Em Wave Equation Prerequisites

Given: The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$; times $t_0<T$; a $C^{\mathrm{\infty }}$ function $h$ on an open set ${\mathbb{R}}^3\times J$ with $[t_0,T]\subset J$; and the surface measure $d\mathrm{\Omega }$ on the unit sphere $S^2\subset {\mathbb{R}}^3$ of total mass ${\int }_{S^2}d\mathrm{\Omega }=4\pi$.

A map is of class $C^k$ when every word iterated coordinate partial derivative of order at most $k$ exists and is continuous, and of class $C^{\mathrm{\infty }}$ when it is $C^k$ for every $k$; $D^{\alpha }f={\mathrm{\partial }}_0^{{\alpha }_0}\cdots {\mathrm{\partial }}_{m-1}^{{\alpha }_{m-1}}f$ is the canonical-order multi-index derivative, and for a coordinate word $w=(i_1,\dots ,i_r)$ write $D^w:={\mathrm{\partial }}_{i_r}\cdots {\mathrm{\partial }}_{i_1}$, with the empty word denoting the identity, and let $w\cdot j$ denote appending the coordinate $j$, so $D^{w\cdot j}={\mathrm{\partial }}_j{D}^w$; partial derivatives are the one-variable derivatives along coordinate lines ($C^k$ Euclidean maps and diffeomorphisms, $C^k$ maps and multi-index derivative notation in Euclidean space).

Chain rule: if $f$ is totally differentiable at $a$ and $g$ at $f(a)$, then $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

Finite sums, scalar multiples and products of $C^k$ Euclidean maps are $C^k$, and composites of composable $C^k$ maps are $C^k$, for every $k\in \mathbb{N}$ ($C^k$ Euclidean maps are closed under componentwise algebra and composition); sums, scalar multiples and products of differentiable real functions differentiate by the sum, scalar-multiple and product rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

If all first partial derivatives of $q$ exist on a neighbourhood of $a$ and are continuous at $a$, then $q$ is totally differentiable at $a$ with total derivative the Jacobian matrix; if $q$ is totally differentiable at $a$, then for every direction $u$ the directional derivative exists and $D_{u}q(a)=Dq(a)u={\sum }_j{\mathrm{\partial }}_{j}q(a)u_j$, which for scalar $q$ is $⟨\mathrm{\nabla }q(a),u⟩$ with the gradient $\mathrm{\nabla }q$ of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, A total derivative computes every directional derivative, and its matrix is the Jacobian, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

Parameter integrals: if $K\subseteq {\mathbb{R}}^n$ is compact, measurable and of finite Lebesgue measure, $a<b$, and $f$ and ${\mathrm{\partial }}_{t}f$ are continuous on a neighbourhood of $K\times [a,b]$, then $t\mapsto {\int }_{K}f(x,t)dx$ is differentiable on $[a,b]$ with derivative ${\int }_K{\mathrm{\partial }}_{t}f(x,t)dx$; the same conclusion holds, under the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention of Surface integration on compact C1 hypersurfaces, with surface measure on a fixed compact $C^1$ hypersurface in place of $K$; nothing is asserted for moving parameter sets or singularities crossing the integration set (Differentiation under the integral sign over a compact measurable set and over a compact $C^1$ surface).

On a compact embedded $C^1$ hypersurface $S$ the surface integral is defined by the finite chart formula with Gram density; for signed functions with ${\int }_S\mathrm{\mid }f\mathrm{\mid }dS<\mathrm{\infty }$ it is the difference of the integrals of the positive and negative parts; the notation of this item fixes ${\int }_{S^2}1d\mathrm{\Omega }=4\pi$ for the unit sphere $S^2=\{n\in {\mathbb{R}}^3:⟨n,n⟩=1\}$ (Surface integration on compact C1 hypersurfaces, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

For nonnegative measurable $f\le g$ one has $\int f\le \int g$, and $\int cf=c\int f$ for $c\ge 0$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

Mean value theorem: if $f:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$ and differentiable at every point of $(a,b)$, then there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$ (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 $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$).

