Math 6410 • ODE

Consider the linear partial differential equation
\[
  Lu\equiv\sum_{\lvert\alpha\rvert\le k} a_\alpha(x)\partial^\alpha u=f
  \tag{6.1}
\]
where \(\alpha=(\alpha_1,\ldots,\alpha_n)\) is a multi-index. Let
\(\Omega\subset\R^n\) be a bounded domain with piecewise smooth boundary and suppose
throughout this problem that the coefficients in the equations are smooth.
\begin{enumerate}
\item Define what is meant by a weak solution of (6.1).
\item Define the principal part of \(L\), the characteristic variety of \(L\), and a
characteristic surface for \(L\).
\item Obtain conditions on \(a\), \(b\) and \(c\) under which
\[
  Lu=a(x,y)u_{xx}+2b(x,y)u_{xy}+c(x,y)u_{yy}=0
\]
is
\begin{enumerate}
\item hyperbolic
\item elliptic
\item parabolic.
\end{enumerate}
\item Define what is meant by a well-posed problem for a partial differential equation
with boundary and/or initial data specified. Give an example of a well-posed problem
for one of the three types of equation in (c) and briefly justify your assertions.
\end{enumerate}