function $f(t)$ is the temporal profile defined in (81). The solution of the
problem (60) is again singular at the end points of the arc and, as observed
in the previous experiment, the components of $\boldsymbol{\Phi}$ increase as
$\mathcal{O}(r^{-1/2})$ when the distance $r$ tends to zero (see Figure 8).

Figure 8: Behaviour of the horizontal component $\Phi_{1}$ on $\Gamma$ and
w.r.t the distance towards the right endpoint at $T=1$. Both plots are
obtained imposing on $\Gamma$ an algebraically $3$-graded mesh of $80$
segments. Figure 9: Squared error of the energy norm for various
discretization methods

We again study the decay of the error in energy norm for this new Dirichlet
condition, leading to similar considerations for the rate of convergence of
the different discretization methods. The spatial and temporal discretization
parameters for the h, p and hp version are chosen as in the previous
experiment. The results are shown in Figure 9. The squared error for the h
version is $\mathcal{O}\left(h^{{\tilde{\beta}}}\right)$ on the algebraically
${\tilde{\beta}}$-graded mesh, as in Corollary 5.4. The corresponding result
for the p version is $\mathcal{O}\left(p^{-2}\right)$, in agreement with
Corollary 5.8. Faster than algebraic convergence is achieved by the hp version
on a geometrically graded mesh.

###  7.2 Soft scattering problems on polygonal obstacles

In the following we consider the weakly singular integral equation (15) on
different types of closed obstacles $\Gamma$, as shown in Figure 10(b), where
the four considered convex polygonal geometries are collected.

Recalling the notation stated in Section 2, a closed arc $\Gamma$ determines a
partition of $\mathbb{R}^{2}$ made by the bounded interior domain
$\Omega^{\prime}$, with $\partial\Omega^{\prime}=\Gamma$, and its complement
$\Omega=\mathbb{R}^{2}\setminus\overline{\Omega^{\prime}}$. From Section 3.1
we know that the solution u in the exterior set $\Omega$ and near a corner
point of $\Gamma$ locally behaves like a power of the distance $r$ to the
vertex:

$u_{i}\approx C_{i,\omega_{ext}}(t)r^{\nu^{*}(\omega_{ext})},\quad\>r\to 0,$

where $\omega_{ext}$ is the considered exterior angle (with complement
$\omega_{int}$) and the exponent $\nu^{*}(\omega_{ext})$ is the smallest
solution of the equation (25), namely

$\sin^{2}(\omega_{ext}\>\nu^{*})=\left(\frac{\nu^{*}}{k}\sin{\omega_{ext}}\right)^{2},$
(83)

with positive real part, where $k=3-2\lambda/(\lambda+\mu)$. The prefactor
$C_{i,\omega_{ext}}(t)$ is a smooth function in $t$, independent of $r$, so
the leading singular behaviour does not change with time. The solution
$\boldsymbol{\Phi}=\textbf{p}(\textbf{u})|_{\Gamma}$ of the boundary integral
equation (15) represents the traction at the obstacle and, from the discussion
in Section 3, its asymptotic behaviour a the vertex can be expressed as

$\Phi_{i}\approx\widetilde{C}_{i,\omega_{ext}}(t)r^{\nu^{*}(\omega_{ext})-1},\>r\to
0.$

For Lamé parameters $\lambda=2$, $\mu=1$ and mass density $\varrho=1$, Figure
10(a) shows the exterior and interior exponents,
$\nu^{*}(\omega_{ext})=\nu^{*}(2\pi-\omega_{int})$ and
$\nu^{*}(\omega_{int})$, as a function of $\omega_{int}$. Red crosses indicate
the exponents $\nu^{*}$ corresponding to the red corners of the polygons
depicted on the right of Figure 10(b), for interior angles $\frac{7\pi}{24}$
($\nu^{*}=0.5372$), $\frac{\pi}{3}$ ($\nu^{*}=0.5451$), $\frac{3\pi}{8}$
($\nu^{*}=0.542$) and $\frac{3\pi}{5}$ ($\nu^{*}=0.6306$).

