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{ \large \textbf{\textsc{Math 5588 -- Homework 3 (Due Thursday February 2)}}}
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\begin{enumerate}
\item Find and solve the Euler-Lagrange equation for the functional
\[I(u) = \int_0^{\log(2)} u(x)^2 + (u'(x))^2 \, dx\]
subject to boundary conditions $u(0)=0$ and $u(\log(2))=1$, where $\log$ is the natural logarithm. Do you think your solution is a minimum or a maximum (or neither)?
\item Find the Euler-Lagrange equation for the functional
\[I(u) = \int_0^\pi u(x)^2 - (u'(x))^2 \, dx\]
subject to boundary conditions $u(0)=u(\pi)=0$. Then find \emph{all} solutions of the Euler-Lagrange equation (as a one-parameter family) and evaluate $I$ on all solutions.
\item Find the Euler-Lagrange equation for the functional
\[I(u) = \int_U (f(x) - u(x))^2 + \lambda \phi(|\nabla u(x)|) \, dx\]
where $f:U \to \R$, $\phi:\R \to \R$, and $\lambda$ is a constant.
\item Show that the Euler-Lagrange equation for the functional
\[I(u) = \int_U \frac{1}{p}|\nabla u(x)|^p -u(x)f(x)\, dx\]
is the $p$-Laplace equation 
\[-\Delta_p u = f \ \ \text{in } U.\]
[Hint: The $p$-Laplacian was defined in Homework 2.]
\item Find the Euler-Lagrange equation for the functional
\[I(u) = \int_U (\Delta u(x))^2 \, dx.\]
[Hint:  Proceed as in the proof of the Euler-Lagrange equation from class. That is, let $\phi$ be smooth with compact support in $U$ and compute
\[\frac{d}{dt}\Big\vert_{t=0} I(u+t\phi)=0.\]
Use integration by parts and the vanishing lemma to find the Euler-Lagrange equation.]
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