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\centerline{Ph.D. Qualifying Examination}
\centerline{Spring 2002}
\bigskip
\noindent{\bf Instructions:}
\begin{itemize}
\item[1.] If you think that there is a mistake ask the proctor.
If the proctor's explanation is not satisfactory, interpret the
problem as you see fit, but not in such a way that the answer is
trivial.
\item[2.] From each part solve 3 of 5 problems.
\item[3.] If you solve more that three problems from a part
indicate the problems that you wish to have graded.
\end{itemize}
\bigskip
\centerline{\bf Part A}
\bigskip
\noindent 1. Find the fundamental solution of the system
\[
\left[\begin{array}{c}
\dot x \\
\dot y \\
\dot z
\end{array}\right]
=
\left[\begin{array}{ccc}
3x & & +4z \\
2x & & +3z \\
-2x & +y & -2z
\end{array}\right].
\]
\bigskip
\noindent 2. Consider the system
\begin{eqnarray*}
\dot x&=&\sin(t)y \\
\dot y&=&-\cos(t)x
\end{eqnarray*}
Suppose a solution $u(t)=(x(t),y(t))$ has initial values
$u(0)=u_0= (0,1)$. Show that for $0\leq t\leq\pi$
\[
||u(t)-u_0||\leq{1\over 2}(1-\sqrt{2}\cos(t-{\pi\over 4})).
\]
\bigskip
\noindent 3. Consider the equation $\dot x = Ax+B(t)x$ where $A$
is an $n\times n$ matrix all of whose eigenvalues have strictly
negative real part and $B(t)$ is a continuous $n\times n$-matrix
valued function that satisfies $||B(t)||\leq ce^{-\beta t}$. Prove
that $x(t)=0$ is an asymptotically stable solution.
\bigskip
\noindent 4. Suppose that $X\colon\mathbf{R}^2\to\mathbf{R}^2$ is
a Lipschitz continuous dynamical system on $\mathbf{R}\mathbf{}^2$
with flow $\varphi_t$. Let $K$ be a compact subset of
$\mathbf{R}^2$ that contains no singular points. Prove that if
for some $p\in\mathbf{R}^2$, $\varphi_t(p)$ enters $K$, then it
must leave $K$ in finite time.
\bigskip
\noindent 5. Consider the system $\dot x = (1-x^2)a(t)$ for a
continuous function $a(t)$. What condition on $a(t)$ implies that
the solution $x(t)=1$ is Lyapunov stable. What condition implies
that $x(t)=1$ is asymptotically stable. Find an $a(t)$ so that
$x(t)=1$ is uniformly asymptotically stable.
\bigskip
\centerline{\bf Part B}
\bigskip
\noindent 1. Find the canonical form and the general solution of
the equation
\[
2xu_{xx}+2(1+xy)u_{xy}+2yu_{yy}+{2(1-x)\over 1-xy}u_x+{2(1-y)\over
1-xy}u_y=0.
\]
\bigskip
\noindent 2. Consider the linear system for two vector valued
functions $\mathbf{E}\colon\mathbf{R}\times\mathbf{R}^3\to
\mathbf{R}^3$ and $\mathbf{B}\colon\mathbf{R}\times\mathbf{R}^3\to
\mathbf{R}^3$. If $p\in\mathbf{R}\times\mathbf{R}^3$ is
represented by $p=(t,\mathbf{x})$ then $\mathbf{E}(t,\mathbf{x})$
and $\mathbf{B}(t,\mathbf{x})$ satisfy the linear system
\[
\begin{array}{ll}
\nabla\cdot\mathbf{E}=\rho & \nabla\cdot\mathbf{B}=0 \\
\nabla\times\mathbf{E}=-{\partial\mathbf{B}\over\partial t}
&\nabla\times\mathbf{B}=\mathbf{j}+{\partial\mathbf{E}\over\partial
t }
\end{array}
\]
Describe the symbol of this equation and show that it is of
hyperbolic type.
\bigskip
\noindent 3. Let $A_1$ and $A_2$ be $2\times 2$ matrices with
$A_1=\left[\begin{array}{c} A_{11} \\ A_{12}\end{array}\right]$
and $A_2=\left[\begin{array}{c} A_{21} \\
A_{22}\end{array}\right]$ where $A_{11}$, $A_{12}$, $A_{21}$, and
$A_{22}$ are nonzero elements of $\mathbf{R}^2$ representing the
rows of $A_1$ and $A_2$. Suppose that
$u(x,y)=\left[\begin{array}{c}u_1(x,y) \\
u_2(x,y)\end{array}\right]$ satisfies the system
\[
A_1{\partial\over\partial x}u(x,u)+A_2{\partial\over\partial
y}u(x,y)=0.
\]
Also suppose that $(A_{11},A_{21})$ and $(A_{12},A_{22})$ are
independent in $\mathbf{R}^4$ and that either $A_{11}$ and
$A_{21}$ are independent or $A_{12}$ and $A_{22}$ independent in
$\mathbf{R}^2$. Show that by a change of variables of the form
$w(x,y)=Cu(B\left[\begin{array}{c} x \\ y \end{array}\right])$
this system can be placed in Cauchy-Kowalewski form. Here $B$ and
$C$ are $2\times 2$ matrices.
\newpage
\bigskip
\noindent4.
\begin{itemize}
\item[(a)] Give an example of a subharmonic function $u$ on $B_R(0)$ that
is not harmonic and has the boundary values $u|_{\partial B_R(0)}
= \varphi$ for some continuous function $\varphi$ on $\partial
B_R(0)$.
\item[(b)] Let $\Omega\subset \mathbf{R}^n$ be an open subset and
let $u$ be a harmonic function on $\Omega$. Suppose the for $x\in
\Omega$, $B_R(x)$ is a open ball about $x$ with
$B_R(x)\subset\Omega$ and let $v$ be a subharmonic function
defined on $B_R(x)$ with $u|_{\partial B_R(x)}=v|_{\partial
B_R(x)}$ Show that the function
\[
V(x)=\left\{\begin{array}{ll}
u(x) & x\in B_R(x)^\sim \\
v(x) & x\in B_R(x)
\end{array}\right.
\]
is subharmonic.
\end{itemize}
\bigskip
\noindent 5. Consider the 2-dimensional wave equation in
cylindrical coordinates $u_{tt}=u_{rr}+{1\over r}u_r+{1\over
r^2}u_{\theta\theta} $ with domain the unit disk
$D^1=\{(r,\theta)|0\leq r<1\}$ and $t>0$ and with the boundary
conditions $u(r,\theta,0)=1-r^2$ and $u_t(r,\theta,0)=0$.
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