\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 266, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/266\hfil Convergence of exterior solutions]
{Convergence of exterior solutions to radial Cauchy solutions for 
$\partial_t^2U-c^2\Delta U=0$}

\author[H. K. Jenssen, C. Tsikkou \hfil EJDE-2016/266\hfilneg]
{Helge Kristian Jenssen, Charis Tsikkou}

\address{Helge Kristian Jenssen\newline
Department of Mathematics,
Penn State University, University Park,
State College, PA 16802, USA}
\email{jenssen@math.psu.edu}

\address{Charis Tsikkou \newline
Department of Mathematics,
 West Virginia University,
Morgantown, WV 26506, USA}
\email{tsikkou@math.wvu.edu}


\thanks{Submitted July 22, 2016. Published September 30, 2016.}
\subjclass[2010]{35L05, 35L15, 35L20}
\keywords{Linear wave equation; Cauchy problem; radial solutions;
\hfill\break\indent  exterior solutions; Neumann and Dirichlet conditions}

\begin{abstract}
 Consider the Cauchy problem for the 3-D linear wave equation
 $\partial_t^2U-c^2\Delta U=0$ with radial initial data
 $U(0,x)=\Phi(x)=\varphi(|x|)$,
 $U_t(0,x)=\Psi(x)=\psi(|x|)$. A standard result states that $U$ belongs
 to $C([0,T];H^s(\mathbb{R}^3))$ whenever
 $(\Phi,\Psi)\in H^s\times H^{s-1}(\mathbb{R}^3)$.
 In this article we are interested in the question of how $U$ can be realized
 as a limit of solutions to initial-boundary value problems on the
 exterior of vanishing balls $B_\varepsilon$ about the origin. We note that,
 as the solutions we compare are defined on different domains, the answer
 is not an immediate consequence of $H^s$ well-posedness for the wave equation.

 We show how explicit solution formulae yield convergence and optimal
 regularity for the Cauchy solution via exterior solutions, when the latter are
 extended continuously as constants on $B_\varepsilon$ at each time.
 We establish that for $s=2$ the solution $U$ can be realized
 as an $H^2$-limit (uniformly in time) of exterior solutions on
 $\mathbb{R}^3\setminus B_\varepsilon$ satisfying vanishing Neumann conditions
 along $|x|=\varepsilon$, as $\varepsilon\downarrow 0$. Similarly for $s=1$:
 $U$ is then an $H^1$-limit of exterior solutions satisfying vanishing Dirichlet
 conditions along $|x|=\varepsilon$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}
\subsection*{Notation}
 We use the $\mathbb{R}^+ =(0,\infty)$ and $\mathbb{R}_0^+=[0,\infty)$.
 For function of time and spatial position, the time variable $t$
 is always listed first, and the spatial variable ($x$ or $r$) is listed last.
 We indicate by subscript ``rad'' that the functions under consideration
 are spherically symmetric, e.g.\ $H^2_{\rm rad}(\mathbb{R}^3)$ denotes the set
 of $H^2(\mathbb{R}^3)$-functions $\Phi$ with the property that
 $\Phi(x)=\varphi(|x|)$ for some function $\varphi:\mathbb{R}_0^+\to\mathbb{R}$. For a radial
 function we use the same symbol whether it is considered as a function
 of $x$ or of $r=|x|$.

 Throughout we fix $T>0$ and $c>0$ and set
\[
\square_{1+1}:=\partial_t^2-c^2\partial_r^2,
 \quad \square_{1+3}:=\partial_t^2-c^2\Delta,
\]
 where $\Delta$ is the 3-D Laplacian.
 The open ball of radius $r$ about the origin in $\mathbb{R}^3$ is denoted $B_r$.
 We write $\partial_r$ for the directional derivative in the (outward) radial direction
 while $\partial_i$ denotes $\partial_{x_i}$.
 Finally, for two functions $A(t)$ and $B(t)$ we write
\[
A(t)\lesssim B(t)
\]
 to mean that there is a number $C$, possibly depending
 on the time $T$, $c$, the fixed cutoff functions $\beta$ and $\chi$ (see
 \eqref{approx_neum_data_1}-\eqref{approx_neum_data_2} and
 \eqref{approx_dir_data_1}-\eqref{approx_dir_data_2}), as well as
 the initial data $\Phi$, $\Psi$, but independent of the vanishing radii
 $\varepsilon$, such that
\[
A(t)\leq C\cdot B(t)\quad\text{holds for all $t\in [0,T]$.}
\]

\section{Radial Cauchy solutions as limits of exterior solutions}\label{rad_solns}

Consider the Cauchy problem for the 3-D linear wave equation
with radial initial data:
\begin{equation} \label{eCP}
\begin{gathered}
 \square_{1+3}U=0 \quad \text{on $(0,T)\times\mathbb{R}^3$}\\
 U(0,x)=\Phi(x) \quad \text{on $\mathbb{R}^3$}\\
 U_t(0,x)=\Psi(x) \quad \text{on $\mathbb{R}^3$,}
\end{gathered}
\end{equation}
where
\[
\Phi\in H^s_{\rm rad}(\mathbb{R}^3), \quad
\Psi\in H^{s-1}_{\rm rad}(\mathbb{R}^3),
\]
with
\begin{equation}\label{rad_versns}
 \Phi(x)=\varphi(|x|)\quad \Psi(x)=\psi(|x|).
\end{equation}
Throughout we refer to the unique solution $U$ of \eqref{eCP}
 as the {\em Cauchy solution}.

In this work we consider how the radial Cauchy solution $U$ can be realized
as a limit of solutions to initial-boundary value problems posed on the
exterior of vanishing balls $B_\varepsilon$ ($\varepsilon\downarrow 0$)
about the origin.
The precise issue will be formulated below.
We shall consider exterior solutions satisfying either a vanishing Neumann
condition or a vanishing Dirichlet condition along $|x|=\varepsilon$.

It is well known that the Sobolev spaces $H^s$ provide a natural
setting for the Cauchy problem for the wave equation; see \cite{ra} and
\eqref{opt_reg_1}-\eqref{opt_reg_2} below.
The choice of space dimension $3$ is for convenience: it is particularly
easy to generate radial solutions in this case. Next, both the choice of
spaces for the initial data for \eqref{eCP}, as well as the boundary condition
imposed on the exterior solutions, will influence the convergence of
exterior solutions toward the Cauchy solution.
For the wave equation in $\mathbb{R}^3$ the different convergence behavior of
exterior Neumann and exterior Dirichlet solutions is brought out by
considering $H^2$ vs.\ $H^1$ initial data; see Remark \ref{1st_rmk} below.


The scheme of generating radial solutions to Cauchy problems as limits of
exterior solutions has been applied to a variety of evolutionary PDE problems;
see \cite{jt1} for references and discussion.
In our earlier work \cite{jt1} we used the 3-D wave equation to gauge
the effectiveness of this general scheme in a case where
``everything is known.'' In order that the results be relevant
to other (possibly nonlinear) problems, the analysis in \cite{jt1} deliberately
avoided any use of explicit solution formulae. Based on energy arguments
and strong convergence alone, it was found that the exterior solutions do
converge to the Cauchy solution as the balls vanish. However,
the arguments did not yield optimal information about the regularity of the
limiting Cauchy solution. Specifically,
for $s=2$ we obtained the Cauchy solution as a limit only in
$H^1$ (via exterior Neumann solutions) or in $L^2$ (via exterior Dirichlet
solutions). This is strictly less regularity than what is known to be the case, see
\eqref{opt_reg_2}.

Thus, in general, while limits of exterior solutions to evolutionary PDEs
may be used to establish existence for radial Cauchy problems, one should
not expect optimal regularity information about the Cauchy solution via this
approach.

On the other hand, for the particular case of the 3-D wave equation
with radial data, it is natural to ask what type of convergence we can
establish if we exploit solution formulae (for the Cauchy
solution as well as for the exterior solutions). The present work
addresses this question, and our findings are summarized
in Theorem \ref{main_result} below.

We stress that while \cite{jt1} dealt with the issue of using exterior
solutions as a stand-alone method for obtaining existence of radial
Cauchy problems, the setting for the present work is different.
We are now exploiting what is known about the solution of the
Cauchy solutions as well as exterior solutions for the radial 3-D
linear wave equation, and the only issue is how the former
solutions are approximated by the latter.

\begin{remark}\label{1st_rmk} \rm
 Before starting the detailed analysis we comment on a
 slightly subtle point. As recorded in our main result (Theorem
 \ref{main_result}), we
 establish $H^2$-regularity of the limiting Cauchy solution $U$
 when the initial data $(\Phi,\Psi)$ belong to $H^2\times H^1$,
 and $H^1$-regularity when the data belong to $H^1\times L^2$.
 This is as it should be according to
 \eqref{opt_reg_1}. Now, in the former case $U$
 is obtained as a limit of exterior Neumann solutions, while in the
 latter case it is obtained as a limit of exterior Dirichlet solutions.
 This raises a natural question: what regularity is obtained for
 $U$ in the case of $H^2\times H^1$-data, if we insist on
 approximating by exterior Dirichlet solutions?