Closed boxes in ${\mathbb{R}}^m$ are compact in the Euclidean metric, and a continuous real-valued function on a nonempty compact metric space attains a maximum and a minimum (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Cauchy-Schwarz: $\mathrm{\mid }{\sum }_j{u}_j{v}_j\mathrm{\mid }\le ({\sum }_j{u}_j^2{)}^{1\mathrm{/}2}({\sum }_j{v}_j^2{)}^{1\mathrm{/}2}$ for real vectors, the right-hand factors being Euclidean norms (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$: every at most countable family of nonempty sets has a choice function (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Given: The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$; a speed $c>0$; times $t_0<T$; data $a,b\in C^{\mathrm{\infty }}({\mathbb{R}}^3)$; a function $h$ of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times [t_0,T]$ in the extension sense, that is, the restriction of a $C^{\mathrm{\infty }}$ function on ${\mathbb{R}}^3\times J$ for some open interval $J\supset [t_0,T]$; the operators $W=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$ and ${\mathrm{\square }}_c={\mathrm{\partial }}_t^2-c^2\mathrm{\Delta }$; and the surface measure $d\mathrm{\Omega }$ on the unit sphere $S^2\subset {\mathbb{R}}^3$ of total mass $4\pi$.

A map is of class $C^k$ when every word iterated coordinate partial derivative of order at most $k$ exists and is continuous, and of class $C^{\mathrm{\infty }}$ when it is $C^k$ for every $k$; the Laplacian is $\mathrm{\Delta }f=\mathrm{div}\mathrm{\nabla }f={\sum }_{i<3}{\mathrm{\partial }}_i{\mathrm{\partial }}_{i}f$ ($C^k$ Euclidean maps and diffeomorphisms, The Laplacian of a $C^2$ function and of a $C^2$ vector field).

Finite sums, scalar multiples and products of $C^k$ maps are $C^k$ and composites of composable $C^k$ maps are $C^k$ ($C^k$ Euclidean maps are closed under componentwise algebra and composition); the chain rule holds for total derivatives and continuous first partial derivatives imply total differentiability (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative); mixed second partials of a $C^2$ function commute (Clairaut--Schwarz theorem for continuous second partial derivatives); and sums, scalar multiples and products of differentiable real functions differentiate by the usual rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

Spherical-mean family of a smooth spacetime function: with $h$ as given and $d\mathrm{\Omega }$ of total mass $4\pi$, the family $m(x,s,v):=\frac{1}{4\pi }{\int }_{S^2}h(x+sn,v)d\mathrm{\Omega }(n)$ is of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times \mathbb{R}\times [t_0,T]$ in the extension sense, and every word iterated partial derivative is obtained by differentiating the composed integrand under the sphere integral; the statement is local, assumes no decay, compact support or boundedness of $h$, and uses uniform bounds only on compact parameter sets (Joint smoothness of the spherical-mean family of a smooth spacetime function).

Spherical means of a fixed smooth datum: for $f\in C^{\mathrm{\infty }}({\mathbb{R}}^3)$ and $M_{r}f(x)=\frac{1}{4\pi }{\int }_{S^2}f(x+rn)d\mathrm{\Omega }(n)$, $r\ge 0$, the map $r\mapsto M_{r}f(x)$ is smooth for $r\ge 0$ and satisfies the Euler-Poisson-Darboux identity ${\mathrm{\partial }}_r^2{M}_{r}f+\frac{2}{r}{\mathrm{\partial }}_r{M}_{r}f=\mathrm{\Delta }{M}_{r}f$ for $r>0$, and $H_f(x,t):=t{M}_{ct}f(x)$, $t\ge 0$, satisfies ${\mathrm{\partial }}_t^2{H}_f=c^2\mathrm{\Delta }{H}_f$, $H_f(\cdot ,0)=0$ and ${\mathrm{\partial }}_t{H}_f(\cdot ,0)=f$; all identities are local and assume no decay of $f$ (The spherical mean identity behind the wave equation).

Parameter integrals: if $K\subseteq {\mathbb{R}}^n$ is compact, measurable and of finite Lebesgue measure, $a<b$, and $f,{\mathrm{\partial }}_{t}f$ are continuous on a neighbourhood of $K\times [a,b]$, then $t\mapsto {\int }_{K}f(x,t)dx$ is differentiable on $[a,b]$ with derivative ${\int }_K{\mathrm{\partial }}_{t}f(x,t)dx$; the same holds, under the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention of Surface integration on compact C1 hypersurfaces, with surface measure on a fixed compact $C^1$ hypersurface in place of $K$; moving parameter sets and singularities crossing the integration set are excluded (Differentiation under the integral sign over a compact measurable set and over a compact $C^1$ surface).

