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\title{Math 581 Assignment 5}
\author{Due Friday March 16}
\date{Winter 2012}
\begin{document}
\maketitle
\begin{enumerate}[1.]
\item
Prove the followings.
The results in b) and c) are part of the {\em Paley-Wiener theorem}.
\begin{enumerate}[a)]
\item
For a compactly supported distribution $u\in\tstE'$,
its Fourier transform is equal to
$$
\hat{u}(\xi) = \langle u(x), e^{-i\xi\cdot x}\rangle,
$$
where the notation $u(x)$ is to indicate that the distribution $u$ acts on $e^{-i\xi\cdot x}$ as a function of $x$.
The above expression also makes sense for $\xi\in\C^n$,
defining an entire analytic function $\hat{u}$.
This is called the {\em Fourier-Laplace transform} of $u$.
\item
In this setting, if $u$ is supported in a ball of radius $r$ centred at the origin, then $\hat{u}$ satisfies the growth estimate
$$
|\hat{u}(\xi)| \leq C(1+|\xi|)^N e^{r|\mathrm{Im}\,\xi|},
$$
with some constants $C$ and $N$.
Hence the Fourier-Laplace transform of a compactly supported distribution is an entire function of growth order at most $1$.
\item
If $u\in\tstD$ and is supported in a ball of radius $r$ centred at the origin, then for every integer $N$ there is a constant $C_N$ such that
$$
|\hat{u}(\xi)| \leq C_N(1+|\xi|)^{-N} e^{r|\mathrm{Im}\,\xi|}.
$$
Hence the Fourier-Laplace transform of a compactly supported smooth function is an entire function of growth order at most $1$,
with rapid decay in the real directions.
\item
If the set of real zeroes of $p$ is bounded, then every tempered distribution solution of $p(D)u=0$ is an entire function of growth order at most $1$.
\end{enumerate}
\item
In this exercise, we will construct a fundamental solution for an arbitrary (nontrivial) constant coefficient operator by using a construction
known as {\em H\"ormander's staircase}.
Let $p$ be a nontrivial polynomial in $2$ variables.
\begin{enumerate}[a)]
\item
Show that without loss of generality we can assume
$$
p(\xi) = \xi_1^m + \sum_{k=0}^{m-1}q_k(\xi_2)\xi_1^k,
$$
where $m$ is the degree of $p$, and $q_k$ are polynomials of a single variable.
\item
Consider a subdivision of the $\xi_2$-axis into a countably many disjoint intervals $\{I_k\}$,
and consider a sequence $\{\eta_1^{(k)}\}$ of real numbers.
Then the set of points $(\xi_1,\xi_2,\eta_1)\in\R^3$ where $(\xi_1,\xi_2)\in\R^2$ and $\eta_1=\eta_1^{(k)}$ for $\xi_2\in I_k$,
can be visualized as a staircase, with steps of heights $\eta_1^{(k)}$ and widths $|I_k|$ and that are infinitely long in the $\xi_1$-direction.
Under the identification of $\R^3$ with $\C\times\R$ by $(\xi_1,\xi_2,\eta_1)\mapsto(\xi_1+i\eta_1,\xi_2)$,
let us denote by $H\subset\C\times\R$ the above described staircase.
Construct a staircase $H$ (that is, sequences $\{I_k\}$ and $\{\eta_1^{(k)}\}$) such that
$|p|\geq\alpha$ on $H$ and $|\eta_1^{(k)}|\leq\beta$ for all $k$,
with some constants $\alpha>0$ and $\beta$.
\item
Let $E:\tstD\to\C$ be defined by
$$
\langle E,\varphi\rangle
=
\frac1{4\pi^2}\int_{H} \frac{\tilde{\hat\varphi}(\zeta)}{p(\zeta)}\exd\zeta
\equiv
\frac1{4\pi^2}\sum_{k}\int_{I_k}\int_\R \frac{\tilde{\hat\varphi}(\xi_1+i\eta_1^{(k)},\xi_2)}{p(\xi_1+i\eta_1^{(k)},\xi_2)}\exd\xi_1\exd\xi_2,
$$
where $\varphi\in\tstD$, and the tilde $\tilde{}$ denotes the reflection.
Prove that indeed $E\in\tstD'$ and that $E$ is a fundamental solution of $p(D)$.
\item
Extend the construction to $n$ dimensions.
\end{enumerate}
\item
Let $p$ be a nonzero polynomial. Show the followings.
\begin{enumerate}[a)]
\item
The equation $p(D)u=f$ has at least one smooth solution for every $f\in\tstD$.
\item
If all solutions of $p(D)u=0$ are smooth,
then $\singsupp\,u\subset\singsupp\,p(D)u$ for any $u\in\tstD'$.
So hypoelliptic operators can be defined as those $p(D)$ such that all solutions of $p(D)u=0$ are smooth.
\end{enumerate}
\item
Recall H\"ormander's theorem that $p(D)$ is hypoelliptic if and only if
for any $\eta\in\R^n$ one has $p(\xi+i\eta)\neq0$ for all sufficiently large $\xi\in\R$.
Apply this criterion to show the followings.
\begin{enumerate}[a)]
\item
All elliptic operators are hypoelliptic.
\item
The wave operator is not hypoelliptic.
\item
The heat operator is hypoelliptic.
\end{enumerate}
\item
Construct a non-hypoelliptic polynomial $p$ in dimension $n>1$ such that
$|p(\xi)|\to\infty$ as $|\xi|\to\infty$ for $\xi\in\R^n$.
\item
Consider the Cauchy problem
$$
\partial_t u = \sum_{k=1}^{n} A_k \partial_k u,
\qquad
u|_{t=0} = f,
$$
where $u$ is a vector function with $m$ components, each $A_k$ is a (possibly complex) $m\times m$ matrix,
and $f$ is a given (vector) function.
We say that the problem is {\em strongly well-posed} if
for any $f\in L^2$, there exists a solution $u\in C^0(\overline{\R}_+,L^2)$, which satisfies the estimate
$$
\|u(t)\|_{L^2} \leq C e^{\alpha t} \|f\|_{L^2},
\qquad t\geq0,
$$
with some constants $\alpha$ and $C$, and $u$ is the only solution in $C^0(\overline{\R}_+,L^2)$.
In each of the following cases, prove that the corresponding Cauchy problem is strongly well-posed.
\begin{enumerate}[a)]
\item
Symmetric hyperbolic: All $A_k$ are Hermitian.
\item
Strictly hyperbolic: For all nonzero $\xi\in\R^n$, the eigenvalues of $P(\xi)=\sum_{k=1}^n A_k\xi_k$ are real and distinct.
\end{enumerate}
\item
Prove the strong well-posedness of the Cauchy problem for the system
\begin{equation*}
\begin{split}
\partial_t u &= P(\partial) u + Q(\partial) v,\\
\partial_t v &= H(\partial) v + Mu,
\end{split}
\end{equation*}
where $u$ and $v$ are vector functions,
$P(\partial)$ is a second order parabolic operator,
$Q(\partial)$ is an arbitrary first order operator,
$H(\partial)$ is a first order symmetric hyperbolic operator,
and $M$ is simply a matrix (i.e., a zeroth order operator).
The operators $P(\partial)$, $Q(\partial)$, and $H(\partial)$ may contain lower order terms, and the spatial dimension is $n$.
\end{enumerate}
\end{document}