 To answer this we need to specify how we compare the
 everywhere defined Cauchy solution $U$ to exterior solutions
 $U^\varepsilon$, which are defined only on the exterior domains
 $\mathbb{R}^3_\varepsilon:=\mathbb{R}^3\setminus B_\varepsilon$.
 There are at least two ways to do this: 
 \begin{itemize}
 \item[(a)] by calculating $\|U(t)-U^\varepsilon(t)\|_{H^s(\mathbb{R}^3_\varepsilon)}$;
 \item[(b)] by first defining a suitable extension $\tilde U^\varepsilon$ of
$U^\varepsilon$
 to all of $\mathbb{R}^3$, and then calculating
 $\|U(t)-\tilde U^\varepsilon(t)\|_{H^s(\mathbb{R}^3)}$.
 \end{itemize}
(When using
 exterior solutions to establish existence for \eqref{eCP} (as in \cite{jt1}),
 there is no such choice: one must produce approximations to
 $U$ that are everywhere defined.)
 With (b), which is what we do in this paper, the natural choice
 is to extend $U^\varepsilon(t)$ continuously as a constant on $B_\varepsilon$
 at each time. I.e., for
 exterior Dirichlet solutions, we let $\tilde U^\varepsilon(t,x)$ vanish identically on
 $B_\varepsilon$, while for exterior Neumann solutions its value there
 is that of $U^\varepsilon(t,x)$ along the $|x|=\varepsilon$.

 It turns out that regardless of whether we use (a) or (b) to compare
 the Cauchy solution to the exterior solutions, the answer to the question
 above is that we obtain only $H^1$-convergence when
 exterior Dirichlet solutions are used. In fact, for (b) this is immediate:
 the exterior Dirichlet solution $\tilde U^\varepsilon$ will typically 
have a nonzero radial derivative at $r=\varepsilon+$ so that its extension 
$\tilde U^\varepsilon$
 contains a ``kink'' along $|x|=\varepsilon$. Thus, second derivatives of
 $\tilde U^\varepsilon$ will typically contain a $\delta$-function along 
$|x|=\varepsilon$,
 and $\tilde U^\varepsilon$ does not even belong to $H^2(\mathbb{R}^3)$ 
in this case.
 For (a) it suffices to consider the situation at time zero.
 With $\Phi$ as above we consider smooth cutoffs $\Phi^\varepsilon$
 (see \eqref{approx_dir_data_1} below).
 A careful calculation, carried out in \cite{jt1}, shows that
 $\|\Phi-\Phi^\varepsilon\|_{H^2(\mathbb{R}^3_\varepsilon)}$ blows up as
 $\varepsilon\downarrow 0$.

 These remarks highlight the unsurprising but relevant fact that
 exterior Dirichlet solutions are more singular than exterior Neumann
 solutions; see \cite{jt1} for a discussion.
\end{remark}

The goal is to show that the Cauchy solution $U$ of \eqref{eCP} can be approximated,
uniformly on compact time intervals, in $H^2$-norm by suitably chosen
exterior Neumann solutions and in $H^1$-norm by
suitably chosen exterior Dirichlet solutions.

As indicated we shall use explicit solution formulae for both the Cauchy
problem \eqref{eCP} as well as for the exterior Neumann and Dirichlet problems.
These formulae for radial solutions are readily available in 3 dimensions
and exploits the fact that radial solutions of $\square_{1+3}U=0$
admit the representation
\[
U(t,x)=\frac{u(t,|x|)}{|x|}
\]
where $u(t,r)$ solves $\square_{1+1}u=0$ on the half-line $\mathbb{R}^+$.
(Exterior Neumann solutions require a little work to write down
explicitly; see \eqref{U_eps}.)

Of course, with the explicit formulae in place, it is a matter of
computation to analyze the required norm differences. However,
it is a rather involved computation since the formulae involve different
expressions in several different regions.
Also, the answers do not follow by appealing to well-posedness
for the wave equation (see \eqref{opt_reg_2} below): the Cauchy solution
and the exterior solutions are defined on different domains.
As noted above we opt to extend the exterior solutions to the interior
of the balls $B_\varepsilon$, before comparing them to the Cauchy solution.

Instead of a direct comparison we prefer to estimate the
$H^2$- and $H^1$-differences in question by employing the
natural energies for the wave equation. These energies will majorize
the $L^2$-distances of the first and second derivatives, and will
also provide an estimate on the $L^2$-distance of the functions
themselves.

There are two advantages of this approach: first, it is straightforward
to calculate the exact rates of change of the energies in question, and second,
these rates depend only on what takes place at or within radius $r=\varepsilon$.
The upshot is that it suffices to analyze fewer terms than required by
a direct approach. Finally, to estimate the rates of change of the relevant
energies we make use of the explicit solution formulae.


\section{Setup and statement of main result}

\subsection{Cauchy solution}
A standard result (see e.g.\ \cite{bjs,ra}) shows that the radial Cauchy
solution $U$ of \eqref{eCP} may be calculated explicitly by using the representation
\[
U(t,x)=\frac{u(t,|x|)}{|x|},
\]
where $u(t,r)$ solves the half-line problem (Half-line)
\begin{equation}\label{half_line_dir}
\begin{gathered}
 \square_{1+1}u =0 \quad \text{on $(0,T)\times\mathbb{R}^+$}\\
 u(0,r)=r\varphi(r) \quad \text{for $r\in\mathbb{R}^+$}\\
 u_t(0,r)=r\psi(r) \quad \text{for $r\in\mathbb{R}^+$}\\
 u(t,0)\equiv 0 \quad \text{for $t>0$,}
\end{gathered}
\end{equation}
where $\varphi$ and $\psi$ are as in \eqref{rad_versns}.
By using the d'Alembert formula for the half-line problem (see \cite{bjs})
we obtain that
\begin{equation}\label{U}
 U(t,r)= \begin{cases}
\frac{1}{2r}[ (ct+r) \varphi(ct+r)-(ct-r)\varphi(ct-r)]
  +\frac{1}{2cr}\int_{ct-r}^{ct+r} s\psi(s)\, ds \\
\quad\text{if $0\leq r\leq ct$} \\[4pt]
\frac{1}{2r}[ (r+ct) \varphi(r+ct)+(r-ct)\varphi(r-ct)]
  +\frac{1}{2cr}\int_{r-ct}^{r+ct} s\psi(s)\, ds. \\
\quad\text{if $r\geq ct$}
\end{cases}
\end{equation}
In addition to the solution formula \eqref{U} we shall also exploit the
following well-known stability property \cite{ra,sel}:
with data $\Phi\in H^s(\mathbb{R}^3)$ and $\Psi\in H^{s-1}(\mathbb{R}^3)$
(radial or not), the Cauchy problem \eqref{eCP} admits a unique solution
$U$ which satisfies
\begin{gather} \label{opt_reg_1}
 U\in C([0,T];H^s(\mathbb{R}^3))\cap C^1([0,T];H^{s-1}(\mathbb{R}^3)), \\
\label{opt_reg_2}
 \|U(t)\|_{H^s(\mathbb{R}^3)}+\|U_t(t)\|_{H^{s-1}(\mathbb{R}^3)}
 \leq C_T\left( \|\Phi\|_{H^s(\mathbb{R}^3)}+\|\Psi\|_{H^{s-1}(\mathbb{R}^3)}\right),
\end{gather}
for each $T>0$, where $C_T$ is a number of the form
$C_T=\bar C\cdot(1+T)$, and $\bar C$ a universal constant.

\subsection{Exterior solutions and their extensions}\label{ext_solns}

With $\Phi\in H^s_{\rm rad}(\mathbb{R}^3)$ and
$\Psi\in H^{s-1}_{\rm rad}(\mathbb{R}^3)$, $s=1$
or $2$, the goal is to show that the solution $U$ of \eqref{eCP} can be
``realized as a limit of exterior solutions'' defined outside of $B^\varepsilon$
as $\varepsilon\downarrow 0$.
To make this precise we need to specify:
\begin{enumerate}
 \item precisely which exterior solutions we consider: which boundary
 conditions do they satisfy along $\partial B_\varepsilon$, and how are their
 initial data related to the given Cauchy data $\Phi$, $\Psi$;
 \item how we compare the everywhere defined Cauchy solution $U$
 with exterior solutions $U^\varepsilon$, that are defined only outside of
 $B_\varepsilon$; and
 \item which norm we use for comparing $U$ and $U^\varepsilon$.
\end{enumerate}
Concerning (1) we shall consider exterior solutions that satisfy either
vanishing Neumann or vanishing Dirichlet conditions along $\partial B^\varepsilon$.
In either case, the initial data for the exterior problem are generated
by a two-step procedure: we first approximate the original
Cauchy data by $C^\infty_{c,\rm rad}(\mathbb{R}^3)$-functions,
and then apply an appropriate modification of these
smooth approximations near the origin. These modifications
use smooth cut-off functions and are made so that the result
satisfies vanishing Neumann or Dirichlet conditions along $|x|=\varepsilon$.
(See \eqref{approx_neum_data_1}-\eqref{approx_neum_data_2} and
\eqref{approx_dir_data_1}-\eqref{approx_dir_data_2} for details.)
In either case we denote the exterior, radial solutions corresponding to
the approximate, smooth data by $U^\varepsilon(t,x)\equiv U^\varepsilon(t,r)$;
they are given explicitly in \eqref{U_eps} and \eqref{U_eps_dir} below.

