Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 1 / a / Solution

WriteAs a differential operator of order two, its principal symbol isIf $\nu =({\nu }_t,{\nu }_x,{\nu }_y)$ is a unit normal to the real-analytic hypersurface $\Sigma$, the non-characteristic hypersurface condition is thereforeUnder this condition the equation can be solved for the second derivative normal to $\Sigma$. The Cauchy-Kovalevskaya theorem then gives a unique real-analytic solution in a neighbourhood of $\Sigma$ for the prescribed analytic Cauchy data $u{|}_{\Sigma }$ and ${\partial }_{\nu }u{|}_{\Sigma }$.