Example 3. We consider the Galerkin solution of the weakly singular integral
equation (60) on the polygons represented in Figure 10(b) up to time $T=1$. In
all cases the right hand side imposed is
$\widetilde{g}_{1}(\textbf{x},t)=0,\>\widetilde{g}_{2}(\textbf{x},t)=H[t]f(t)100|x|^{9.5}$.
An example of the solution produced by the boundary condition is in Figure
10(c), where the vertical component of $\boldsymbol{\Phi}$ is plotted at the
base of the equilateral triangle $\Gamma_{1}$. The solution is characterized
by a high gradient near the corners on the base. The mesh on each side of
polygons $\Gamma_{i}$, $i=1,\ldots,4$, is algebraically graded towards the
corners following (62), for given grading parameter ${\tilde{\beta}}=1,\ 2,\
3$. The polygons $\Gamma_{1}$ and $\Gamma_{4}$, which are both equilateral,
are discretized with $80$ segments per side, while for $\Gamma_{2}$ and
$\Gamma_{3}$ we use $80$ segments on the two sides which are of equal length
and $75$ and $87$ segments on the base, respectively. The time step is chosen
as $\Delta t=0.00625$ for all experiments.

(a)  (b)  (c)

Figure 10: Expected exponent with dependence on $\omega_{int}$ and its
complementary ($k=5/3$) and tested polygonal meshes (a and b); plot of the
vertical component of $\boldsymbol{\Phi}$ on the base of $\Gamma_{1}$ (c).

(a)  (b)

Figure 11: Asymptotic behavior towards the vertices

In Figure 11, for each geometry the Euclidean norm of $\boldsymbol{\Phi}$ is
plotted with respect to the distance $r$ towards the angle indicated in Figure
10. We observe that the solution follows the expected behavior
$\mathcal{O}(r^{-(1-\nu^{*})})$ for all the considered geometries. In
particular, the asymptotic behavior for acute corners leads to stronger
singularities ($1-\nu^{*}\approx 0.5$) than for the obtuse angle of the
pentagon ($1-\nu^{*}\approx 0.37$). This confirms the theoretical discussion
in Section 3.1.

We finally consider the convergence in energy on the polygonal obstacles. In
particular, we examine the equilateral triangle $\Gamma_{1}$ and report in
Table 4 the value of the energy for each level of the space discretization.
The energy tends to a benchmark value with increasing DOF (also in this case
the number refers to one component of the vector solution), and the squared
error in energy norm is shown in Figure 12. The decay of the squared error in
a log scale plot is linear, corresponding to
$\mathcal{O}($DOF${}^{-2\nu^{*}{\tilde{\beta}}})$ in each experiment as in
Corollary 5.4.

Table 4: Energy norm squared of the approximate solution for $T=1$ $\Delta t$ | DOF |  $\boldsymbol{\alpha}^{\top}{E}_{\mathcal{V}}\boldsymbol{\alpha}$, ${\tilde{\beta}}=1$ |  $\boldsymbol{\alpha}^{\top}{E}_{\mathcal{V}}\boldsymbol{\alpha}$, ${\tilde{\beta}}=2$ |  $\boldsymbol{\alpha}^{\top}{E}_{\mathcal{V}}\boldsymbol{\alpha}$, ${\tilde{\beta}}=3$
---|---|---|---|---
$5.00\cdot 10^{-2}$ | $30$ | $5.7394\cdot 10^{-2}$ | $7.4875\cdot 10^{-2}$ | $7.6829\cdot 10^{-2}$
$2.50\cdot 10^{-2}$ | $60$ | $6.8490\cdot 10^{-2}$ | $7.6828\cdot 10^{-2}$ | $7.7460\cdot 10^{-2}$
$1.25\cdot 10^{-2}$ | $120$ | $7.3821\cdot 10^{-2}$ | $7.7448\cdot 10^{-2}$ | $7.7566\cdot 10^{-2}$
$6.25\cdot 10^{-3}$ | $240$ | $7.5989\cdot 10^{-2}$ | $7.7558\cdot 10^{-2}$ | $7.7582\cdot 10^{-2}$
Figure 12: Squared error of the energy norm with h version on $\Gamma_{1}$,
${\tilde{\beta}}$-graded mesh