As mentioned in Remark \ref{1st_rmk}, for (2) we opt to compare
the Cauchy solution $U$ to the natural extensions $\tilde U^\varepsilon$ of
the smooth exterior solution $U^\varepsilon$: at each time $t$,
$\tilde U^\varepsilon(t,x)$ takes the constant value $U^\varepsilon(t,\varepsilon)$ on
$B_\varepsilon$, and coincides with $U^\varepsilon(t,x)$ for $|x|\geq \varepsilon$.
Thus, in the case of Dirichlet data, $\tilde U^\varepsilon(t,x)$ vanishes
identically on $B_\varepsilon$, while for Neumann data its value there
is that of $U^\varepsilon(t,x)$ along the boundary $|x|=\varepsilon$.

Finally, concerning (3), Remark \ref{1st_rmk} above also explains
the choice of $H^2$-norm for comparing the Cauchy solution $U$
to exterior Neumann solutions, and $H^1$-norm for comparison
to exterior Dirichlet solutions.
Our main result is as follows.

\begin{theorem}\label{main_result}
 Let $T>0$ be given and let $U$ denote the solution of the
 radial Cauchy problem \eqref{eCP} for the linear wave
 equation in three space dimensions with initial data $(\Phi,\Psi)$.
 \begin{itemize}
 \item[(i)] For initial data in $H^2_{\rm rad}(\mathbb{R}^3)\times H^1_{\rm rad}(\mathbb{R}^3)$ the
 Cauchy solution $U$ can be realized as a $C([0,T];H^2(\mathbb{R}^3))$-limit
 of suitable extended exterior Neumann solutions as
 $\varepsilon\downarrow 0$.
 \item[(ii)] For initial data in $H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ the
 Cauchy solution $U$ can be realized as a $C([0,T];H^1(\mathbb{R}^3))$-limit
 of suitable extended exterior Dirichlet solutions as
 $\varepsilon\downarrow 0$.
 \end{itemize}
\end{theorem}

We point out that, e.g.\ in part (i), we do not claim that the
extended Neumann solutions $\tilde U^\varepsilon$ converge to $U$
in $H^2$-norm. In fact, we establish this latter property only for
the case with $C^\infty_c(\mathbb{R}^3)$ initial data. However, thanks
to the stability property \eqref{opt_reg_2}, this is sufficient to obtain
(i); see Proposition \ref{smooth_case} below.

The rest of the paper is organized as follows. After reducing to
the case with smooth and compactly supported data in Section
\eqref{smooth_c}, we treat $H^2$-convergence of exterior Neumann
solutions in Sections \eqref{gen_neu_soln}-\eqref{comp_neu}, while
$H^1$-convergence of exterior Dirichlet solutions is established
in Sections \eqref{gen_dir_soln}-\eqref{comp_dir}.


\subsection{Reduction to smooth case}\label{smooth_c}
The first step of the proof is to use well-posedness \eqref{opt_reg_2}
for the Cauchy problem to reduce to the case of smooth initial data.

\begin{proposition}\label{smooth_case}
 With the setup in Theorem \ref{main_result},
 let $\tilde U^{N,\varepsilon}$ and $\tilde U^{D,\varepsilon}$ denote the
 extensions of the exterior Neumann and Dirichlet solutions,
 respectively, as described in Section \eqref{ext_solns}.
 Then, Theorem \ref{main_result} follows once it is established that
 \begin{gather} \label{smooth_case1}
 \sup_{0\leq t\leq T}\|U(t)- \tilde U^{N,\varepsilon}(t)\|_{H^2(\mathbb{R}^3)}\to 0
 \quad\text{as $\varepsilon\downarrow 0$}, \\
\label{smooth_case2}
 \sup_{0\leq t\leq T}\|U(t)- \tilde U^{D,\varepsilon}(t)\|_{H^1(\mathbb{R}^3)}\to 0
 \quad\text{as $\varepsilon\downarrow 0$,}
 \end{gather}
for any initial data $\Phi,\Psi\in C_{c,\rm rad}^\infty(\mathbb{R}^3)$.
\end{proposition}

\begin{proof}
 For concreteness consider the case of exterior
 Neumann solutions, and let arbitrary data $\Phi\in H^2_{\rm rad}(\mathbb{R}^3)$,
 $\Psi\in H^1_{\rm rad}(\mathbb{R}^3)$ be given. Fix any $\delta>0$.
 We first choose $\Phi_0$, $\Psi_0$ in $C_{c,\rm rad}^\infty(\mathbb{R}^3)$ with
\[
\|\Phi-\Phi_0\|_{H^2}+ \|\Psi-\Psi_0\|_{H^1}< \frac{\delta}{2C_T},
\]
 where $C_T$ is as in \eqref{opt_reg_2}.
 The existence of such $\Phi_0$, $\Psi_0$ may be established in a
 standard manner via convolution (using a radial mollifier)
 and smooth cutoff at large radii. Let $U_0$ denote the solution of
 \eqref{eCP} with data $\Phi_0$, $\Psi_0$. Also, for any $\varepsilon>0$
 let $\tilde U^{N,\varepsilon}_0(t,x)$ denote the extension of the exterior Neumann
 solution with data $\Phi_0^\varepsilon$, $\Psi_0^\varepsilon$, as described in
 Section \eqref{ext_solns}.
 Then, assuming that \eqref{smooth_case1} has been
 established, we can choose $\varepsilon>0$
 sufficiently small to guarantee that
\[
\sup_{0\leq t\leq T}\|U_0(t)- \tilde U^{N,\varepsilon}_0(t)\|_{H^2}
<\frac{\delta}{2}.
\]
 Hence, for any $t\in[0,T]$ we have
 \begin{align*}
 \|U(t)- \tilde U^{N,\varepsilon}_0(t)\|_{H^2}
 &\leq \|U(t)- U_0(t)\|_{H^2}+\|U_0(t)- \tilde U^{N,\varepsilon}_0(t)\|_{H^2}\\
 &\overset{\eqref{opt_reg_2}}{\leq}
 C_T\left(\|\Phi-\Phi_0\|_{H^2}+ \|\Psi-\Psi_0\|_{H^1}\right)+\frac{\delta}{2}
 <\delta,
 \end{align*}
 by the choice of $\Phi_0$, $\Psi_0$.
\end{proof}

From now on we therefore consider an arbitrary but fixed pair of
functions $\Phi,\, \Psi\in C^\infty_{c,\rm rad}(\mathbb{R}^3)$.
Note that we then have that the functions $\varphi$ and $\psi$
in \eqref{rad_versns} are smooth on $\mathbb{R}_0^+$ and satisfy
$\varphi'(0+)=\psi'(0+)=0$.

\section{Exterior Neumann solutions}\label{gen_neu_soln}

In this section and the next we consider the case of exterior Neumann
solutions.
For the fixed initial data $\Phi,\, \Psi\in C^\infty_{c,\rm rad}(\mathbb{R}^3)$ and
any $\varepsilon>0$ we derive a formula for
$U^\varepsilon(t,x)\equiv U^{N,\varepsilon}(t,x)$, defined for
$|x|\geq\varepsilon$ and satisfying $\partial_rU^\varepsilon|_{r=\varepsilon}=0$.
We refer to $U^\varepsilon$ as the {\em exterior Neumann solution} corresponding
to the solution $U$ of \eqref{eCP} with data $\Phi$, $\Psi$. In Section
\eqref{comp_neu} we will then estimate how it (really, its extension
$\tilde U^\varepsilon(t,x)$ to all of $\mathbb{R}^3$) approximates the solution
$U(t)$ in $H^2(\mathbb{R}^3)$ at fixed times.

To generate the exterior Neumann solution $U^\varepsilon$ we
fix a smooth, nondecreasing function $\beta:\mathbb{R}_0^+\to\mathbb{R}_0^+$ with
\begin{equation}\label{beta_props_1}
 \beta\equiv 1 \quad\text{on $[0,1]$,}\quad \beta(s)=s\quad\text{for $s\geq 2$.}
\end{equation}
Then, with $\varphi$ and $\psi$ as in \eqref{rad_versns}, we define
\begin{gather} \label{approx_neum_data_1}
 \Phi^\varepsilon(x)\equiv \varphi^\varepsilon(|x|):=\varphi\big(\varepsilon\beta
 \big(\textstyle\frac{|x|}{\varepsilon}\big)\big), \\
\label{approx_neum_data_2}
 \Psi^\varepsilon(x)\equiv \psi^\varepsilon(|x|):=\psi\big(\varepsilon\beta
 \big(\textstyle\frac{|x|}{\varepsilon}\big)\big).
\end{gather}
We refer to $(\Phi^\varepsilon,\Psi^\varepsilon)$ as the {\em Neumann data}
corresponding to the original Cauchy data $(\Phi,\Psi)$ for \eqref{eCP}.
Note that the Neumann data are actually defined on all of $\mathbb{R}^3$,
that they are constant (equal to $\varphi(\varepsilon)$ and $\psi(\varepsilon)$,
respectively) on $B_\varepsilon$, and that their restrictions to the
exterior domain $\{x\in\mathbb{R}^3: |x|\geq\varepsilon\}$
satisfy homogeneous Neumann conditions along $|x|=\varepsilon$.

