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Sphere-supported versus interior-supported free wave kernels

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ and let $u$ be the free solution assigned to admissible data $(u_{0}, u_{1})$ by the formulas of the page (Kirchhoff, Poisson, odd- and even-dimensional formulas). (i) If $n \geq 3$ is odd and $t > 0$ , the value $u (x, t)$ depends on the displacement and velocity data only through their restrictions to a neighbourhood of the sphere $\partial B_{c t} (x)$ : if two admissible data pairs agree on such a neighbourhood, their solutions agree at $(x, t)$ . Pointwise agreement only on the sphere is not asserted. (ii) If $n \geq 2$ is even, $t > 0$ and $u_{1}$ is supported in a compact subset of the open ball $B_{c t} (x)$ , then the velocity contribution has the kernel form $c^{1 - n} D_{t}^{k - 1} W_{u_{1}} (x, c t) = \int_{B_{c t} (x)} u_{1} (y) K (∣ y - x ∣, t) d y, K (ρ, t) : = c^{1 - n} D_{t}^{k - 1} [\frac{1}{n!! V_{n}} (c^{2} t^{2} - ρ^{2})^{- 1 / 2}]$ for $0 \leq ρ < c t$ , where $K$ is smooth on that region and $K (0, t) = a t^{- (n - 1)}$ with $a = c^{- n} (- 1)^{k - 1} (2 k - 3)!! / (n!! V_{n}) \neq 0$ (read $(2 k - 3)!! = 1$ for $k = 1$ ); consequently some data supported strictly inside $B_{c t} (x)$ give a nonzero velocity contribution, so the kernel fills the interior of the ball rather than sitting on the sphere. For displacement data also supported in a compact subset of the open ball, the displacement term has the corresponding kernel $\partial_{t} K (ρ, t)$ .

Facts & Assumptions

Given: Countable Choice, $c > 0$ , even $n = 2 k \geq 2$ , and velocity data $u_{1}$ supported in a compact subset of $B_{c t} (x)$ .

[F1]

The odd-dimensional formula expresses $u (x, t)$ as a finite combination of $t$ -derivatives of $t \mapsto t^{n - 2} M_{u_{0}} (x, c t)$ and $t \mapsto t^{n - 2} M_{u_{1}} (x, c t)$ (The odd-dimensional wave formula by iterated spherical means).

[F2]

For $m \geq 1$ and $h \in C^{m} (R^{n})$ , $(x, r) \mapsto M_{h} (x, r)$ is $C^{m}$ with all derivatives obtained by differentiating $h$ under the sphere integral (Smoothness, parity and zero-radius limits of spherical means).

[F3]

The even-dimensional formula reads $u = c^{1 - n} [\partial_{t} D_{t}^{k - 1} W_{u_{0}} (x, c t) + D_{t}^{k - 1} W_{u_{1}} (x, c t)]$ , with $W_{f} (x, c t) = (n!! V_{n})^{- 1} \int_{B_{c t} (x)} f (y) (c^{2} t^{2} - ∣ y - x ∣^{2})^{- 1 / 2} d y$ (The even-dimensional wave formula by descent, Spherical means and the weighted ball integral of space-dependent data).

[F4]

If $G (y, s)$ is continuous with continuous $\partial_{s} G$ on a compact rectangle, then $s \mapsto \int G (y, s) d y$ is $C^{1}$ with derivative $\int \partial_{s} G$ ; iterating gives the higher derivatives when they are continuous (Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral).

[F5]

Sums, constant multiples of differentiable functions are differentiable with the usual rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

1.1F1F2algebra

Part (i). By [F1] the value $u (x, t)$ is computed from the functions $r \mapsto M_{u_{0}} (x, r)$ and $r \mapsto M_{u_{1}} (x, r)$ and their $r$ -derivatives at $r = c t$ , and by [F2] these are obtained by differentiating the defining sphere integrals. For $r$ in a neighbourhood of $c t$ , $M_{f} (x, r)$ is the average of $f$ over $\partial B_{r} (x)$ , a sphere contained in the chosen neighbourhood of $\partial B_{c t} (x)$ ; hence all these quantities depend on $f$ only through its restriction to that neighbourhood, and so does $u (x, t)$ .

