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\begin{document}

\begin{center}
{\Large \em Math 560 (1998)}\\
\vspace{.1in}
{\Large Midterm Exam (April 1, 1999)}\\
\end{center}


\vspace{.2in}
\begin{center}
EACH QUESTION COUNTS FOR 25 POINTS. YOUR EXAM GRADE IS \\
\vspace{.1in}
GRADE \% = min(100, TOTAL POINTS)
\end{center}

\vspace{.2in}
\large
\begin{itemize}
\item[1.] Define the differential operator
\eq
Lu = x u''(x)
\endeq
with domain
\eq
D(L) =\{ u\in L^2[0,1]: Lu \in L^2[0,1], u(0)=u(1)=0 \} \ .
\endeq
\begin{itemize}
\item[a)] Define the adjoint operator $L^*$ operator of $L$ using the
standard $L^2[0,1]$ inner product.
\item[b)] For what $\alpha \in \reals$ (if any) is $f \notin R(L)$ if $f(x)=x^\alpha$?
\end{itemize}
\item[2.] Give examples of linear second-order differential operators
  $L$ defined on a domain $D(L) \subset L^2[0,1]$ for which
\begin{itemize}
\item[a)] $L=L^*$ and $D(L)=D(L^*)$
\item[b)] $L\ne L^*$ and $D(L)\ne D(L^*)$
\item[c)] $L=L^*$ and $D(L)\ne D(L^*)$
\end{itemize}

\item[3.] 
Define the inner product
\eq
<u,v> = \int_1^2 x\left(u_1(x)v_1(x) +u_2(x) v_2(x)\right) dx
\endeq
and Hilbert space $H=\{u\in L^2[1,2]\times L^2[1,2] :
\parallel u \parallel < \infty \}$.
Given
\begin{eqnarray}
Lu & \equiv &
\left[
\begin{array}{ll}
\frac{d^2}{dx^2} & x \\
0 & \frac{d}{dx} 
\end{array}
\right] u , \quad u=(u_1,u_2)^T 
\end{eqnarray}
and domain
\[
D(L) =\{u\in H: Lu\in H, u_1(1)=u_1(2)=0, u_2'(1)=0 \}
\]
find the adjoint operator $L^*$ and its associated domain $D(L^*)$.
Find a (simple) $f(x)=(f_1(x),f_2(x))^T$ for which $Lu=f$ has no
solution $u\in D(L)$.

\item[4.] For what $\lambda$ does
\[
u'' + \lambda^2 u = 1 + \sum_{n=1}^{5} a_n
cos \left( n x\right) \quad , \quad  a_n =\frac{n^2-3n+2}{n^3}
\]
have a solution $u\in D(L)$ where
\[
D(L) =\{u\in L^2[0,\pi]: Lu\in L^2[0,\pi], u'(0)=u'(\pi)=0\}
\]

\item[5.] Define
\begin{eqnarray}
J(u) & = & \int_0^1   \frac{1}{x} u''(x)^2  dx \\
{\cal A} & = & \{u \in C^4[0,1]: u(0)=u'(0)=0,u(1)=u'(1)=1 \}
\end{eqnarray}
Assuming there exists a $\bar{u}(x) \in{\cal{A}}$ such that
\[J(\bar{u}) = \min_{u\in {\cal{A}}} J(u) \ , \] 
find $\bar{u}$.

\item[6.] Derive the Euler-Lagrange equations and natural boundary
conditions associated with the extremization of the functional
\[
J(u) = \int \int \int_R \left( u_x^4 + u_y^4 +u_z^4 \right) dV 
+ \int \int_{\partial R} u^2 dS
\]
where $u=u(x,y,z) \in C^2(\bar{R})$ and $R\subset \reals^3$.
\end{itemize}


\end{document}