The exterior Neumann solution $U^\varepsilon$ is now defined as
the unique radial solution of the initial-boundary value problem
\begin{gather*}
 \square_{1+3}V =0 \quad \text{on $(0,T)\times\{|x|>\varepsilon\}$}\\
 V(0,x)=\Phi^\varepsilon(x) \quad \text{for $|x|>\varepsilon$}\\
 V_t(0,x)=\Psi^\varepsilon(x) \quad \text{for $|x|>\varepsilon$}\\
 \partial_r V(t,x)=0 \quad \text{along $|x|=\varepsilon$ for $t>0$.}
\end{gather*}

To obtain a formula for $U^\varepsilon$ we exploit the fact that $V$ is a radial
solution of the 3-D wave equation if and only if $v=rV$ solves the 1-D
 wave equation. Setting
\[
u^\varepsilon(t,r):=rU^\varepsilon(t,r),
\]
we obtain that $u^\varepsilon$ solves the corresponding 1-D problem on
$\{r>\varepsilon\}$: ($\varepsilon$-Half-line)
\begin{gather*}
 \square_{1+1}u=0 \quad \text{on $(0,T)\times\{r>\varepsilon\}$}\\
 u(0,r)=r\varphi^\varepsilon(r) \quad \text{for $r>\varepsilon$}\\
 u_t(0,r)=r\psi^\varepsilon(r) \quad \text{for $r>\varepsilon$}\\
 u_r(t,\varepsilon)= \frac{1}{\varepsilon}u(t,\varepsilon) \quad \text{for $t>0$.}
\end{gather*}
Note that the Neumann condition for the 3-D solution corresponds to a Robin
condition for the 1-D solution. (A direct calculation shows that the initial data
for $u^\varepsilon$ and $u_t^\varepsilon$ both satisfy this Robin condition.)

The solution $u^\varepsilon$ to the $\varepsilon$-Half-line problem is explicitly
given via d'Alembert's formula: 
\begin{equation}\label{u_eps}
 u^\varepsilon(t,r)
= \begin{cases}
\frac{1}{2}[ (ct+r) \varphi^\varepsilon(ct+r)
 +(ct-r+2\varepsilon)\varphi^\varepsilon(ct-r+2\varepsilon)]\\
 +\frac{1}{2c}\int_{ct-r+2\varepsilon}^{ct+r} s\psi^\varepsilon(s)\, ds \\
 +e^{\frac{r-ct-2\varepsilon}{\varepsilon}}
 \int_\varepsilon^{ct-r+2\varepsilon}[\frac{s\psi^\varepsilon(s)}{c}
 -\frac{s\varphi^\varepsilon(s)}{\varepsilon}]e^{s/\varepsilon}\, ds\\
\quad\text{if }  \varepsilon\leq r\leq ct+\varepsilon,
\\[4pt]
\frac{1}{2}[ (r+ct) \varphi^\varepsilon(r+ct)
 +(r-ct)\varphi^\varepsilon(r-ct)]\\
 +\frac{1}{2c}\int_{r-ct}^{r+ct} s\psi^\varepsilon(s)\, ds \\
\quad\text{if } r\geq ct+\varepsilon\,.
 \end{cases}
\end{equation}
(One way to solve the 1-Dimensional
Robin IBVP is to first solve the IBVP with general Dirichlet data
$u^\varepsilon(t,\varepsilon)=h(t)$ along $r=\varepsilon$, 
for which a d'Alembert formula is
readily available (see John \cite{bjs}, p.\ 8); one may then identify the $h$
which gives $u_r= \frac{1}{\varepsilon}u$ along $r=\varepsilon$.)
A direct calculation shows that $u^\varepsilon$ is a classical solution on
$\mathbb{R}_t\times \{r>\varepsilon\}$. From this we obtain the radial exterior
Neumann solution $U^\varepsilon(t,r):=\frac{u^\varepsilon(t,r)}{r}$:
\begin{equation}\label{U_eps}
 U^\varepsilon(t,r)
=\begin{cases}
\frac{1}{2r}[ (ct+r) \varphi^\varepsilon(ct+r)
 +(ct-r+2\varepsilon)\varphi^\varepsilon(ct-r+2\varepsilon)]\\
 +\frac{1}{2cr}\int_{ct-r+2\varepsilon}^{ct+r} s\psi^\varepsilon(s)\, ds\\
 +\frac{1}{r}e^{\frac{r-ct-2\varepsilon}{\varepsilon}}
 \int_\varepsilon^{ct-r+2\varepsilon}[\frac{s\psi^\varepsilon(s)}{c}
 -\frac{s\varphi^\varepsilon(s)}{\varepsilon}]
 e^{s/\varepsilon}\, ds\\
\quad\text{if }  \varepsilon\leq r\leq ct+\varepsilon \\[4pt]
\frac{1}{2r}[ (r+ct) \varphi^\varepsilon(r+ct)
 +(r-ct)\varphi^\varepsilon(r-ct)]\\
 +\frac{1}{2cr}\int_{r-ct}^{r+ct} s\psi^\varepsilon(s)\, ds \\
\quad\text{if }  r\geq ct+\varepsilon\,.
 \end{cases}
\end{equation}
We finally extend $U^\varepsilon$ at each time to obtain an everywhere
defined approximation of the Cauchy solution $U(t,x)$. As discussed
earlier we use the natural choice of extending $U^\varepsilon$ continuously
as a constant on $B_\varepsilon$ at each time:
\begin{equation}\label{tilde_U_eps}
 \tilde U^\varepsilon(t,x)
= \begin{cases}
 U^\varepsilon(t,\varepsilon) & \text{for $0\leq |x|\leq \varepsilon$}\\
 U^\varepsilon(t,x) & \text{for $|x|\geq \varepsilon$.}
 \end{cases}
\end{equation}
For later use we record that the value along the boundary is explicitly given as
\begin{equation}\label{U_eps_bndry_val}
 U^\varepsilon(t,\varepsilon)
=\frac{1}{\varepsilon}(ct+\varepsilon)\varphi^\varepsilon(ct+\varepsilon)
 +\frac{1}{\varepsilon}e^{-\frac{ct+\varepsilon}{\varepsilon}}
 \int_\varepsilon^{ct+\varepsilon}
[\frac{s\psi^\varepsilon(s)}{c}-\frac{s\varphi^\varepsilon(s)}{\varepsilon}]
 e^{s/\varepsilon}\, ds,
\end{equation}
and we also note that
\begin{equation}\label{init_approx_data}
 \tilde U^\varepsilon(0,x)=\Phi^\varepsilon(x),\quad
 \tilde U_t^\varepsilon(0,x)=\Psi^\varepsilon(x)\quad
\text{for all $x\in\mathbb{R}^3$.}
\end{equation}

\section{Comparing Cauchy and exterior Neumann solutions}\label{comp_neu}

The issue now is to estimate the $H^2$-distance
\[
\|U(t)-\tilde U^\varepsilon(t)\|_{H^2(\mathbb{R}^3)}
\]
as $\varepsilon\downarrow 0$. As explained in Section \eqref{rad_solns}
we prefer to estimate this $H^2$-difference by employing the
natural energies for the wave equation.
These energies will majorize the $L^2$-distances of the
first and second derivatives of $U(t)$ and $\tilde U^\varepsilon(t)$, and
also provide control of the $L^2$-distance of the functions themselves.

\subsection{Energies}\label{energies}

For any function $W(t,x)$ which is twice weakly differentiable on
$\mathbb{R}\times\mathbb{R}^3$ we define the following 1st and 2nd order energies
(note their domains of integration):
\begin{gather*}
\mathcal E_{W}(t):={\frac{1}{2}}\int_{\mathbb{R}^3}
|\partial_t W(t,x)|^2+c^2|\nabla W(t,x)|^2\, dx,\\
\mathcal E^\varepsilon_{W}(t):={\frac{1}{2}}\int_{|x|\geq \varepsilon}
|\partial_t W(t,x)|^2+c^2|\nabla W(t,x)|^2\, dx,
\end{gather*}
and
\begin{gather*}
\mathbb{E}_{W}(t):=\sum_{i=1}^3 \mathcal E_{\partial_i W}(t)
=\sum_{i=1}^3{\frac{1}{2}}\int_{\mathbb{R}^3}
|\partial_t \partial_i W(t,x)|^2 +c^2|\nabla \partial_i W(t,x)|^2\, dx,\\
\mathbb{E}^\varepsilon_{W}(t):=\sum_{i=1}^3 \mathcal E^\varepsilon_{\partial_i W}(t)
=\sum_{i=1}^3{\frac{1}{2}}\int_{|x|>\varepsilon}
|\partial_t \partial_i W(t,x)|^2 +c^2|\nabla \partial_i W(t,x)|^2\, dx.
\end{gather*}
The first goal is to estimate the energies
\begin{gather*} \label{ult_energ_1}
 \mathcal E^\varepsilon(t):=\mathcal E_{U-\tilde U^\varepsilon}(t), \\
\label{ult_energ_2}
 \mathbb{E}^\varepsilon(t):=\mathbb{E}_{U-\tilde U^\varepsilon}(t),
\end{gather*}
which majorizes the $L^2$-distances between the 1st and 2nd
derivatives of $U$ and $\tilde U^\varepsilon$, respectively.
As a first step we observe the following facts.

