NOTATION:
The symbol $$\boldsymbol{-\kern-11.2pt}\int$$ Is used to mean "the average value of", e.g for a $k$ dimensional subset $E$ of $\mathbb R^m$ , $$\boldsymbol{-\kern-11.2pt}\int_E f~\mathrm d\mu^k=\frac{1}{\mu^{k}(E)}\int_E f~\mathrm d\mu^k$$ Where $\mu^k$ is an appropriate measure (most commonly, and in this case, the standard Lesbesgue measure).
Actual start of my post.
Hi,
As mentioned in some of my previous posts I am currently reading "Partial Differential Equations" by Lawrence C Evans. We are currently focused on solving the wave equation Cauchy problem in $\mathbb R^m$, $$\begin{cases}\partial_t^2u-\Delta u=0 & \text{in}~\mathbb R^m\times (0,\infty) \\ u=g,~\partial_t u=h & \text{on}~\mathbb R^m\times \{0\}\end{cases}$$
This is a PDE in $m+1$ variables $(t,x_1,\dots,x_m)$. By introducing the "spherical averages"
$$U(x,r,t):=\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}u(y,t)~\mathrm d^{m-1}y \\ G(x,r):=\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}g(y)~\mathrm d^{m-1}y \\ H(x,r):=\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}h(y)~\mathrm d^{m-1}y$$
One can now show that $U$ obeys Euler-Poisson-Darboux equation:
$$\begin{cases}\partial_t^2 U-\partial_r^2 U -\frac{m-1}{r}\partial_r U=0 & \text{in}~\mathbb R_+\times (0,\infty) \\ U=G, ~\partial_t U=H & \text{on}~\mathbb R_+\times \{0\}\end{cases}$$
This is useful as we have reduced an equation in $m+1$ variables to one of just $2$ variables. One can notice that this is just the wave equation in spherical coordinates in $\mathbb R^m$ with no dependence on the angular coordinates $\theta_1,\dots,\theta_{m-1}$. Our original function $u$ can be recovered by the limit $u(x,t)=\lim_{r\to 0^+}U(x,r,t).$
When our number of dimensions is odd, i.e $m=2k+1$ where $k\geq 1$, we introduce the transformation
$$\tilde{U}(x,r,t)=\left(\frac{1}{r}\frac{\partial}{\partial r}\right)^{k-1}\big(r^{2k-1}U(x,r,t)\big)$$ Similarly to construct $\tilde{G}$ and $\tilde{H}$. Evans then claims that that this new function $\tilde U$ obeys the one dimensional wave equation on the half line with homogeneous Dirichlet BC at the origin (which is useful because we've already constructed a solution for that): $$\begin{cases}\partial_t^2\tilde{U}-\partial_r^2\tilde{U}=0 & \text{in}~\mathbb R_+\times (0,\infty) &(\text{i})\\ \tilde{U}=\tilde{G}, \partial_t\tilde{U}=\tilde{H} & \text{on}~\mathbb R_+\times \{0\} &\text{(ii)} \\ \tilde{U}=0 & \text{on}~\{0\}\times (0,\infty)&\text{(iii)}\end{cases}$$
$(\text{ii})$ is obvious, and Evans explicitly verifies $(\text{i})$ so I have no problems with those. However, he does not offer much justification for $(\text{iii})$ and I'm having a hard time convincing myself it is true.
He does provide a lemma, for functions $\phi:\mathbb R\to \mathbb R$, $\phi: x\mapsto \phi(x)$ we have
$$\left(\frac{1}{x}\frac{\mathrm d}{\mathrm dx}\right)^{k-1}\big(x^{2k-1}\phi(x)\big)=\sum_{\ell=0}^{k-1}\beta_{k,\ell}x^{\ell+1}\phi^{(\ell)}(x)$$ Where $\beta_{k,\ell}$ are constants. This means that $$\tilde{U}(x,r,t)=\sum_{\ell=0}^{k-1}\beta_{k,\ell} r^{\ell+1}\partial_r^\ell U(x,r,t)\tag{*}$$
So, in order to verify that $\lim_{r\to 0^+}\tilde{U}(x,r,t)=0$ we need to verify that $\left|\lim_{r\to 0^+}\partial_r^\ell U(x,r,t)\right|<\infty~~~\forall \ell \leq k-1$.
