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The spherical mean identity behind the wave equation

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the cited sphere, parameter, polar and divergence suppliers. Let $c > 0$ , let $f \in C^{\infty} (R^{3})$ (unbounded $f$ is allowed), and define the spherical mean

$M_{r} f (x) = \frac{1}{4 π} \int_{S^{2}} f (x + r n) d Ω (n), r \geq 0,$

with the outward sphere measure of total area $4 π$ (Surface integration on compact C1 hypersurfaces, Agreement with the existing polar sphere measure). Then $r \mapsto M_{r} f (x)$ is smooth for $r \geq 0$ and satisfies the Euler-Poisson-Darboux identity

$\partial_{r}^{2} M_{r} f (x) + \frac{2}{r} \partial_{r} M_{r} f (x) = Δ M_{r} f (x) (r > 0),$

and the function $H_{f} (x, t) = t M_{c t} f (x)$ , $t \geq 0$ , satisfies

$\partial_{t}^{2} H_{f} = c^{2} Δ H_{f}, H_{f} (x, 0) = 0, \partial_{t} H_{f} (x, 0) = f (x) .$

All identities are local and need no decay of $f$ , since every integration set is compact. This is the pure spherical-mean identity; it makes no claim about finite-domain boundary data or about general rough data.

Facts & Assumptions

Given: The Axiom of Countable Choice ${A C}_{ω}$ ; a constant $c > 0$ , a function $f \in C^{\infty} (R^{3})$ , and the spherical mean $M_{r} f (x)$ over the unit sphere with chart/polar measure $d Ω$ of total mass $4 π$ .

[F1]

Parameter integrals over the sphere. Differentiating a compact surface integral in a parameter is legitimate when the integrand is smooth and all derivatives are continuous on the compact parameter and surface sets; the parameter-integral lemma supplies the surface-measure form and the interchange of the parameter derivative with the integral (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface, Surface integration on compact C1 hypersurfaces).

[F2]

Divergence theorem and the Laplacian. On a bounded piecewise $C^{1}$ domain with outward normal, $\int_{Ω} div F = \int_{\partial Ω} F \cdot ν d S$ for $C^{1}$ fields (Divergence for finite piecewise C1 presentations); the Laplacian is $Δ f = \sum_{i} \partial_{i} \partial_{i} f$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field) and $\int_{B_{r} (x)} Δ f d y = \int_{S_{r} (x)} \partial_{ν} f d S$ .

[F3]

Radial differentiation and polar coordinates. The ball integral $J (r) = \int_{B_{r} (x)} g d y$ of a continuous $g$ has derivative $J^{'} (r) = \int_{S_{r} (x)} g d S$ , by polar integration and the one-variable fundamental theorem (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma); on $S_{r} (x)$ the outward normal is $ν = (y - x) / r$ and the substitution $y = x + r n$ gives $d S = r^{2} d Ω (n)$ with the chart/polar measure (Agreement with the existing polar sphere measure).

[F4]

Calculus rules. The product, chain and algebra rules for smooth functions (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 $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F5]

Choice accounting. The Axiom of Countable Choice states that every at most countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )). It is assumed for the sphere/parameter conventions in [F1], the divergence theorem in [F2], and the polar/surface measure suppliers in [F3]. The elementary differentiation in [F4] uses no further choice.

Proof

technique · direct

1.1givenF1F4F5algebra

For fixed $x$ the function $r \mapsto f (x + r n)$ is smooth on the compact sphere and the sphere measure is finite, so [F1] permits differentiation under the integral sign and gives $\partial_{r} M_{r} f (x) = \frac{1}{4 π} \int_{S^{2}} n \cdot \nabla f (x + r n) d Ω (n)$ , with all higher $r$ -derivatives obtained by iteration.

2.1step 1.1F2F3F5algebra

For $r > 0$ apply the divergence theorem [F2] on $B_{r} (x)$ to the $C^{1}$ field $\nabla f$ : $\int_{B_{r} (x)} Δ f (y) d y = \int_{S_{r} (x)} \partial_{ν} f d S$ ; substituting $y = x + r n$ with outward normal $n$ and $d S = r^{2} d Ω$ by [F3] gives $\int_{B_{r} (x)} Δ f = 4 π r^{2} \partial_{r} M_{r} f (x)$ .

3.1step 1.1step 2.1F1F2F3F4F5algebra

Differentiate the identity of step 2.1 in $r$ : by [F3] the derivative of the left side is $\int_{S_{r} (x)} Δ f d S = r^{2} \int_{S^{2}} Δ f (x + r n) d Ω (n) = 4 π r^{2} M_{r} (Δ f) (x)$ , while the derivative of the right side is $4 π (2 r \partial_{r} M_{r} f + r^{2} \partial_{r}^{2} M_{r} f)$ ; moving the Laplacian through the sphere integral by [F1] gives $M_{r} (Δ f) = Δ M_{r} f$ , so dividing by $4 π r^{2}$ yields $\partial_{r}^{2} M_{r} f + \frac{2}{r} \partial_{r} M_{r} f = Δ M_{r} f$ for $r > 0$ .

4.1step 3.1F4algebra

Set $H_{f} (x, t) = t M_{c t} f (x)$ for $t > 0$ ; differentiating twice with [F4] and the chain rule gives $\partial_{t} H_{f} = M_{c t} f + c t \partial_{r} M_{c t} f$ and $\partial_{t}^{2} H_{f} = 2 c \partial_{r} M_{c t} f + c^{2} t \partial_{r}^{2} M_{c t} f$ ; substituting step 3.1 at $r = c t$ cancels the terms with $\partial_{r} M_{c t} f$ and leaves $\partial_{t}^{2} H_{f} = c^{2} t Δ M_{c t} f = c^{2} Δ H_{f}$ .

5.1step 4.1F1F3algebra

At $t = 0$ one has $M_{0} f (x) = \frac{1}{4 π} \int_{S^{2}} f (x) d Ω = f (x)$ , and $\partial_{r} M_{0} f (x) = \frac{1}{4 π} \int_{S^{2}} n \cdot \nabla f (x) d Ω = 0$ because the chart/polar measure is invariant under $n \mapsto - n$ while $n \mapsto n \cdot \nabla f (x)$ is odd (Agreement with the existing polar sphere measure); hence $H_{f} (x, 0) = 0$ and $\partial_{t} H_{f} (x, 0) = f (x)$ , while the displayed equation at $t = 0$ reduces to $2 c \partial_{r} M_{0} f = 0 = c^{2} \cdot 0$ .

6.1step 4.1step 5.1F1F2F3F5algebra∎

The stated Countable Choice premise is used only through [F1]-[F3] as recorded in [F5]; no choice-free proof is claimed. Every identity was proved on compact parameter and spatial sets with no decay of $f$ used, and the defining integral is smooth in signed $r$ as well, so the equation, the initial value and the initial velocity extend to $t = 0$ ; this proves the stated spherical-mean identity.

Depends on

Dependency tree · 2 levels

Sources

J. K. Hunter, Notes on Partial Differential Equations (University of California, Davis; revised 6/18/2014), Chapter 7 (the wave equation) and Chapter 2.1 (mean values)