\begin{lemma}\label{energy_1}
 With $U$ and $U^\varepsilon$ as defined above we have: each of the energies
\[
\mathcal E_{U}(t),\quad \mathcal E^\varepsilon_{U^\varepsilon}(t),\quad
 \mathcal E_{\partial_i U}(t),\quad\text{and}\quad
\mathcal E^\varepsilon_{\partial_i U^\varepsilon}(t)
\]
 are constant in time.
\end{lemma}

\begin{proof}
 The constancy of the first three energies is standard, while the
 constancy of $\mathcal E^\varepsilon_{\partial_i U^\varepsilon}(t)$ is a consequence
 of the fact that we consider radial solutions. Indeed, as $U^\varepsilon$
 is radial and satisfies vanishing Neumann conditions along $|x|=\varepsilon$,
 we have that $\nabla U^\varepsilon(t,x)\equiv 0$ along $|x|=\varepsilon$. Thus,
 $U^\varepsilon_{x_it}\equiv 0$ for each $i=1,2,3$ along $|x|=\varepsilon$.
 Differentiating in time, using that $U^\varepsilon$ is a solution of the wave equation,
 and integrating by parts, we therefore have
 \begin{align*}
 \dot{\mathcal E}^\varepsilon_{\partial_i U^\varepsilon}(t)
 &= \int_{|x|>\varepsilon} U^\varepsilon_{x_it}U^\varepsilon_{x_itt}
 +c^2\nabla U^\varepsilon_{x_i}\cdot\nabla U^\varepsilon_{x_it}\, dx\\
 &=c^2\int_{|x|>\varepsilon} U^\varepsilon_{x_it}\Delta U^\varepsilon_{x_i}
 +\nabla U^\varepsilon_{x_i}\cdot\nabla U^\varepsilon_{x_it}\, dx \\
&= c^2\int_{\partial\{|x|>\varepsilon\}}U^\varepsilon_{x_it}
 \frac{\partial U^\varepsilon_{x_i}}{\partial\nu}\, dS=0.
 \end{align*}
\end{proof}

Next, to estimate $\mathcal E^\varepsilon(t)$, we expand the integrand and use that
$\nabla \tilde U^\varepsilon$ vanishes on $B_\varepsilon$ (by our choice of extension),
to get
\begin{align*}
 \mathcal E^\varepsilon(t)
&= \mathcal E_{U- \tilde U^\varepsilon}(t)
 =\frac{1}{2} \int_{\mathbb{R}^3} |U_t-\tilde U^\varepsilon_t|^2
 + c^2|\nabla U-\nabla \tilde U^\varepsilon|^2\, dx\\
 &=\mathcal E_U(t)+\mathcal E_{\tilde U^\varepsilon}(t)
 -\int_{\mathbb{R}^3} U_t\tilde U^\varepsilon_t
 +c^2\nabla U\cdot\nabla \tilde U^\varepsilon\, dx\\
 &= \mathcal E_U(t)+\mathcal E^\varepsilon_{U^\varepsilon}(t)
 +\frac{\text{vol}(B_\varepsilon)}{2} |U^\varepsilon_t(t,\varepsilon)|^2
 -U^\varepsilon_t(t,\varepsilon)\int_{|x|<\varepsilon} U_t(t,x)\, dx\\
 &\quad-\int_{|x|>\varepsilon} U_tU^\varepsilon_t
 +c^2\nabla U\cdot\nabla U^\varepsilon\, dx.
\end{align*}
Differentiating in time, applying Lemma \ref{energy_1}, integrating by parts,
and using the boundary condition $\partial_r U^\varepsilon(t,\varepsilon)\equiv 0$,
then yield
\[
\dot{\mathcal E}^\varepsilon(t)
= \frac{d}{dt}\Big[\frac{\text{vol}(B_\varepsilon)}{2}
 |U^\varepsilon_t(t,\varepsilon)|^2
-U^\varepsilon_t(t,\varepsilon)\int_{|x|<\varepsilon} U_t(t,x)\, dx\Big]
+c^2\int_{|x|=\varepsilon} U^\varepsilon_t \partial_rU\, dS.
\]
Integrating back up in time, and recalling that $U^\varepsilon$ and $U$
are radial, we obtain
\begin{equation} \label{1_st_energy_diff}
\begin{aligned}
 \mathcal E^\varepsilon(T)
&= \mathcal E^\varepsilon(0)
 +\Big[\frac{\text{vol}(B_\varepsilon)}{2} |U^\varepsilon_t(t,\varepsilon)|^2
 -U^\varepsilon_t(t,\varepsilon)\int_{|x|<\varepsilon} U_t(t,x)\, dx
 \Big]_{t=0}^{t=T} \\
 &\quad + c^2\operatorname{area}(B_\varepsilon)\int_0^T U^\varepsilon_t(t,\varepsilon)
 \partial_rU(t,\varepsilon)\, dt.
\end{aligned}
\end{equation}
Below we shall carefully estimate the terms on the right-hand side to show that
$\mathcal E^\varepsilon(T)\to 0$ as $\varepsilon\downarrow 0$.

Before carrying out a similar representation of the 2nd order energy
difference $\mathbb{E}^\varepsilon(t)$, we observe how $\mathcal E^\varepsilon(t)$
controls the $L^2$-distance between $U$ and $\tilde U^\varepsilon$.
Setting
\begin{equation}\label{L2_dist}
 \mathcal D^\varepsilon(t):=\frac{1}{2}\int_{\mathbb{R}^3}|U(t,x)
-\tilde U^\varepsilon(t,x)|^2\, dx,
\end{equation}
the Cauchy-Schwarz inequality gives
\[
\dot{\mathcal D}^\varepsilon(t)\leq 2\mathcal D^\varepsilon(t)^{1/2}
\mathcal E^\varepsilon(t)^{1/2},
\]
such that
\begin{equation}\label{1_vs_0_energy}
 \mathcal D^\varepsilon(T)\lesssim \mathcal D^\varepsilon(0)
+\int_0^T \mathcal E^\varepsilon(t)\, dt.
\end{equation}

We now consider how $\mathbb{E}^\varepsilon(t)$ changes in time. Arguing as above, using
Lemma \ref{energy_1} and the fact that $U^\varepsilon_{x_i}\equiv 0$
on $B_\varepsilon$, we have
\begin{equation}
\begin{aligned}
 \mathcal E_{\partial_i U-\partial_i \tilde U^\varepsilon}(t)
&= \frac{1}{2}\int_{\mathbb{R}^3} |U_{x_it}-\tilde U^\varepsilon_{x_it}|^2
 +c^2|\nabla U_{x_it}-\nabla \tilde U^\varepsilon_{x_it}|^2\, dx \\
 &=\mathcal E_{\partial_i U}(t)+\mathcal E_{\partial_i \tilde U^\varepsilon}(t)
 -\int_{\mathbb{R}^3} U_{x_i,t}\tilde U^\varepsilon_{x_it}+c^2\nabla U_{x_i}
 \cdot\nabla \tilde U^\varepsilon_{x_i}\, dx \\
 &=\mathcal E_{\partial_i U}(0)+\mathcal E^\varepsilon_{\partial_i U^\varepsilon}(0)
 - \int_{|x|>\varepsilon}U_{x_i,t} U^\varepsilon_{x_it}+c^2\nabla U_{x_i}
 \cdot\nabla U^\varepsilon_{x_i}\, dx.
\end{aligned}
\end{equation}
Differentiating in time and integrating by parts in the last integral, give
\begin{equation}\label{indiv_term}
 \dot{\mathcal E}_{\partial_i U-\partial_i \tilde U^\varepsilon}(t)
 =c^2\int_{|x|=\varepsilon}\big(U_{x_it}\big)\big(\partial_r U^\varepsilon_{x_i}\big)\, dS.
\end{equation}
Observing that we have
\[
\sum_{i=1}^3U_{x_it}\partial_r U^\varepsilon_{x_i}=\big(\partial_r U_t\big)
\big(\partial_{rr}U^\varepsilon\big)
\]
along $\{|x|=\varepsilon\}$ (recall that $\partial_r U^\varepsilon$
vanishes along $\{|x|=\varepsilon\}$),
we obtain from \eqref{indiv_term} that
\begin{equation}\label{2_nd_energy_diff}
 \mathbb{E}^\varepsilon(T)=\mathbb{E}^\varepsilon(0)+c^2\operatorname{area}(B_\varepsilon)
 \int_0^T \big(\partial_r U_t(t,\varepsilon)\big)\big(\partial_{rr}
U^\varepsilon(t,\varepsilon)\big)\, dt.
\end{equation}
To estimate $\mathcal E^\varepsilon(T)$ and $\mathbb{E}^\varepsilon(T)$, and hence also
$\mathcal D^\varepsilon(T)$ according to \eqref{1_vs_0_energy}, we employ
the solution formulae \eqref{U} and \eqref{U_eps}.

