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

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.

The wave operator has principal symbolThe conormal to the initial plane is $dt$, and $p(dt)=-1\ne 0$. The plane is therefore non-characteristic.

The principal symbol is ${\tau }^2+|\xi {|}^2$, so its value on $dt$ is $1$. The initial plane is non-characteristic.

When the heat equation is viewed as a second-order equation, its principal symbol contains only the spatial second derivatives:up to an irrelevant sign. It vanishes on $dt$, so the initial plane is characteristic in this second-order sense. The equation remains a well-posed first-order evolution equation in time; these are different notions of order.

The principal symbol isIts value on $dt$ is $1$, independently of the data, so the initial plane is non-characteristic everywhere.

For $\varphi (x)=|x{|}^2-1$, the conormal on the unit sphere is $d\varphi =(0,2x)$. Henceon the sphere. It is non-characteristic at every point.

For $\varphi =t-x_1$, the conormal is $dt-d{x}_1$, andThis null hyperplane is a characteristic hypersurface of the wave equation.

The second-order principal symbol of the heat operator evaluated on $dt-d{x}_1$ is $-1$, up to the overall sign convention. Thus this tilted hypersurface is non-characteristic.

Introduce the null coordinatesThen $\square =-4{\partial }_{\xi }{\partial }_{\eta }$, so the equation becomesWrite the compatible boundary values asTwice integrating the equation gives the equivalent Volterra integral equation

Let $T$ denote the double-integral operator including the factor $-1/4$, and put $g=a+b-c$. Successive approximation gives the Neumann seriesOn a rectangle $|\xi |\le R$, $|\eta |\le R$,The series and its differentiated series converge locally uniformly. Since $a$ and $b$ are analytic, the sum is analytic and solves the equation and data near the origin.

If two solutions have the same data, their difference $w=Tw$. Iterating and using the same factorial estimate gives $\Vert w{\Vert }_{\infty }=0$ on every sufficiently small rectangle. This proves uniqueness. The argument is the Analytic Goursat problem for a Klein--Gordon equation.

The same Volterra series converges uniformly on the entire compact characteristic square $0\le \xi ,\eta \le 1$, becauseThe boundary functions are analytic on neighbourhoods of the compact axis segments, so finitely many complex neighbourhoods give uniform Cauchy estimates for their derivatives. Applying $T^m$ adds the two factorial denominators above, and the corresponding derivative series converges on a neighbourhood of every point of the closed square. Thus the local analytic solutions continue across the whole square and agree on overlaps by uniqueness.

Equivalently, the integral equation bounds $u$ and every differentiated equation on each smaller rectangle; no norm can blow up at a first missing corner. The local analytic existence theorem therefore extends the solution through that corner. This is Global continuation for the analytic Goursat problem.

For compatible $C^2$ boundary functions $a$ and $b$, use exactly the same Volterra series. The factorial estimate holds in the $C^2$ norm after differentiating the integral formula, so the series converges to a $C^2$ function on the full square. It satisfiesand the two boundary values. This directly proves existence. One can equivalently approximate $a,b$ in $C^2$ by compatible analytic functions; the same estimates make their analytic solutions Cauchy in $C^2$.