Differentiating an integral with moving endpoints: if $I$ is an open interval, $\alpha ,\beta \in C^1(I)$ with $\alpha <\beta$, $J$ is an open interval containing the closures of the intervals $[\alpha (t),\beta (t)]$ and $F:I\times J\to \mathbb{R}$ is continuous with continuous ${\mathrm{\partial }}_{t}F$, then $G(t)={\int }_{\alpha (t)}^{\beta (t)}F(t,y)dy$ is $C^1$ on $I$ with $G^{\mathrm{\prime }}(t)=F(t,\beta (t)){\beta }^{\mathrm{\prime }}(t)-F(t,\alpha (t)){\alpha }^{\mathrm{\prime }}(t)+{\int }_{\alpha (t)}^{\beta (t)}{\mathrm{\partial }}_{t}F(t,y)dy$ (Differentiating an integral with moving endpoints). Continuous integrands on nondegenerate compact intervals are Riemann integrable and, under ${\mathrm{A}\mathrm{C}}_{\omega }$, their Riemann and Lebesgue integrals agree (Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable, Borel Darboux integrands in finite dimension); thus this rule applies to the continuous Lebesgue integrands used here.

A continuous function on a compact metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous). For integrable functions, the Lebesgue integral is linear and $\mathrm{\mid }\int f\mathrm{\mid }\le \int \mathrm{\mid }f\mathrm{\mid }$ (The Lebesgue integral is linear on $L^1(\mu )$, The modulus of an integral is bounded by the integral of the modulus); the nonnegative integral is monotone (Monotonicity and nonnegative homogeneity of the nonnegative integral) and ${\lambda }_1([0,1])=1$ under ${\mathrm{A}\mathrm{C}}_{\omega }$ (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included). Consequently $\mathrm{\mid }{\int }_0^1(f-g)dz\mathrm{\mid }\le {\sup }_{z\in [0,1]}\mathrm{\mid }f(z)-g(z)\mathrm{\mid }$ for continuous $f,g$.

Closed boxes in ${\mathbb{R}}^m$ are compact in the Euclidean metric (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), and a continuous real-valued function on a nonempty compact metric space attains a maximum and a minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

The reflection $n\mapsto -n$ of $S^2$ preserves the surface measure $d\mathrm{\Omega }$ and ${\int }_{S^2}nd\mathrm{\Omega }(n)=0$; orthogonal transformations of ${\mathbb{R}}^3$ preserve the surface measure (Reflection invariance and vanishing first moment of the sphere measure, Agreement with the existing polar sphere measure).

For the Euclidean norm on ${\mathbb{R}}^m$, $\parallel u+v{\parallel }_2\le \parallel u{\parallel }_2+\parallel v{\parallel }_2$ and $\parallel \lambda u{\parallel }_2=\mathrm{\mid }\lambda \mathrm{\mid }\parallel u{\parallel }_2$, so also $\parallel u{\parallel }_2\ge \parallel u+v{\parallel }_2-\parallel v{\parallel }_2$; and $B_r(x_0)=\{x:\mathrm{\mid }x-x_0\mathrm{\mid }<r\}$, ${\bar{B}}_r(x_0)=\{x:\mathrm{\mid }x-x_0\mathrm{\mid }\le r\}$ are the open and closed Euclidean balls (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

Wave operator and Cauchy data: ${\mathrm{\square }}_c={\mathrm{\partial }}_t^2-c^2\mathrm{\Delta }$ and $W=c^{-2}{\mathrm{\partial }}_t^2-\mathrm{\Delta }$, so ${\mathrm{\square }}_c=c^{2}W$; a classical solution on a time slab is $C^2$ on the open slab and satisfies the equation there, and Cauchy data are the limits as $t\downarrow t_0$; the closed backward cone with vertex $(x_{\ast },t_{\ast })$ and speed $c$ is $K(x_{\ast },t_{\ast })=\{(y,v):t_0\le v\le t_{\ast },\mathrm{\mid }y-x_{\ast }\mathrm{\mid }\le c(t_{\ast }-v)\}$ with base ball ${\bar{B}}_{c(t_{\ast }-t_0)}(x_{\ast })$ (Wave equation, Cauchy data and wave speed, Forward and backward wave cones, domain of dependence and influence).

Local uniqueness and domain of dependence: under ${\mathrm{A}\mathrm{C}}_{\omega }$, if $u,v\in C^2$ solve ${\mathrm{\square }}_{c}u=f$ and ${\mathrm{\square }}_{c}v=g$ on a neighbourhood of the closed backward cone $K^{-}(x_0,t_0)$ with $t_0>0$, if $f=g$ on the cone and $u(\cdot ,0)=v(\cdot ,0)$, $u_t(\cdot ,0)=v_t(\cdot ,0)$ on the base ball $B_{c{t}_0}(x_0)$, then $u=v$ on $K^{-}(x_0,t_0)$ (Domain of dependence and local uniqueness).

The Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$: every at most countable family of nonempty sets has a choice function (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).