\subsection{Initial differences in energy}

The details of estimating the initial differences of the first and second
order energies, i.e.\ $\mathcal E^\varepsilon(0)$ and $\mathbb{E}^\varepsilon(0)$, were
carried out in \cite[Section 3.2]{jt1} (and makes use of
\eqref{init_approx_data}). Translating to our present notation
we have that
\begin{gather} \label{initial_0th_energy_diff}
 \mathcal D^\varepsilon(0)\lesssim \varepsilon^2\|\Phi\|_{H^1(B_{2\varepsilon})}^2, \\
\label{initial_1st_energy_diff}
 \mathcal E^\varepsilon(0)\lesssim \varepsilon^2\|\Psi\|_{H^1(B_{2\varepsilon})}^2
 +\|\Phi\|_{H^1(B_{2\varepsilon})}^2, \\
\label{initial_2nd_energy_diff}
 \mathbb{E}^\varepsilon(0)\lesssim \|\Psi\|_{H^1(B_{2\varepsilon})}^2
 +\|\Phi\|_{H^2(B_{2\varepsilon})}^2.
\end{gather}


\subsection{Estimating growth of first order energy difference}

According to \eqref{1_st_energy_diff}, to estimate $\mathcal E^\varepsilon(T)$
we need to estimate the quantities $U^\varepsilon_t$ and $\partial_r U$ along
$|x|=\varepsilon$.
For the remaining term involving $U_t(t,x)$ in \eqref{1_st_energy_diff}
(for $|x|\leq \varepsilon$),
it will suffice to employ an energy estimate that does not require formulae.

Before considering these terms in detail we record the following fact.
For any $k\in \mathbb{R}$ and for any $t>0$ let
\[
Q_k^\varepsilon(t):=\frac{1}{\varepsilon^k}
\Big(e^{-\frac{ct+\varepsilon}{\varepsilon}}
\int_\varepsilon^{ct+\varepsilon}[\frac{s\psi^\varepsilon(s)}{c}
-\frac{s\varphi^\varepsilon(s)}{\varepsilon}]e^{s/\varepsilon}\, ds
+ (ct+\varepsilon)\varphi^\varepsilon(ct+\varepsilon)\Big);
\]
then
\begin{equation}\label{Q}
 Q_k^\varepsilon(t)\to 0\quad\text{as $\varepsilon\downarrow 0$}.
\end{equation}
To see this, integrate by parts in the $\varphi^\varepsilon$-term to get that
\begin{align*}
 Q_k^\varepsilon(t)
&=\frac{1}{c}
 \int_\varepsilon^{ct+\varepsilon} s\psi^\varepsilon(s)\varepsilon^{-k}
 e^{\frac{s-ct-\varepsilon}{\varepsilon}}\, ds
 +\varphi^\varepsilon(\varepsilon)\varepsilon^{1-k}e^{-\frac{ct}{\varepsilon}}\\
&\quad +\int_\varepsilon^{ct+\varepsilon}[\varphi^\varepsilon(s)
 +s{\varphi^\varepsilon}'(s)]\varepsilon^{-k}e^{\frac{s-ct-\varepsilon}{\varepsilon}}
\, ds.
\end{align*}
Recalling \eqref{approx_neum_data_1}-\eqref{approx_neum_data_2} and using that
$\varphi$ and $\psi$ are fixed, smooth functions, the Dominated Convergence Theorem
yields $Q_k^\varepsilon(t)\to 0$ as $\varepsilon\downarrow 0$.

\subsubsection{Estimating $U^\varepsilon_t(t,\varepsilon)$}

According to \eqref{U_eps_bndry_val} we have
\[
U^\varepsilon_t(t,\varepsilon)
=\frac{c}{\varepsilon}\Big(\varphi^\varepsilon(ct+\varepsilon)
+(ct+\varepsilon){\varphi^\varepsilon}'(ct+\varepsilon)
+\frac{(ct+\varepsilon)}{c}\psi^\varepsilon(ct+\varepsilon)\big)
-cQ^\varepsilon_2(t).
\]
As $Q^\varepsilon_2(t)$ tends to zero while $\varphi^\varepsilon$ and
$\psi^\varepsilon$ remain bounded, we conclude that
\begin{equation}\label{U^veps_t_bound}
 |U^\varepsilon_t(t,\varepsilon)|\lesssim \frac{1}{\varepsilon}\quad
\text{for all $t\in[0,T]$ as $\varepsilon\downarrow 0$.}
\end{equation}

\subsubsection{Estimating $\partial_rU(t,\varepsilon)$}

According to \eqref{U} we have
\begin{equation}\label{U_r}
 \partial_rU(t,\varepsilon)
= \begin{cases}
-\frac{1}{2\varepsilon^2}[ (ct+\varepsilon) \varphi(ct+\varepsilon)
 -(ct-\varepsilon)\varphi(ct-\varepsilon)]\\
 +\frac{1}{2\varepsilon}[\varphi(ct+\varepsilon)
 + (ct+\varepsilon)\varphi'(ct+\varepsilon)+\varphi(ct-\varepsilon)
 +(ct-\varepsilon)\varphi'(ct-\varepsilon)]\\
 -\frac{1}{2c\varepsilon^2}\int_{ct-\varepsilon}^{ct+\varepsilon} s\psi(s)\, ds
 +\frac{1}{2c\varepsilon}[(ct+\varepsilon) \psi(ct+\varepsilon)
 +(ct-\varepsilon)\psi(ct-\varepsilon)]\\
\quad\text{if }  t\geq \frac{\varepsilon}{c}
\\[4pt]
-\frac{1}{2\varepsilon^2}[ (\varepsilon+ct) \varphi(\varepsilon+ct)
 +(\varepsilon-ct)\varphi(\varepsilon-ct)]\\
 +\frac{1}{2\varepsilon} [ \varphi(\varepsilon+ct)
 +(\varepsilon+ct)\varphi'(\varepsilon+ct)+\varphi(\varepsilon-ct)
 +(\varepsilon-ct)\varphi'(\varepsilon-ct)]\\
 -\frac{1}{2c\varepsilon^2}\int_{\varepsilon-ct}^{\varepsilon+ct} s\psi(s)\, ds
 +\frac{1}{2c\varepsilon}[(\varepsilon+ct) \psi(\varepsilon+ct)
 -(\varepsilon-ct) \psi(\varepsilon-ct)] \\
\quad\text{if }  t\leq \frac{\varepsilon}{c}
 \end{cases}
\end{equation}
The terms for $t\geq\frac{\varepsilon}{c}$ are estimated by 2nd order
Taylor expansion of $\varphi(ct\pm\varepsilon)$ and $\psi(ct\pm\varepsilon)$
about $\varepsilon=0$. The terms for $t\leq \frac{\varepsilon}{c}$ are estimated
by 2nd order Taylor expansion of $\varphi$ and $\psi$ about zero,
and then using that $\varphi'(0)=\psi'(0)=0$. (As observed earlier, this
holds since $\varphi$ and $\psi$ are profile functions of the {\em smooth},
radial functions $\Phi$ and $\Psi$, respectively).
These expansions are straightforward and we omit them. The end
result is that the leading order terms in \eqref{U_r} cancel, leaving terms
of size at most $O(\varepsilon)$. We thus have
\begin{equation}\label{U_r_bound}
 |\partial_r U(t,\varepsilon)|\lesssim \varepsilon
 \quad\text{for all $t\in[0,T]$ as $\varepsilon\downarrow 0$.}
\end{equation}
(Note: this is actually obvious since we know that $U$ is a smooth,
radial solution satisfying $\partial_r U(t,0)\equiv 0$ and with fixed data
independent of $\varepsilon$.)

Finally, the Cauchy-Schwarz inequality and Lemma \ref{energy_1}
give
\[
\Big|\int_{|x|<\varepsilon} U_t(t,x)\, dx\Big|
\lesssim \varepsilon^{3/2}\mathcal E_U(0)^{1/2}.
\]
Applying this together with \eqref{U^veps_t_bound} and \eqref{U_r_bound}
in \eqref{1_st_energy_diff}, we conclude that
\begin{equation}\label{1_st_energy_diff_1}
 \mathcal E^\varepsilon(T) \lesssim \mathcal E^\varepsilon(0) +\varepsilon^{1/2}.
\end{equation}


\subsection{Estimating growth of second order energy differences}

Next, according to \eqref{2_nd_energy_diff}, to estimate $\mathbb{E}^\varepsilon(T)$,
we need to estimate the quantities $\partial_rU_t$ and $\partial_{rr} U^\varepsilon$
along $|x|=\varepsilon$.


\subsubsection{Estimating $\partial_rU_t(t,\varepsilon)$}

By taking the time derivative of \eqref{U_r} and then Taylor expanding
the various terms as outlined above, we deduce that
\begin{equation}\label{U^_rt_bound}
 |\partial_r U_t(t,\varepsilon)|\lesssim \varepsilon\quad
\text{for all $t\in[0,T]$ as $\varepsilon\downarrow 0$.}
\end{equation}

\subsubsection{Estimating $\partial_{rr}U^\varepsilon(t,\varepsilon)$}

This estimate again requires a direct, but rather long, calculation
(which we omit), followed by a careful analysis of the resulting expression.

