Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 1 / a / Solution

Consider an analytic quasilinear partial differential equationLet the analytic initial hypersurface be $\Gamma =\{\varphi =0\}$ and prescribe $u=u_0$ and one transverse derivative ${\partial }_{N}u=u_1$ on $\Gamma$. The tangential derivatives of $u_0$ together with $u_1$ determine the full first jet $Du$ on $\Gamma$. The hypersurface is non-characteristic at $x_0$ with respect to these data whenThis is precisely the principal symbol of a partial differential equation evaluated on the conormal $d\varphi (x_0)$.

The Cauchy-Kovalevskaya theorem then gives a unique real-analytic solution near $x_0$. In coordinates flattening $\Gamma$ to $x_n=0$, non-characteristicity lets the equation solve analytically for ${\partial }_n^{2}u$, after which the analytic equation and the two initial jets determine every higher Taylor coefficient.