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The even-dimensional wave formula by descent

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n = 2 k \geq 2$ be even, $c > 0$ , $u_{0} \in C^{k + 2} (R^{n})$ , $u_{1} \in C^{k + 1} (R^{n})$ , let $W$ be the weighted ball integral of Spherical means and the weighted ball integral of space-dependent data and put $D_{t} = t^{- 1} \partial_{t}$ . Then $u (x, t) : = c^{1 - n} [\frac{\partial}{\partial t} D_{t}^{k - 1} W_{u_{0}} (x, c t) + D_{t}^{k - 1} W_{u_{1}} (x, c t)]$ defines a $C^{2}$ function on $R^{n} \times (0, \infty)$ solving $u_{t t} = c^{2} Δ u$ . For $n = 2$ ( $k = 1$ ) this is exactly Poisson's formula Poisson's formula in two dimensions by descent, and the factor $c^{1 - n}$ is the exact rescaling of the unit-speed formula obtained by substituting $s = c t$ in the $(n + 1)$ -dimensional odd-dimensional formula.

Facts & Assumptions

Given: Countable Choice, $n = 2 k \geq 2$ , $c > 0$ , $u_{0} \in C^{k + 2} (R^{n})$ , $u_{1} \in C^{k + 1} (R^{n})$ , and the cylindrical extensions $U_{j} (ξ, z) : = u_{j} (ξ)$ to $R^{n + 1}$ .

[F1]

For odd $m = 2 k + 1$ and data in $C^{k + 2}$ respectively $C^{k + 1}$ , the formula of The odd-dimensional wave formula by iterated spherical means with $D_{t} = t^{- 1} \partial_{t}$ defines a $C^{2}$ solution of $v_{t t} = c^{2} Δ v$ on $R^{m} \times (0, \infty)$ .

[F2]

For even $n$ , $t^{n - 1} M_{U}^{(n + 1)} ((x, 0), c t) = \frac{(n - 1)!!}{c^{n - 1}} W_{u} (x, c t)$ for the cylindrical extension of $u$ , where $M^{(n + 1)}$ is the spherical mean in $n + 1$ variables (Sphere integrals of a cylindrical function project to weighted ball integrals).

[F3]

For $n = 2$ and $k = 1$ the formula reduces to Poisson's formula Poisson's formula in two dimensions by descent with $W_{f} (x, c t) = \frac{1}{2 π} \int_{B_{c t} (x)} f (y) (c^{2} t^{2} - ∣ y - x ∣^{2})^{- 1 / 2} d y$ (Spherical means and the weighted ball integral of space-dependent data).

[F4]

Sums, products, constant multiples of differentiable functions are differentiable with the usual rules, and constants commute with differentiation (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

1.1F1algebra

Descent. The number $n + 1 = 2 k + 1$ is odd, and $U_{0} \in C^{k + 2} (R^{n + 1})$ , $U_{1} \in C^{k + 1} (R^{n + 1})$ , so [F1] applies in dimension $n + 1$ with the same $k$ : $V (ξ, z, t) : = \frac{1}{(n - 1)!!} [\partial_{t} D_{t}^{k - 1} (t^{n - 1} M_{U_{0}}^{(n + 1)} ((ξ, z), c t)) + D_{t}^{k - 1} (t^{n - 1} M_{U_{1}}^{(n + 1)} ((ξ, z), c t))]$ is $C^{2}$ on $R^{n + 1} \times (0, \infty)$ with $V_{t t} = c^{2} Δ_{n + 1} V$ . Since $U_{j} (ξ + c t ω, z + c t ω_{n + 1}) = u_{j} (ξ + c t ω)$ is independent of $z$ , the means of $U_{0}, U_{1}$ at centre $(ξ, z)$ do not depend on $z$ ; hence $V$ does not depend on $z$ , so $\partial_{z}^{2} V = 0$ , $Δ_{n + 1} V = Δ_{n} V$ , and the restriction $u (x, t) : = V (x, 0, t)$ is a $C^{2}$ function on $R^{n} \times (0, \infty)$ with $u_{t t} = c^{2} Δ_{n} u$ .

1.2F2F4algebra

Rewriting with the weighted ball integral. By [F2] with the cylindrical extensions, $t^{n - 1} M_{U_{j}}^{(n + 1)} ((x, 0), c t) = \frac{(n - 1)!!}{c^{n - 1}} W_{u_{j}} (x, c t)$ for $j = 0, 1$ ; substituting into the definition of $V$ and using [F4] to move the constant $\frac{(n - 1)!!}{c^{n - 1}}$ and the factor $\frac{1}{(n - 1)!!}$ through the $t$ -derivatives gives $u (x, t) = c^{1 - n} [\partial_{t} D_{t}^{k - 1} W_{u_{0}} (x, c t) + D_{t}^{k - 1} W_{u_{1}} (x, c t)]$ .

1.3F3algebra

The two-dimensional case. For $n = 2$ , $k = 1$ and $c^{1 - n} = c^{- 1}$ , so the formula reads $c^{- 1} [\partial_{t} W_{u_{0}} (x, c t) + W_{u_{1}} (x, c t)]$ , which by [F3] is exactly the Poisson expression of Poisson's formula in two dimensions by descent.

2.1given∎

Therefore the displayed even-dimensional formula defines a $C^{2}$ solution of the homogeneous wave equation, it reduces to Poisson's formula when $n = 2$ , and its prefactor $c^{1 - n}$ is the one produced by substituting $s = c t$ in the $(n + 1)$ -dimensional odd formula.

Depends on

Used by

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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)