The first step is to calculate $\partial_{rr}U^\varepsilon(t,r)$ for
$\varepsilon\leq r\leq ct+\varepsilon$,
by using the first part of formula \eqref{U_eps}. A number of cancelations occur
when the resulting expression is evaluated at $r=\varepsilon$, and we are left with
\begin{align*}
 \partial_{rr}U^\varepsilon(t,\varepsilon)
&= Q^\varepsilon_3(t)
 -\frac{1}{\varepsilon^2}[\varphi^\varepsilon+(ct-\varepsilon){\varphi^\varepsilon}'
 -\varepsilon(ct+\varepsilon){\varphi^\varepsilon}''] \\
&\quad -\frac{1}{c\varepsilon^2}[ct\psi^\varepsilon
 -\varepsilon(ct+\varepsilon){\psi^\varepsilon}'],
\end{align*}
where $\varphi^\varepsilon$, $\psi^\varepsilon$, and their derivatives are
evaluated at $ct+\varepsilon$.
According to \eqref{approx_neum_data_1}-\eqref{approx_neum_data_2} we have that
$\varphi^\varepsilon$, $\psi^\varepsilon$, and their first derivatives remain
bounded independently
of $\varepsilon$, while ${\varphi^\varepsilon}''$ is at most of order
$\frac{1}{\varepsilon}$.
Since $Q^\varepsilon_3(t)\to 0$ as $\varepsilon\downarrow 0$ by \eqref{Q},
we therefore have that
\begin{equation}\label{U^eps_rr_bound}
 |\partial_{rr}U^\varepsilon(t,\varepsilon)|\lesssim \frac{1}{\varepsilon^2}
 \quad\text{for all $t\in[0,T]$ as $\varepsilon\downarrow 0$.}
\end{equation}
Finally, by using \eqref{U^_rt_bound} and \eqref{U^eps_rr_bound} in
\eqref{2_nd_energy_diff}, we conclude that
\begin{equation}\label{2_nd_energy_diff_1}
 \mathbb{E}^\varepsilon(T) \lesssim \mathbb{E}^\varepsilon(0) +\varepsilon.
\end{equation}

\subsection{Convergence of exterior Neumann solutions}

According to the definitions of $\mathcal D^\varepsilon(t)$,
$\mathcal E^\varepsilon(t)$, and
$\mathbb{E}^\varepsilon(t)$, together with the estimates \eqref{1_vs_0_energy},
\eqref{1_st_energy_diff_1}, \eqref{2_nd_energy_diff_1} we have
\begin{align*}
 \|U(t)-\tilde U^\varepsilon(t)\|_{H^2(\mathbb{R}^3)}^2
 &\lesssim \mathcal D^\varepsilon(t) + \mathcal E^\varepsilon(t)+\mathbb{E}^\varepsilon(t)\\
 &\lesssim \mathcal D^\varepsilon(0)+\mathcal E^\varepsilon(0)+\mathbb{E}^\varepsilon(0)
 +\varepsilon^{1/2},
\end{align*}
at any time $t\in[0,T]$.
Applying the bounds \eqref{initial_0th_energy_diff}, and
\eqref{initial_1st_energy_diff},
\eqref{initial_2nd_energy_diff}, we conclude that the (extended) Neumann solutions
$\tilde U^\varepsilon(t)$ converge to the Cauchy solution $U(t)$ in $H^2(\mathbb{R}^3)$,
uniformly on bounded time intervals, as $\varepsilon\downarrow 0$. 
Thanks to Proposition \ref{smooth_case}, this concludes
the proof of part (i) of Theorem \ref{main_result}.

\section{Exterior Dirichlet solutions}\label{gen_dir_soln}

In this and the next section $U^\varepsilon$ refers to the exterior Dirichlet
solutions; similarly for their extensions $\tilde U^\varepsilon(t,x)$.


For fixed initial data $\Phi,\, \Psi\in C^\infty_{c,\rm rad}(\mathbb{R}^3)$ and any
$\varepsilon>0$ we shall derive a formula for the exterior, radial Dirichlet
solution $U^\varepsilon(t,x)$, defined for $|x|\geq\varepsilon$ and satisfying
$U^\varepsilon|_{r=\varepsilon}=0$. We refer to $U^\varepsilon$ as the {\em exterior
Dirichlet solution} corresponding to the solution $U$ of \eqref{eCP} with data
$\Phi$, $\Psi$. In Section \eqref{comp_dir} we will then estimate how it
(really, its extension $\tilde U^\varepsilon(t,x)$ to all of $\mathbb{R}^3$)
approximates the solution $U(t)$ in $H^1(\mathbb{R}^3)$ at fixed times.


To generate the exterior Dirichlet solution $U^\varepsilon(t,x)$ and its
extension we use the following scheme.
To smoothly approximate the original data $(\Phi,\Psi)$ with exterior
Dirichlet data we fix a smooth, nondecreasing cutoff function
$\chi:\mathbb{R}_0^+\to\mathbb{R}_0^+$ with
\begin{equation}\label{eta_props}
 \chi\equiv 0 \quad\text{on $[0,1]$,}
 \quad \chi\equiv 1\quad\text{on $[2,\infty)$.}
\end{equation}
Then, with $\varphi$ and $\psi$ as in \eqref{rad_versns} we define
\begin{gather} \label{approx_dir_data_1}
 \Phi^\varepsilon(x)\equiv \varphi^\varepsilon(|x|)
 :=\chi\big(\textstyle\frac{|x|}{\varepsilon}\big)\varphi(|x|), \\
\label{approx_dir_data_2}
 \Psi^\varepsilon(x)\equiv \psi^\varepsilon(|x|)
 :=\chi\big(\textstyle\frac{|x|}{\varepsilon}\big)\psi(|x|).
\end{gather}
We refer to $(\Phi^\varepsilon,\Psi^\varepsilon)$ as the {\em Dirichlet data}
corresponding to the original Cauchy data $(\Phi,\Psi)$ for \eqref{eCP}.
Note that the Dirichlet data are actually defined on all of $\mathbb{R}^3$,
that they vanish identically on $B_\varepsilon$, and hence their restrictions
to the exterior domain $\{x\in\mathbb{R}^3\,:\, |x|\geq\varepsilon\}$
satisfy homogeneous Dirichlet conditions along $|x|=\varepsilon$.

The exterior Dirichlet solution $U^\varepsilon(t,x)$ is then the unique radial
solution of the initial-boundary value problem
\begin{gather*}
 \square_{1+3}V =0 \quad \text{on $(0,T)\times\{|x|>\varepsilon\}$}\\
 V(0,x)=\Phi^\varepsilon(x) \quad \text{for $|x|>\varepsilon$}\\
 V_t(0,x)=\Psi^\varepsilon(x) \quad \text{for $|x|>\varepsilon$}\\
 V(t,x)=0 \quad \text{along $|x|=\varepsilon$ for $t>0$.}
\end{gather*}

We next record the solution formula for the exterior, radial Dirichlet solution
$U^\varepsilon(t,r)$ (which is simpler to derive than the formula for the
exterior Neumann solution):
\begin{equation}\label{U_eps_dir}
 U^\varepsilon(t,r)
= \begin{cases}
\frac{1}{2r}[ (ct+r) \varphi^\varepsilon(ct+r)
 -(ct-r+2\varepsilon)\varphi^\varepsilon(ct-r+2\varepsilon)]\\
  +\frac{1}{2cr}\int_{ct-r+2\varepsilon}^{ct+r} s\psi^\varepsilon(s)\, ds\\
\quad\text{if }  \varepsilon\leq r\leq ct+\varepsilon \\[4pt]

\frac{1}{2r}\left[ (r+ct) \varphi^\varepsilon(r+ct)+(r-ct)\varphi^\varepsilon(r-ct)\right]\\
 +\frac{1}{2cr}\int_{r-ct}^{r+ct} s\psi^\varepsilon(s)\, ds\\
\quad\text{if }  r\geq ct+\varepsilon.
 \end{cases}
\end{equation}
We finally extend $U^\varepsilon$ at each time to obtain an everywhere
defined approximation of the Cauchy solution $U$. The natural choice is
to extend $U^\varepsilon$ continuously as zero
on $B_\varepsilon$ at each time
\begin{equation}\label{tilde_U_eps_dir}
 \tilde U^\varepsilon(t,x)
= \begin{cases}
 0 & \text{for $0\leq |x|\leq \varepsilon$}\\
 U^\varepsilon(t,x) & \text{for $|x|\geq \varepsilon$.}
 \end{cases}
\end{equation}
We note that
\begin{equation}\label{initial_approx_N_data}
 \tilde U^\varepsilon(0,x)=\Phi^\varepsilon(x),\quad
\tilde U_t^\varepsilon(0,x)=\Psi^\varepsilon(x)
 \quad\text{for all $x\in\mathbb{R}^3$.}
\end{equation}

\section{Comparing the Cauchy and exterior Dirichlet solutions}\label{comp_dir}

We proceed to estimating the $H^1$-distance
\[
\|U(t)-\tilde U^\varepsilon(t)\|_{H^1(\mathbb{R}^3)},
\]
and show that it vanishes as $\varepsilon\downarrow 0$.
As for exterior Neumann solutions we prefer to estimate this
difference by estimating the first order energy
\[\mathcal E^\varepsilon(t)=\mathcal E_{U-\tilde U^\varepsilon}(t)\]
as defined in \eqref{ult_energ_1}. This energy bounds the $L^2$-norm
of the gradient of the difference $U-\tilde U^\varepsilon$, and it also
controls the $L^2$-norm of $U-\tilde U^\varepsilon$ itself.
The calculations for these estimates are similar to the
ones for the Neumann case in Section \eqref{energies}, and will only be outlined.

