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The constructed classical solutions are locally determined by the Cauchy data

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ and let $u$ be a free solution constructed by the formulas of Kirchhoff's formula in three dimensions, Poisson's formula in two dimensions by descent, The odd-dimensional wave formula by iterated spherical means or The even-dimensional wave formula by descent from compactly supported admissible data. Then for every $(x, t)$ with $t > 0$ the value $u (x, t)$ is determined by the data restricted to ${\overline{B}}_{c t} (x)$ : two admissible data pairs agreeing there produce the same value at $(x, t)$ . For the forced solution of The forced three-dimensional version as a retarded potential, the value is determined by the source on the backward cone ${(y, s) : 0 \leq s \leq t, ∣ y - x ∣ \leq c (t - s)}$ ; two sources agreeing there produce the same value at $(x, t)$ .

Facts & Assumptions

Given: Countable Choice, $c > 0$ , a point $(x, t)$ with $t > 0$ , and two admissible configurations agreeing on the stated set.

[F1]

The four free formulas express $u (x, t)$ through the spherical means $M_{u_{j}} (x, c t)$ and their $r$ -derivatives (odd case) or through $W_{u_{j}} (x, c t)$ and its $t$ -derivatives with the substitution $y = x + c t z$ (even case), and the forced solution is the retarded potential (The odd-dimensional wave formula by iterated spherical means, The even-dimensional wave formula by descent, Kirchhoff's formula in three dimensions, Poisson's formula in two dimensions by descent, The forced three-dimensional version as a retarded potential).

[F2]

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

[F3]

For even $n$ , writing $w (z) = (1 - ∣ z ∣^{2})^{- 1 / 2}$ on $B_{1} \subset R^{n}$ gives $W_{h} (x, c t) = (n!! V_{n})^{- 1} (c t)^{n - 1} \int_{B_{1}} h (x + c t z) w (z) d z$ , and $w$ is integrable (Spherical means and the weighted ball integral of space-dependent data, A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

[F4]

Difference quotients converging pointwise almost everywhere under one integrable majorant have convergent integrals (Dominated convergence).

[F5]

A scalar function continuous on a closed interval and differentiable on its interior has a difference quotient equal to one of its derivatives on the interior (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F6]

Continuous derivatives are bounded on compact Euclidean balls (For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Proof

1.1F1F2F3F4F5F6algebra

Free case. Let $(u_{0}, u_{1})$ and $({\tilde{u}}_{0}, {\tilde{u}}_{1})$ be admissible data agreeing on ${\overline{B}}_{c t} (x)$ and put $δ_{j} : = u_{j} - {\tilde{u}}_{j}$ . Each $δ_{j}$ vanishes on the open ball, so all its derivatives through the orders in the formulas vanish on the closed ball by continuity. In the odd-dimensional formula, every sphere-average ingredient is an average of a derivative of $δ_{j}$ evaluated on $\partial B_{c t} (x)$ , hence is zero by [F1, F2]. For even $n$ , put $G_{δ_{j}} (t^{'}) : = \int_{B_{1}} δ_{j} (x + c t^{'} z) w (z) d z$ . Fix a compact interval $J$ about $t$ and a closed ball containing all $x + c t^{'} z$ for $t^{'} \in J$ , $∣ z ∣ \leq 1$ . For each derivative order $0 \leq m < k$ , the difference quotients in $t^{'}$ of $c^{m} D^{m} δ_{j} (x + c t^{'} z) [z, \dots, z] w (z)$ are bounded by $c^{m + 1} C_{m + 1} w (z)$ on $J$ , where $C_{m + 1}$ bounds the next derivative on that ball by [F5, F6]. This is an integrable majorant independent of $t^{'}$ ; [F4] therefore justifies differentiating under the integral successively through order $k$ . At $t^{'} = t$ , all integrand derivatives vanish because $x + c t z \in {\overline{B}}_{c t} (x)$ , so $G_{δ_{j}}^{(m)} (t) = 0$ for $m \leq k$ . By [F3] the even-formula terms are finite combinations of these derivatives and hence vanish. Thus replacing the data by $({\tilde{u}}_{0}, {\tilde{u}}_{1})$ changes no term of the formula and leaves $u (x, t)$ unchanged.

2.1F1given∎

Forced case and conclusion. The retarded potential of the forced three-dimensional formula is an integral of the source over the backward cone $∣ y - x ∣ \leq c (t - s)$ , $0 \leq s \leq t$ ; sources agreeing there give equal integrals, hence equal values at $(x, t)$ . This proves the local determination of the constructed solutions by the stated data or source; no uniqueness claim for arbitrary $C^{2}$ solutions is made, that being the energy statement of the wave-energy page.

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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)
John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)