Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 105 / 1

Only the second-order terms contribute to the principal symbol. For a covector $\xi =({\xi }_t,{\xi }_r,{\xi }_z)$ it isThe conormal to the hypersurface $\Sigma =\{\varphi =0\}$ is $d\varphi =({\varphi }_t,{\varphi }_r,{\varphi }_z)$. After evaluating the coefficient at the prescribed boundary value of $u_z$, the non-characteristic condition is therefore

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At $x\in \Sigma$, keep the unit normal $N(x)$ fixed and define the $j$th normal derivative byFor $j=1$, the chain rule and $N=\nabla \varphi /|\nabla \varphi |$ give

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WriteDifferentiating the prescribed identity $u(\psi (r,z),r,z)=0$ in its two tangential directions givesThe unit normal is $(1,-{\psi }_r,-{\psi }_z)/\sqrt{D}$, so the second item of Cauchy data becomesConsequentlySubstitution into the principal symbol from part a shows that the graph is non-characteristic exactly where

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The Cauchy-Kovalevskaya theorem says that an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients has a unique local real-analytic solution near each point of a real-analytic non-characteristic hypersurface, provided the prescribed Cauchy dataare real analytic there. The uniqueness is among local real-analytic solutions agreeing with all of those data.

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For ${\psi }_0=(r^2+z^2)/2$,The prescribed function is $g=\sqrt{D}$, so part c gives $u_z=-z$. Hence the non-characteristic condition reduces toThe Cauchy-Kovalevskaya theorem therefore guarantees a unique local real-analytic solution at exactly those pointsfor which

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