First, a direct calculation similar to what was done above (using that the
energies $\mathcal E_U(t)$ and $\mathcal E^\varepsilon_{U^\varepsilon}(t)$ are both
conserved in time), shows that
\[
\mathcal E^\varepsilon(t)=\mathcal E^\varepsilon_{U^\varepsilon}(0)+\mathcal E_U(0)
-\int_{|x|>\varepsilon}U_tU^\varepsilon_t+c^2\nabla U^\varepsilon\cdot\nabla U\, dx.
\]
Differentiating with respect to time, integrating by parts, and applying the
Dirichlet condition for $U^\varepsilon$ yield
\begin{equation}\label{E_dot}
 \dot{\mathcal E}^\varepsilon(t)=c^2\int_{|x|=\varepsilon}
 U_t(t,\varepsilon)\partial_r U^\varepsilon(t,\varepsilon)\, dS.
\end{equation}
Also, with $\mathcal D^\varepsilon(t)$ defined as in \eqref{L2_dist}, we have that
\eqref{1_vs_0_energy} holds also in the case of Dirichlet boundary conditions.

To estimate the $H^1$-distance between the Cauchy
solution $U(t)$ and the exterior Dirichlet solution $\tilde U^\varepsilon(t)$,
we proceed to provide bounds for the initial terms $\mathcal D^\varepsilon(0)$ and
$\mathcal E^\varepsilon(0)$, as well as for the surface integral in \eqref{E_dot}.
It is immediate to verify that
\[
\mathcal D^\varepsilon(0)\lesssim \|\Phi^\varepsilon
-\Phi\|_{L^2(\mathbb{R}^3)}^2\leq \|\Phi\|_{L^2(B_{2\varepsilon})}^2,
\]
and similarly that
\[
\mathcal E^\varepsilon(0) \lesssim \|\Psi\|_{L^2(B_{2\varepsilon})}^2
+\|\nabla \Phi^\varepsilon-\nabla\Phi\|_{L^2(\mathbb{R}^3)}^2.
\]
To bound the last term we recall the definition of $\Phi^\varepsilon$
in \eqref{approx_dir_data_1} to calculate that
\begin{equation}
\begin{aligned}
 \|\nabla \Phi^\varepsilon-\nabla\Phi\|_{L^2(\mathbb{R}^3)}^2
 &\lesssim \int_{|x|<2\varepsilon}|\nabla \Phi|^2\, dx
 + \frac{1}{\varepsilon^2}\int_{\varepsilon<|x|<2\varepsilon}|\Phi(x)|^2\, dx  \\
 &\lesssim \|\nabla \Phi\|_{L^2(B_{2\varepsilon})}^2
 + \int_{\varepsilon<|x|<2\varepsilon}\frac{|\Phi(x)|^2}{|x|^2}\, dx.
\end{aligned}\label{intrmed_energy}
\end{equation}
As $\Phi$ belongs to $H^1(\mathbb{R}^3)$, Hardy's inequality (as formulated in
\cite[Lemma 17.1]{tar}) shows that $\frac{|\Phi(x)|^2}{|x|^2}$ belongs to
$L^1(\mathbb{R}^3)$, so that the Dominated Convergence Theorem yields
\[
\|\nabla \Phi^\varepsilon-\nabla\Phi\|_{L^2(\mathbb{R}^3)}^2 \to 0
\quad\text{as $\varepsilon\downarrow 0$.}
\]
We have thus established that
\begin{equation}\label{init_enegies}
 \mathcal D^\varepsilon(0)\to 0\quad\text{and}\quad \mathcal E^\varepsilon(0)
 \to 0 \quad\text{as $\varepsilon\downarrow 0$.}
\end{equation}

\subsubsection{Estimating $\partial_tU(t,\varepsilon)$}
According to \eqref{U} we have
\begin{equation}\label{U_t}
 \partial_tU(t,\varepsilon)
= \begin{cases}
\frac{c}{2\varepsilon}[\varphi(ct+\varepsilon)
 + (ct+\varepsilon)\varphi'(ct+\varepsilon)-\varphi(ct-\varepsilon)
 -(ct-\varepsilon)\varphi'(ct-\varepsilon)]\\
\frac{1}{2\varepsilon}[(ct+\varepsilon)\psi(ct+\varepsilon)
 -(ct-\varepsilon)\psi(ct-\varepsilon)]\\
\quad\text{if }  t\geq \frac{\varepsilon}{c} \\[4pt]
\frac{c}{2\varepsilon}[ \varphi(\varepsilon+ct)
+(\varepsilon+ct)\varphi'(\varepsilon+ct)-\varphi(\varepsilon-ct)
 -(\varepsilon-ct)\varphi'(\varepsilon-ct)]\\
 +\frac{1}{2\varepsilon}[(\varepsilon+ct) \psi(\varepsilon+ct)
+(\varepsilon-ct) \psi(\varepsilon-ct)] \\
\quad\text{if }  t\leq \frac{\varepsilon}{c}\,.
\end{cases}
\end{equation}
We estimate the terms for $t\geq\frac{\varepsilon}{c}$ by 2nd order
Taylor expansion of $\varphi(ct\pm\varepsilon)$ and $\psi(ct\pm\varepsilon)$
about $\varepsilon=0$. The terms for $t\leq \frac{\varepsilon}{c}$ are estimated
by 2nd order Taylor expansion of $\varphi$ and $\psi$ about zero,
and then using that $\varphi'(0)=\psi'(0)=0$.
These expansions are straightforward and are omitted. The
result is that the leading order term in \eqref{U_t} for all times is $O(1)$.
We thus have that
\begin{equation}\label{U_t_bound}
 |U_t(t,\varepsilon)|\lesssim 1
 \quad\text{for all $t\in[0,T]$ as $\varepsilon\downarrow 0$.}
\end{equation}

\subsubsection{Estimating $\partial_{r}U^\varepsilon(t,\varepsilon)$}

We first calculate $\partial_{r}U^\varepsilon(t,r)$ for
$\varepsilon\leq r\leq ct+\varepsilon$
by using the first part of formula \eqref{U_eps_dir}. Evaluating at $r=\varepsilon$
gives that
\[
 \partial_{r}U^\varepsilon(t,\varepsilon) =
 \frac{1}{\varepsilon}[\varphi^\varepsilon(ct+\varepsilon)
 +(ct+\varepsilon){\varphi^\varepsilon}'(ct+\varepsilon)]
 +\frac{(ct+\varepsilon)}{c\varepsilon}{\psi^\varepsilon}(ct+\varepsilon).
\]
Recalling the definitions of $\varphi^\varepsilon$ and $\psi^\varepsilon$ in
\eqref{approx_dir_data_1}-\eqref{approx_dir_data_2}, and
splitting the calculations into $t\gtrless \frac{\varepsilon}{c}$, we obtain 
\begin{equation}\label{U^eps_r_bound}
 |\partial_{r}U^\varepsilon(t,\varepsilon)|\lesssim \frac{1}{\varepsilon}
 \quad\text{for all $t\in[0,T]$ as $\varepsilon\downarrow 0$.}
\end{equation}

\subsection{Convergence of exterior Dirichlet solutions}

By using \eqref{U_t_bound} and \eqref{U^eps_r_bound} in \eqref{E_dot} 
we obtain that
\[
|\dot{\mathcal E}^\varepsilon(t)|\lesssim \varepsilon \quad\text{for all $t\in[0,T]$ as
$\varepsilon\downarrow 0$,}
\]
such that \eqref{init_enegies}${}_2$ gives
\[
\mathcal E^\varepsilon(t)\to 0 \quad\text{uniformly for $t\in[0,T]$ as
$\varepsilon\downarrow 0$.}
\]
Finally, recalling that \eqref{1_vs_0_energy} also holds in the Dirichlet case,
we have that \eqref{init_enegies}${}_1$ yields
\[
\mathcal D^\varepsilon(t)\to 0 \quad\text{uniformly for $t\in[0,T]$ as
$\varepsilon\downarrow 0$,}
\]
as well. We thus conclude that
\[
\|U(t)-\tilde U^\varepsilon(t)\|_{H^1(\mathbb{R}^3)}
\lesssim \mathcal D^\varepsilon(t)+
\mathcal E^\varepsilon(t)\to 0 \quad\text{uniformly for $t\in[0,T]$ as
$\varepsilon\downarrow 0$.}
\]
Thanks to Proposition \ref{smooth_case}, this concludes
the proof of part (ii) of Theorem \ref{main_result}.

\subsection*{Acknowledgments}
Jenssen was supported by the National Science Foundation under
Grant DMS-1311353.
Tsikkou was supported by the WVU ADVANCE Sponsorship Program.

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\end{document}