) $(\lambda F)^*(u^*)=\lambda F^*(\frac{1}{\lambda}u^*)$ for every $\lambda >0$. 

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\noindent{\bf [2]} Let $V=V^*=\rit^n$. Let $Q$ be a symmetric positive definite $n\times n$ matrix, $b\in\rit^n$, and consider $f(x):=\frac{1}{2}\langle x,Qx\rangle+\langle b,x\rangle$, for all $x\in\rit^n$. Find the polar (or the conjugate) $f^*$ and deduce that, in particular, $\frac{1}{2}\|\cdot\|^2$ is its own polar (or conjugate). 

\medbreak 

\noindent{\bf [3]} Let $F:V\rightarrow \rit$ and $F^*$ its polar. Then $u^*\in \partial F(u)$ if and only if $F(u)+F^*(u^*)=\langle u^*,u\rangle$. 

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\noindent{\bf [4]} Show that the polar $F^*$ is convex. 

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\noindent{\bf [5]} Let $f\in \rit^{N^2}$ be given, and let $u\in\rit^{N^2}$ be an unknown minimizer of the functional
$$E(w)=\sum_{i,j=0}^{N-1}|\nabla w_{i,j}|^2+\lambda \sum_{i,j=0}^{N-1}(w_{i,j}-f_{i,j})^2,$$
for $w\in \rit^{N^2}$, where 
$$\nabla w_{i,j}=\left(
\begin{array}{cc}
(D_xw)_{i,j}\\
(D_yw)_{i,j}
\end{array}
\right)=
\left(
\begin{array}{cc}
w_{i+1,j}-w_{i,j}\\
w_{i,j+1}-w_{i,j}
\end{array}
\right),
$$
for $(i,j)\in \{0,...,N-1\}^2$ (we assume that all vectors $f$, $w$ are periodic outside of their support). 

(a) Find the adjoint operators $D_x^*$ and $D^*_y$ of $D_x$ and $D_y$. 

(b) Find a linear operator $B:\rit^{N^2}\rightarrow\rit^{N^2}$, a $c\in \rit^{N^2}$, and $C(f)$, independent of $w$, such that for all $w\in \rit^{N^2}$, 
$$E(w)=\langle Bw,w\rangle+\langle c,w\rangle+C(f).$$ 

(c) Show that $B$ is self-adjoint. 

(d) Find the Gateaux differential of $E(w)$ in the direction $v$ and thus give a necessary (and sufficient) condition for $w$ to be a minimizer, by setting this differential to zero (as the zero functional). 

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