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

\noindent{\bf Homework \#2}

\bigbreak 

\noindent{\bf [1]} Consider the 1D length functional minimization problem
$$\min_u F(u)=\int_{0}^1 L(u'(x))dx, \mbox{ or }\min_{u}\int_0^1 \sqrt{1+(u'(x))^2}dx,                                       
$$
for twice differentiable functions $u:[0,1]\rightarrow \rit$ with boundary conditions $u(0)=0$, $u(1)=1$.

(a) Show that the functional $u\mapsto F(u)$ is convex.

(b) Formally compute the Gateaux-differential and then obtain the Euler-Lagrange equation associated with the minimization. 

(c) Find the exact solution of the problem.

\noindent{{\bf [3]} Computation of the Euler-Lagrange equation. 

(a) Consider the minimization problem
$$\inf_{u}F(u)=\int_{x_0}^{x_1} L(x,u(x),u'(x))dx,$$
with $u(x_0)=u_0$, $u(x_1)=u_1$ given constants, and $L$ a sufficiently smooth
function. Obtain formally the Euler-Lagrange equation of the minimization problem that is satisfied by a smooth optimal $u$. 

Hint: Consider smooth test functions $v$, such that $v(x_0)=v(x_1)=0$. Since $u$ is a minimizer, we must 
have $F(u)\leq F(u+\varepsilon v)$ for all such sufficiently smooth functions $v$ and every 
real $\epsilon$. Apply integration by parts to obtain the desired result. You should obtain a 
second-order differential equation.

(b) Let now $u(x,t)$ be a smooth solution of the time-dependent partial differential equation (PDE)
$$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x} L_{u'}(x,u,u')-L_u(x,u,u'),$$
with $u(x,0)=u_0(x)$ on $(x_0,x_1)$ and $u(x_0,t)=U_0$, $u(x_1,t)=U_1$ for $t\geq 0$. 
Show that the function $E(t)=F(u(\cdot,t))$ is decreasing in time, where
$F(u)=\int_{x_0}^{x_1}L(x,u,u')dx.$

\bigbreak 

\noindent{\bf Optional problems}

\medbreak

\noindent{\bf [1]} Consider the minimization problem
$$\inf_{u}F(u)=\int_{x_0}^{x_1} L(x,u(x),u'(x),u''(x))dx,$$ 
with $u(x_0)=u_0$, $u(x_1)=u_1$, $u'(x_0)=U_0$, $u'(x_1)=U_1$ given, and $L$ a sufficiently smooth
function. As in the previous problem, derive the equation satisfied by a smooth optimal $u$. Choose test functions $v$ in $C^{\infty}[x_0,x_1]$ that satisfy $v(x_0)=v(x_1)=v'(x_0)=v'(x_1)=0$. 
(you should obtain a fourth-order differential equation).

\medbreak

\noindent{\bf [2]} Consider the minimization problem in two dimensions $(x,y)$,
$$\inf_{u}E(u)=\int_{\Omega}L(x,y,u,u_x,u_y)dxdy,\ \ \ u=g\mbox{ on }\partial\Omega,$$
where $g$ is a given function on the boundary $\partial\Omega$, with $\Omega$ a bounded and open 
region in the plane. Assume that the integrand $L$ is differentiable.

(i) Show that a sufficiently smooth minimizer $u$ formally satisfies the Euler-Lagrange equation
$$\frac{\partial}{\partial x} L_{u_x}(P)+\frac{\partial}{\partial y}L_{u_y}(P)-L_u(P)=0$$
on $\Omega$, where $P=(x,y,u(x,y),u_x(x,y),u_y(x,y))$.

(ii) Apply the above result to the case when $L(x,y,u_x,u_y)=u_x^2+u_y^2-2fu$.

Hint for (i): consider another test function $v$, such that $v=0$ on $\partial \Omega$. Since $u$ is a 
minimizer, we must have $E(u)\leq E(u+\varepsilon v)$ for all such sufficiently smooth functions $v$ and 
all real $\epsilon$. Apply integration by parts to obtain the desired result. Here, $(u_x,u_y)=\nabla u$.

\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). 

\noindent $\bullet$ {\bf Integration by Parts Formula.} Let $\Omega$ be an open and bounded subset of $R^d$, with Lipschitz-continuous (or sufficiently smooth) boundary $\partial\Omega$. Let $\vec{n}=(n_1,n_2,...,n_d)$ be the exterior unit normal to $\partial\Omega$. 
 Let me recall the following fundamental Green's formula, or integration by parts formula: given two functions $u,v$ (with $u$, $v$, and all their 1st order partial derivatives belonging to $L^2(\Omega)$, or $u,v\in H^1(\Omega)$), then 
$$\int_{\Omega}u v_{x_i} dx=-\int_{\Omega} u_{x_i} v dx +\int_{\partial\Omega} u v n_i dS.$$

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