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For an order-$k$ scalar quasilinear partial differential equation with real-analytic coefficients, real-analytic Cauchy data on a real-analytic non-characteristic hypersurface determine a unique real-analytic solution in a neighbourhood of each point of that hypersurface.

The scalar Cauchy-Kovalevskaya theorem says that a partial differential equation solved for its highest derivative normal to a real-analytic non-characteristic hypersurface, with real-analytic coefficients and Cauchy data, has a unique local real-analytic solution. In coordinates, an equationhas such a solution near the origin when $F$ and the prescribed values of ${\partial }_t^j\varphi (0,x)$ for $0\le j<m$ are real analytic.

Choose a real-analytic primitive $F_0$ of $f$ near zero and apply the theorem to the scalar Laplace equationThe line $t=0$ is non-characteristic because the coefficient of ${\varphi }_{tt}$ is one. DefineThen $u(0,x)=f(x)$ and $v(0,x)=g(x)$, while equality of mixed derivatives and the Laplace equation giveThus $(u,v)$ is the required local real-analytic solution.

Solved by gpt-5.6-sol high.

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

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.