Example 4. In this example we show numerically that the singular behavior at
the corners and the decay of the energy error do not depend on the boundary
data imposed at the obstacle. We specifically consider the triangular obstacle
$\Gamma_{2}$ in Figure 10(b). The solution $\boldsymbol{\Phi}$ of (60) is
calculated for a right hand side with trivial horizontal direction
$\widetilde{g}_{1}(\textbf{x},t)=0$ and different vertical components
$\widetilde{g}_{2}(\textbf{x},t)=H[t]f(t)$,
$\widetilde{g}_{2}(\textbf{x},t)=H[t]f(t)x^{4}$ and
$\widetilde{g}_{2}(\textbf{x},t)=100H[t]f(t)|x|^{9.5}$. In Figure 13(a), we
consider the behavior of the Euclidean norm of $\boldsymbol{\Phi}$ for these
different boundary data, plotted as a function of the distance $r$ to the
vertex which is highlighted in red (Figure 10(b), geometry $\Gamma_{2}$). The
singular exponent is expected to be $\nu^{*}\simeq 0.542$ for a base angle of
$3\pi/8$. Indeed, we find that, in log scale, the slope of the norm for $r\to
0$ is parallel to the dashed line corresponding to $r^{-(1-0.542)}$ for each
of the tested boundary data. In Figure 13(b) the vertical component of
$\boldsymbol{\Phi}$ is shown on the base of $\Gamma_{2}$ at time $T=1$,
highlighting the singular behavior at the corners.

(a)  (b)

Figure 13: Asymptotic behavior towards the vertices in $\Gamma_{2}$ for
different boundary conditions (a) and plot of the vertical component of
$\boldsymbol{\Phi}$ on the base of $\Gamma_{2}$ for the indicated boundary
condition (b).

In Figure 14 we consider the equilateral triangle $\Gamma_{1}$ of 10(b) and
study the decay of the error for increasing degrees of freedom for the h
version. The number of segments and the time step are the same as in 4. The
right hand side is here given by $\widetilde{g}_{1}(\textbf{x},t)=0$,
$\widetilde{g}_{2}(\textbf{x},t)=H[t]f(t)x^{4}$. An algebraically
${\tilde{\beta}}$-graded mesh is used on each side, where
${\tilde{\beta}}=1,2$. The energy tends to a benchmark value as the number of
degrees of freedom increases, and the squared error in energy norm in a log
scale plot decays linearly as
$\mathcal{O}($DOF${}^{-2\nu^{*}{\tilde{\beta}}})$, in agreement with Corollary
5.4.

Figure 14: Squared error of the energy norm with h version on $\Gamma_{1}$,
${\tilde{\beta}}$-graded mesh, $\widetilde{g}_{2}(\textbf{x},t)=H[t]f(t)x^{4}$

###  7.3 Hard scattering problems on flat obstacle

In the following we consider the discrete hypersingular integral equation (17)
on the obstacle
$\Gamma=\left\\{(x,0)\in\mathbb{R}\>|\>x\in[-0.5,0.5]\right\\}$ for a time
independent Neumann condition. We focus, in particular, on the solution of the
discrete problem (61) using $h$, $p$ and $hp$ versions.