1.2F3F4algebra

Part (ii), kernel form. Fix $t > 0$ and let $u_{1}$ vanish on a neighbourhood of $\partial B_{c t} (x)$ ; the assumed compact support lies inside the open ball, so there is $ε \in (0, t)$ with $supp u_{1} \subseteq B_{c (t - ε)} (x)$ . For $∣ s - t ∣ < ε / 2$ the integrand $u_{1} (y) (c^{2} s^{2} - ∣ y - x ∣^{2})^{- 1 / 2}$ and all its $s$ -derivatives are continuous on a fixed box containing the support, with the product integrand extended by zero where $u_{1} = 0$ , for $s \in [t - ε / 2, t + ε / 2]$ , so by [F4] applied successively in each coordinate of the box, with Fubini (Fubini's theorem for L^1 functions on a sigma-finite product), the derivative $D_{t}^{k - 1}$ passes under the integral sign: $c^{1 - n} D_{t}^{k - 1} W_{u_{1}} (x, c t) = \int_{B_{c t} (x)} u_{1} (y) K (∣ y - x ∣, t) d y$ with $K (ρ, t) = c^{1 - n} D_{t}^{k - 1} [(n!! V_{n})^{- 1} (c^{2} t^{2} - ρ^{2})^{- 1 / 2}]$ . For displacement data $u_{0}$ with the same compact-interior support condition, one additional time differentiation under the fixed-box integral gives the kernel $\partial_{t} K (∣ y - x ∣, t)$ .

1.3F5algebra

The interior kernel. The identity $D_{t} (c^{2} t^{2} - ρ^{2})^{- p} = - 2 p c^{2} (c^{2} t^{2} - ρ^{2})^{- p - 1}$ gives by induction $K (ρ, t) = b (c^{2} t^{2} - ρ^{2})^{- (2 k - 1) / 2}$ , where $b = c^{1 - n} (- 1)^{k - 1} (2 k - 3)!! c^{2 k - 2} / (n!! V_{n}) \neq 0$ and $(- 1)!! = 1$ for $k = 1$ . Thus $K$ has a fixed nonzero sign throughout $0 \leq ρ < c t$ . In particular $K (0, t) = c^{- n} (- 1)^{k - 1} (2 k - 3)!! t^{- (n - 1)} / (n!! V_{n})$ , the stated value.

2.1F3step 1.2step 1.3algebra

Admissible interior data. Choose $0 < δ < c t / 2$ and, by A smooth bump between concentric Euclidean balls translated to centre $x$ , choose $u_{1} \in C_{c}^{\infty} (B_{2 δ} (x))$ with $0 \leq u_{1} \leq 1$ and $u_{1} = 1$ on ${\overline{B}}_{δ} (x)$ . This is admissible in every dimension here. The actual contribution is $\int_{B_{2 δ} (x)} u_{1} (y) K (∣ y - x ∣, t) d y$ , including the transition annulus. By step 1.3 its integrand has one sign and is strictly of that sign on the inner ball, of positive volume, so the contribution is nonzero. The same construction around any point strictly inside the ball shows that the interior kernel is nonzero throughout, rather than just at the centre.

3.1given∎

Collecting: in odd dimensions the value depends only on data near the sphere $\partial B_{c t} (x)$ (a statement about open neighbourhoods, not about pointwise traces), while in even dimensions the velocity kernel is the explicitly displayed smooth function of $ρ < c t$ , nonzero at the centre, so interior data contribute.

Depends on

Used by

A two-dimensional interior tail Example

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)
Sung-Jin Oh, Lecture Notes for Math 222A (UC Berkeley, 19 March 2024) (standard reference, not scraped)