The strong Huygens principle in the homogeneous Cauchy setting

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $n\ge 1$ and $c>0$, and fix the homogeneous Cauchy setting of this page. Admissible data for dimension $n$ is a pair $(u_0,u_1)$ in the regularity class of the dimension-$n$ representation formula of the preceding pair wave-equation-representation-formulas (Wave equation, Cauchy data and wave speed, Spherical means and the weighted ball integral of space-dependent data), and the solution is the $C^2$ solution on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$ defined by that formula and attaining the data at $t=0$ (The dimension formulas attain the Cauchy data for $n\ge 2$ and d'Alembert's formula and uniqueness in one dimension for $n=1$); in a class closed under differences with sharp energy conservation it is the unique such solution by Energy uniqueness for the wave Cauchy problem.

Fix $x_0\in {\mathbb{R}}^n$, $t_0>0$ and put

$S:=\mathrm{\partial }{B}_{c{t}_0}(x_0)=\{y\in {\mathbb{R}}^n:\mathrm{\mid }y-x_0\mathrm{\mid }=c{t}_0\}\mathrm{.}$

The homogeneous Cauchy problem with speed $c$ satisfies the strong Huygens principle in dimension $n$ when, for every $(x_0,t_0)$ and every admissible data pair $(u_0,u_1)$, the value $u(x_0,t_0)$ is unchanged when $(u_0,u_1)$ is replaced by an admissible pair that agrees with $(u_0,u_1)$ on a neighbourhood of $S$; equivalently, the data-to-value functional is carried by the sphere $S$: admissible perturbations of the data that vanish on a neighbourhood of $S$ do not change the value $u(x_0,t_0)$.

The following formulations are equivalent for these linear representation formulas, which have base-ball locality: data vanishing on a neighbourhood of the closed base ball contribute zero. To check equivalence, choose a smooth radial cutoff equal to one on the closed base ball and supported in a slightly larger ball (A smooth bump between concentric Euclidean balls). Multiplying a perturbation by this cutoff does not change its value in the formula, since the data and all derivatives read at radius $c{t}_0$ are unchanged; for the even formula this follows by writing its weighted ball integral on the fixed unit ball and differentiating the smooth data there. A compactly supported perturbation vanishing near $S$ splits into an interior part, supported compactly in $B_{c{t}_0}(x_0)$, and an exterior part, vanishing near the closed base ball; the split is smooth because the perturbation is zero on a collar of $S$. Thus (i) implies neighbourhood invariance, while the reverse follows because a closed support contained in the open base ball is compact and separated from $S$. Neighbourhood invariance implies (ii) by comparing with zero data; conversely (ii), applied to the cutoff perturbation whose compact support misses $S$, implies neighbourhood invariance. This proves the equivalence of (i), (ii) and (iii).

(i) Strictly-inside form. The value does not depend on the data at points $y$ with $\mathrm{\mid }y-x_0\mathrm{\mid }<c{t}_0$: changes supported in the open base ball $B_{c{t}_0}(x_0)$ do not affect $u(x_0,t_0)$.

(ii) Shell form for compactly supported data. If the data are supported in the compact set $K$, then $u(x_0,t_0)=0$ whenever $S\cap K=\mathrm{\varnothing }$, that is, the value at $(x_0,t_0)$ is carried by the $c{t}_0$-sphere shell of $K$; this is the classical sharp-Huygens picture.

(iii) Germ form. Data agreeing on a neighbourhood of $S$ give the same value. In odd dimensions, the finite radial jets that suffice are identified in The strong Huygens principle in odd spatial dimensions.

Caution. The value may involve finitely many transverse (radial) derivatives of the data, so agreement of the bare restrictions to $S$ is not in general enough. For $n=3$, $c=1$, the radial datum $u_0(y)=(1-\mathrm{\mid }y\mathrm{\mid })\psi (\mathrm{\mid }y\mathrm{\mid })$ with $\psi$ smooth, $\psi \equiv 1$ near $\mathrm{\mid }y\mathrm{\mid }=1$ and $\psi \equiv 0$ near $0$, and $u_1=0$ vanishes on $S=\mathrm{\partial }{B}_1(0)$, yet Kirchhoff's formula (Kirchhoff's formula in three dimensions) gives $u(0,1)=u_0(1)+u_0^{\mathrm{\prime }}(1)=0-1=-1\mathrm{\ne }0$; the data-to-value functional reads the first radial derivative of $u_0$ along $S$, as well as its values, and cannot be represented by integrating only the bare restriction. The precise positive statement is The strong Huygens principle in odd spatial dimensions and the failures are Wave tails in one and even spatial dimensions: strong Huygens fails. This definition asserts no existence or uniqueness beyond the classical class above, and it does not imply that the principle holds; whether it holds is exactly the content of those theorems.