Example 5. We consider Neumann data corresponding to
$\widetilde{h}_{i}(\textbf{x},t)=\eta_{i}$, where $\eta_{i}\in\mathbb{R}$ is
constant for $i=1,2$. The datum at the boundary is independent of time.
Therefore, as time increases, the components $\Psi_{i}(\textbf{x},t)$ of the
solution tend to the stationary functions

$\Psi_{i,\infty}(\textbf{x})=k_{i}\sqrt{1/4-x^{2}},\quad
k_{i}=-\frac{c_{\mathtt{P}}^{2}}{\rho
c_{\mathtt{S}}^{2}\left(c_{\mathtt{P}}^{2}-c_{\mathtt{S}}^{2}\right)}\eta_{i},\quad
i=1,2,$ (84)

representing the components of the solution for the reference _elastostatic_
Neumann problem with boundary datum
$\textbf{h}_{\infty}(\textbf{x})=\eta_{i}$. We specifically set $\eta_{i}=1$
for $i=1,2$, so that both components of $\boldsymbol{\Psi}$ converge to the
same elastostatic function $\Psi_{1,\infty}=\Psi_{2,\infty}$. Two different
sets of velocities are considered, $c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=2$ and
$c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=3$.

Figure 15(a) shows the time history of $\Psi_{1}$ and $\Psi_{2}$, calculated
at the midpoint $(0,0)$ of $\Gamma$, for both sets of velocities on the time
interval $[0,7.5]$. We observe that after an initial transient phase the
solution approaches the stationary value (84). In Figure 15(b) the vertical
component $\Psi_{2}$ is plotted on $\Gamma$ for speeds $c_{\mathtt{P}}=2,3$ at
time $T=7.5$. This time is large enough so that for both problems the
numerical solution closely matches the stationary reference solution in (84).
For the plots in Figure 15 equation (61) is solved on a uniform space-time
mesh with mesh size $h=0.025$ and $\Delta t=0.0125$, respectively.

(a)  (b)

Figure 15: Time history of $\Psi_{1}$ and $\Psi_{2}$ calculated at the middle
point of $\Gamma$ for the couples of velocities
$c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=2$ and
$c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=3$ (a). Vertical component $\Psi_{2}$
calculated at the final time instant $T=7.5$ and the related elastostatic
solution $\Psi_{2,\infty}(\textbf{x},t)=k_{2}\sqrt{1/4-x^{2}}$ ($k_{2}=-4/3$
for $c_{\mathtt{P}}=2$ and $k_{1}=-9/8$ for $c_{\mathtt{P}}=3$) (b).

To illustrate the behaviour of the solution near $\partial\Gamma$, Figure 16
shows the components of $-\boldsymbol{\Psi}$, for
$c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=2$, with respect to the distance $r$
towards the right end point of the segment $(0.5,0)^{\top}$ for various time
instants: one observes that the singular behaviour is independent of time, and
the numerical solutions decrease like $r^{1/2}$ for $r$ tending to zero. The
plots in Figure 16 are obtained using the $h$ version on a
${\tilde{\beta}}$-graded mesh with $81$ nodes, with time step $\Delta
t=0.00625$.

Figure 16: Asymptotic behaviour of $-\Psi_{1}$ and $-\Psi_{2}$ towards the
right end of $\Gamma$ for various time instants
($c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=2$). The arc is discretized by a
${\tilde{\beta}}$-graded mesh with ${\tilde{\beta}}=2$ and $81$ mesh points.

For the case $c_{\mathtt{S}}=1,\>c_{\mathtt{P}}=2$, we study the decay of the
error in energy norm for the approximate solution of (61) up to time $T=2$
analysing the value
$\boldsymbol{\beta}^{\top}{E}_{\mathcal{W}}\boldsymbol{\beta}$, namely the
squared energetic norm of the approximate solution, which increases towards a
common benchmark value for all tested discretization methods. We refer the
reader to section 6 for construction details of $\boldsymbol{\beta}$ and
${E}_{\mathcal{W}}$. The number of spatial DOF in the following, as
previously, corresponds to one component of the vector solution. For the $h$
version we choose a ${\tilde{\beta}}$-graded mesh on $\Gamma$ with
${\tilde{\beta}}=1,2$ and $10,20,40$, respectively $80$ segments. The time
step $\Delta t=0.05$ in the case of $10$ segments is halved at each refinement
of the spatial mesh. The log scale plot in Figure 17 shows a linear decay of
the error for the $h$ version, parallel to the lines
$\mathcal{O}($DOF${}^{-{\tilde{\beta}}})$. The results confirm the prediction
in Corollary 5.4. For the $p$ version we consider a uniform discretization of
the obstacle with $h=0.1$ and a uniform time step $\Delta t=1/(2\cdot$DOF$)$.
The log scale plot shows a linear decay of the error parallel to the expected
line $\mathcal{O}($DOF${}^{-2})$. The $hp$ version with a geometrically graded
mesh is considered for meshes on $\Gamma$ with $4,6,8,10$ and $12$ segments.
At each refinement of the mesh the degree $p$, starting from $2$, increases
uniformly on all the space elements. The time step is chosen as $\Delta
t=0.125$ for $4$ segments and halved at each iteration. Similarly to the soft
scattering problems presented above the $hp$ method shows the fastest decay of
the error with respect to increasing spatial DOF.

