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\begin{document}           


\centerline{{\sc Problem set 3}}
\centerline{Math 218A, Winter 2001}




\bigskip\noindent
{\bf 1.} The Tricomi equation
\[
y u_{xx} - u_{yy} = 0
\]
is a fundamental example of an equation of mixed type, being
hyperbolic in $y > 0$ and elliptic in $y < 0$. It arises
in the modeling of transonic fluid flows. Show that the change of variables
\[
\eta = \frac{2}{3} y^{3/2}
\]
in $y > 0$ transforms the Tricomi equation to an Euler-Poisson-Darboux equation
\[
u_{xx} - u_{\eta\eta} - \frac{1}{3} u_\eta = 0.
\]



\bigskip\noindent
{\bf 2.} Does the maximum principle hold for the wave equation?

\bigskip\noindent
{\bf 3.} Show that the solution of the nonhomogeneous wave equation,
\beann
&&u_{tt} -  \Delta u = f(x,t),\\
&&u(x,0) = u_t(x,0) = 0,
\eeann
in three space dimensions is given by
\[
u(x,t) = \frac{1}{4\pi}\int_{B_t(x)} \frac{f(y, t- |x-y|)}{|x-y|}\,dy.
\]
Give a physical interpretion of this result.


\bigskip\noindent
{\bf 4.} Let $S$ be the following action functional
\[
S[u] = \frac{1}{2} \int_0^\infty \int_{\Rl^n}
\left(u_t^2 - |\nabla u|^2\right)\, dx dt.
\]
Show that if $u$ is stationary point of $S$, meaning that
\[
\frac{d}{d\epsilon} \left.S[u+\epsilon \phi]\right|_{\epsilon=0} = 0
\] 
for all $\phi\in C_c^\infty\left({\Rl^n}\times(0,\infty)\right)$, then
$u$ satisfies the wave equation. You can assume, for simplicity,
that $u \in
C_c^\infty\left(\Rl^n\times[0,T]\right)$ for each $T > 0$.
Does $u$ minimize $S$?

\bigskip\noindent
Note that the action is the time integral of the Lagrangian
\[
L[u] = \frac{1}{2}\int_{\Rl^n}
\left(u_t^2 - |\nabla u|^2\right)\, dx,
\]
which is the \emph{difference}
between the kinetic and potential energies, whereas the conserved
Hamiltonian
\[
H[u] = \frac{1}{2} \int_{\Rl^n}
\left(u_t^2 + |\nabla u|^2\right)\, dx
\]
is the \emph{sum} of the kinetic and potential energies.

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