Math 6410 • ODE

Given \(f\in C^\infty(\R^n)\), and \(g\in C^\infty(\R^{n-1})\), consider the eikonal
equation
\[
  \lvert Du\rvert=f(x),\qquad \text{in }\R^n,
\]
with the boundary condition \(u=g\) on the plane \(\Gamma=\{x_n=0\}\).
\begin{enumerate}
\item[(a)] Write down the characteristic equations
\[
\begin{aligned}
  \dot{p} &= -D_xF-D_zF\,p,\\
  \dot{z} &= D_pF\cdot p,\\
  \dot{x} &= D_pF,
\end{aligned}
\]
associated with this problem.
\item[(b)] Given an initial point \(x^0\in\Gamma\), if possible, determine a set of
initial conditions for the characteristic equations which satisfy the
\textit{compatibility conditions} (in other words, the initial conditions are
\textit{admissible}).
\item[(c)] Determine conditions on \(g\) and \(f\) which guarantee a unique local solution
to the boundary value problem near \(x^0\).
\end{enumerate}