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The radial recursion between dimensions n and n+2

Statement

Let $c > 0$ , $n \geq 1$ and let $w$ be a $C^{3}$ function of $(r, t)$ on a domain with $r > 0$ . Write $L_{m} w : = w_{t t} - c^{2} (w_{r r} + \frac{m - 1}{r} w_{r})$ for the radial $m$ -dimensional wave operator and $Δ_{m} g : = g_{r r} + \frac{m - 1}{r} g_{r}$ for the radial Laplacian, in the sign convention of the wave operator of Wave equation, Cauchy data and wave speed. Then $L_{n + 2} [\frac{1}{r} \partial_{r} w] = \frac{1}{r} \partial_{r} [L_{n} w] (r > 0) .$ Consequently $w \mapsto r^{- 1} \partial_{r} w$ maps radial classical solutions of the $m$ -dimensional homogeneous wave equation to radial classical solutions of the $(m + 2)$ -dimensional one; at $m = 1$ this is the correspondence between one-dimensional waves and radial three-dimensional waves. It is the mechanism by which the kernels in dimension $n + 2$ are radial derivatives of the kernels in dimension $n$ .

Facts & Assumptions

Given: a speed $c > 0$ , an integer $n \geq 1$ , and a $C^{3}$ function $w$ of $(r, t)$ on a domain with $r > 0$ .

[F1]

Sums, products, quotients of differentiable functions are differentiable with the usual product, quotient 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 $g (c) \neq 0$ ).

[F2]

If $f$ is $C^{2}$ on an open subset of $R^{m}$ , then $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ for every pair of coordinate indices (Clairaut--Schwarz theorem for continuous second partial derivatives).

Proof

1.1F1algebra

Put $h : = r^{- 1} w_{r}$ . Differentiating the product and quotient, $h_{r} = r^{- 1} w_{r r} - r^{- 2} w_{r}$ and $h_{r r} = r^{- 1} w_{r r r} - 2 r^{- 2} w_{r r} + 2 r^{- 3} w_{r}$ , so the radial $(n + 2)$ -dimensional Laplacian of $h$ is $Δ_{n + 2} h = h_{r r} + \frac{n + 1}{r} h_{r} = r^{- 1} w_{r r r} + \frac{n - 1}{r^{2}} w_{r r} - \frac{n - 1}{r^{3}} w_{r}$ .

1.2F1algebra

On the other hand $Δ_{n} w = w_{r r} + \frac{n - 1}{r} w_{r}$ , whence $\partial_{r} (Δ_{n} w) = w_{r r r} + \frac{n - 1}{r} w_{r r} - \frac{n - 1}{r^{2}} w_{r}$ and $r^{- 1} \partial_{r} (Δ_{n} w) = r^{- 1} w_{r r r} + \frac{n - 1}{r^{2}} w_{r r} - \frac{n - 1}{r^{3}} w_{r}$ , the same expression as the radial Laplacian of the previous step.

2.1F2algebra∎

Since $w$ is $C^{3}$ , [F2] gives $w_{r t t} = w_{t t r}$ , so time differentiation commutes with $r^{- 1} \partial_{r}$ : $h_{t t} = (r^{- 1} w_{r})_{t t} = r^{- 1} \partial_{r} (w_{t t})$ . Subtracting $c^{2}$ times the identity $Δ_{n + 2} h = r^{- 1} \partial_{r} (Δ_{n} w)$ from this equality gives $L_{n + 2} h = h_{t t} - c^{2} Δ_{n + 2} h = r^{- 1} \partial_{r} (w_{t t}) - c^{2} r^{- 1} \partial_{r} (Δ_{n} w) = r^{- 1} \partial_{r} (w_{t t} - c^{2} Δ_{n} w) = r^{- 1} \partial_{r} (L_{n} w)$ ; if in particular $L_{n} w = 0$ then $L_{n + 2} (r^{- 1} w_{r}) = 0$ , which is the stated mapping of radial solutions.

Depends on

Used by

Dependency tree · two levels

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Sources

Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)