Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 3

For a smooth function $g$ define its spherical meanThe Kirchhoff formuladefines a smooth solution of the three-dimensional wave equation for all positive and negative $t$ and has the prescribed data at $t=0$.

For uniqueness, apply the local energy estimate to the difference of two solutions on a backward light cone. Its energy at the cone tip is bounded by the zero initial energy on the cone base, so the difference and its derivatives vanish. Covering spacetime by such cones proves uniqueness among $C^2$ solutions.

Compact support is unnecessary for existence or uniqueness: the sphere in the Kirchhoff formula is compact for each $(t,x)$, so arbitrary smooth data suffice, and the cone-energy proof is local.

The Strong Huygens principle in three spatial dimensions says that the solution at $(t,x)$ depends only on the initial data on the sphererather than on the full ball bounded by that sphere. This follows immediately from the Kirchhoff formula and its time derivative. Consequently a disturbance has no tail inside the light cone: if the initial data are supported in a compact set $K$, then $u(t,x)=0$ whenever that sphere misses $K$. This is sharper than finite propagation speed, which only excludes influence from outside the ball.

Let $\mathcal{R}g(s,\omega )$ be the Radon transformTaking the large-radius limit in the Kirchhoff formula, with $\xi =t-r$ fixed for the outgoing limit and $\eta =t+r$ fixed for the incoming limit, gives the radiation fieldsIndeed, the expanding spheres converge after multiplication by $r/t$ to the planes $y\cdot \omega =-\xi$ and $y\cdot \omega =\eta$, respectively.

The Radon transforms of smooth compactly supported functions are smooth. If the data are supported in a ball of radius $R$, these transforms vanish for $|s|>R$. Hence both radiation fields are well-defined smooth functions of compact support in the null-time variable.

For radial data, seton $\mathbb{R}$. The conditions at the origin say exactly that these are smooth odd compactly supported functions. The radial reduction $w(t,r)=ru(t,r)$ satisfies the one-dimensional wave equation, and the D'Alembert formula givesIn null coordinates this isThe limits are thereforeThey are smooth and compactly supported because $f_0,f_1$ are odd.

WriteThen $A$ is an arbitrary odd test function and $B$ is an arbitrary even test function. Conversely, every odd $A\in C_c^{\infty }(\mathbb{R})$ gives $f_0=2A$, and every even $B\in C_c^{\infty }(\mathbb{R})$ gives $f_1=-2B^{\prime }$. Thus both maps are injective andSincethe radial scattering map is