Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 3 / b

Locally write the real analytic hypersurface as $\Sigma =\{F=0\}$ with $dF\ne 0$. The conormal bundle is spanned by $dF$. The surface is a characteristic hypersurface at $p$ precisely when the principal symbol vanishes on that conormal:Only the matrix $(a_{ij}+a_{ji})/2$, which is a symmetric matrix, contributes to this expression.

One prescribes analytic Cauchy data: the value of $u$ and one first derivative transverse to $\Sigma$, for examplewhere $\varphi$ and $\psi$ are real analytic on $\Sigma$. Being a non-characteristic hypersurface allows the equation to solve for the second derivative in the transverse direction. The Cauchy-Kovalevskaya theorem then gives one and only one local real analytic solution near $p$.