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\title{Math 581 Assignment 2}
\author{Due Wednesday February 12}
\date{Winter 2014} % Activate to display a given date or no date
\begin{document}
\maketitle
\begin{enumerate}[1.]
\item
For each of the following cases, determine the characteristic cones and characteristic surfaces.
\begin{enumerate}[a)]
\item
Wave equation with wave speed $c>0$: $u_{xx}+u_{yy}=c^{-2}u_{tt}$.
\item
Tricomi-type equation: $u_{xx}+yu_{yy}=0$.
\item
Ultrahyperbolic ``wave'' equation: $u_{xx}+u_{yy}=u_{zz}+u_{tt}$.
\end{enumerate}
\item
Prove that if $\beta\in\R$ and $u\in C^1(\R^2)$ is a solution of $u_t+\beta u_x=0$,
then
$$
\{(x,t):u\in C^k \textrm{ on a neighbourhood of }(x,t)\},
$$
is a union of lines.
\item
Consider the Laplace equation $\Delta u=0$ on the unit disk, given in polar coordinates by $\mathbb{D}=\{(r,\theta):r<1\}$.
Specify the Cauchy data
$$
u(1,\theta)=f(\theta),\qquad
\partial_ru(1,\theta)=g(\theta),
$$
where $f$ and $g$ are $2\pi$-periodic real analytic functions.
Then show that a real analytic solution exists in a neighbourhood of the circle $\partial\mathbb{D}$.
Investigate what happens to the solution as $r\to0$ and $r\to\infty$, if $f$ and $g$ are of the form
$$
a_0+\sum_{n=1}^{m} a_n\cos n\theta + b_n\sin n\theta,
$$
i.e., trigonometric polynomials.
\item
Consider the wave equation
$$
u_{tt} - u_{xx} = f ,
$$
with the initial data
$$
u(x,\alpha x) = \phi(x)
\qquad\textrm{for}\quad x<0,
\qquad\textrm{and}\qquad
u(x,x) = \psi(x)
\qquad\textrm{for}\quad x\geq0,
$$
where $\alpha\neq1$ is a constant, and $\phi$ and $\psi$ are real analytic functions in a neighbourhood of $0\in\R$.
Note that we are specifying the initial condition on the union of two rays, one of which is characteristic, and the other may or may not be characteristic, depending on $\alpha$.
Supposing that $f$ is real analytic in a neighbourhood of $0\in\R^2$,
investigate if and when the problem is locally solvable near $0\in\R^2$.
Do we need to impose compatibility conditions on the data $\phi$ and $\psi$?
\item
Let $p$ be a nontrivial polynomial of $n$ variables, and let $f$ be a real analytic function in a neighbourhood of $0\in\R^n$.
\begin{enumerate}[a)]
\item
Prove that the set $\{\xi\in\R^n:p(\xi)=0\}$ is closed and of measure zero.
\item
Show that there is a neighbourhood of $0\in\R^n$,
on which the equation $p(\partial)u=f$ has a solution.
Supposing that $p(\xi)=\sum_{\alpha} a_\alpha \xi^\alpha$, here the operator $p(\partial)$ is given by
$$
\textstyle p(\partial)=\sum_{\alpha} a_\alpha \partial^\alpha .
$$
\item
Extend this local solvability result to linear operators with analytic coefficients.
That is, assuming that $\{a_\alpha\}$ is a finite collection of real analytic functions in a neighbourhood of $0\in\R^n$,
with the property that $p(\xi)=\sum_\alpha a_\alpha(0)\xi^\alpha$ is a nontrivial polynomial,
show that the equation
$$
\textstyle \sum_{\alpha} a_\alpha \partial^\alpha u = f ,
$$
has a solution on a neighbourhood of $0\in\R^n$.
\end{enumerate}
\item
Let $p$ be a nontrivial polynomial of $n$ variables, and let $H\subset\R^n$ be a (closed) half-space.
\begin{enumerate}[a)]
\item
Show that if $u\in C^\infty(\R^n)$ satisfies $p(\partial)u=0$ in $\R^n$ and $\supp\,u\subset H$,
and if the boundary of $H$ is noncharacteristic for the constant coefficient operator $p(\partial)$,
then $u\equiv0$.
Provide a counterexample when $\partial H$ is characteristic and $p$ is a nonconstant homogeneous polynomial.
\item
Show that if we require that $u$ is compactly supported, then the noncharacteristic condition on $\partial H$ can be dropped,
i.e., prove that if $u\in C_c^\infty(\R^n)$ satisfies $p(\partial)u=0$ in $\R^n$ then $u\equiv0$.
Imply that if $u\in C^\infty_c(\R^n)$ then $\supp\,u$ is contained in the convex hull of $\supp\,p(\partial)u$.
\end{enumerate}
\item
Let $u$ be a $C^2$ solution of the $n$-dimensional wave equation $\partial_t^2 u - \Delta u = 0$,
and assume that $u$ and all its first derivatives vanish on the line segment $\{(0,t)\in\R^{n+1}:0