Evans does show explicitly, for example, that $$U(x,r,t)=\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}u(y,t)\mathrm d^{m-1}y \\ \partial_r U(x,r,t)=\frac{r}{m}\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)}\Delta u(y,t)\mathrm d^m y \\ \partial_r^2 U(x,r,t)=\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}\Delta u(y,t)\mathrm d^{m-1}y+\left(\frac{1}{m}-1\right)\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)}\Delta u(y,t)\mathrm d^m y$$
From which we van verify $$\lim_{r\to 0^+}U(x,r,t)=u(x,t)<\infty \\ \lim_{r\to 0^+}\partial_r U(x,r,t)=0<\infty \\ \lim_{r\to 0^+}\partial_r^2 U(x,r,t)=\frac{1}{m}\Delta u(x,t)<\infty$$ However I'm having a hard time computing $\partial_r^\ell U(x,r,t)$ for general $\ell$. I was able to (correctly, I think!) work out the formulas
$$\frac{\partial}{\partial r}\left(\int_{\mathbb B(x,r)}\phi~\mathrm d\mu^m\right)=\int_{\partial\mathbb B(x,r)}\phi~\mathrm d\mu^{m-1}\tag{A} $$
$$ \frac{\partial}{\partial r}\left(\int_{\partial \mathbb B(x,r)}\phi~\mathrm d\mu^{m-1}\right)=\frac{m-1}{r}\int_{\partial \mathbb B(x,r)}\phi~\mathrm d\mu^{m-1}+\int_{\mathbb B(x,r)}\Delta \phi~\mathrm d\mu^m\tag{B}$$
Which allows us, using the formulas for the volume and surface area of an $m$ ball, to write (again, correctly, I think?!)
$$\frac{\partial}{\partial r}\left(\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)} \phi ~\mathrm d\mu^m\right)=\frac{m}{r}\left(\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}\phi~\mathrm d\mu^{m-1}-\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)}\phi~\mathrm d\mu^m\right)\tag{C}$$
$$\frac{\partial}{\partial r}\left(\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}\phi~\mathrm d\mu^{m-1}\right)=\frac{r}{m}\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)}\Delta \phi~\mathrm d\mu^m\tag{D}$$
However, when I try to use this to calculate e.g $\partial_r^3 U(x,r,t)$, I get $$\partial_r^3 U(x,r,t)=\frac{r}{m}\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)}\Delta^2 u(y,t)\mathrm d^m y+\frac{1-m}{r}\left(\boldsymbol{-\kern-11.2pt}\int_{\partial\mathbb B(x,r)}\Delta u(y,t)\mathrm d^{m-1}y-\boldsymbol{-\kern-11.2pt}\int_{\mathbb B(x,r)}\Delta u(y,t)\mathrm d^m y\right)$$
And since we have no information about $\Delta^2 u$, I don't know how to proceed, nor can I find any reasonable pattern to the derivatives w.r.t $r$ of $U$.
So:
MY QUESTIONS.
- Are the formulas $(A),(B),(C),(D)$ even correct?
- How can I verify that $\lim_{r\to 0^+}\tilde{U}(x,r,t)=0$ using $(*)$ ? If there is a pattern to the derivatives $\partial_r^\ell U(x,r,t)$, what is it?
UPDATE.
As I mentioned in the comments, Evans supposes that $u\in C^{k+1}(\mathbb R^m\times [0,\infty))$ beforehand, reconciling this problem. However, because solutions of the wave equation are not necessarily any smoother than $C^2$, I am troubled by this supposition. Is there any way to derive the representation formula $(31)$ given on page 76 of Evans's book without needing this smoothness assumption?