Figure 17: Squared error of the energy norm calculated up to time instant
$T=2$

##  8 Conclusions

In this work we initiate the study of higher-order versions of the boundary
element method for linear elastodynamics, including $h$, $p$ and $hp$
versions. The asymptotic expansions for the solution obtained near geometric
singularities of the domain give rise to efficient discretizations, with the
same approximation rates as known for $h$, $p$ and $hp$ approximations of time
independent problems.
The quasi-optimal $hp$ explicit estimates in this article complement the
recent analysis for the wave equation, for both finite and boundary element
methods [22, 24, 43], and for linear elastodynamics in 2d [44]. The
convergence is determined by the singular behavior of the solution near the
non-smooth boundary points of the domain. Our analysis relies on the classical
approximation results for time independent problems [16], in combination with
the analysis of the leading singular terms in the time dependent problem [41].
Extensive numerical experiments for a slit and polygonal domains in 2d
illustrate the quasi-optimal convergence rates and confirm the expected
leading asymptotic behavior of the solution near a vertex. On a slit the
energy error $O(p^{-1})$ of the $p$ version converges with the same rate as an
$h$ version on a $2$-graded mesh. For closed polygonal domains the solution is
less singular near the vertices, depending on the material parameters and the
opening angle. Accordingly, higher convergence rates are obtained in both the
analysis and in the numerical experiments.

## Appendix A

In this appendix we introduce space–time anisotropic Sobolev spaces on the
boundary $\Gamma$ as a convenient functional analytic setting for the analysis
of the time dependent boundary integral operators. A detailed exposition may
be found in [30, 20]. Furthermore, we collect mapping properties of the
integral operators $\mathcal{V},\mathcal{W}$ in these space-time anisotropic
spaces (Theorem A.2) and show continuity and coercivity of the associated
bilinear forms (Proposition A.3). The latter imply the stability of the
Galerkin schemes in Section 4. In the case of an open screen or line segment,
$\partial\Gamma\neq\emptyset$, we first extend $\Gamma$ to a closed,
orientable Lipschitz manifold $\widetilde{\Gamma}$. On $\Gamma$ we recall the
usual Sobolev spaces of supported distributions:

$\widetilde{H}^{s}(\Gamma)=\\{u\in H^{s}(\widetilde{\Gamma}):\mathrm{supp}\
u\subset{\overline{\Gamma}}\\}\ ,\quad\ s\in\mathbb{R}\ .$

The Sobolev space ${H}^{s}(\Gamma)$ is the quotient space
$H^{s}(\widetilde{\Gamma})/\widetilde{H}^{s}({\widetilde{\Gamma}\setminus\overline{\Gamma}})$.
To define a family of Sobolev norms, $\alpha_{i}$ be a partition of unity
subordinate to a covering of $\widetilde{\Gamma}$ by open sets $B_{i}$. Given
diffeomorphisms $\varphi_{i}$ from $B_{i}$ to the unit cube in
$\mathbb{R}^{n}$, Sobolev norms are induced from $\mathbb{R}^{n}$, with
parameter $\omega\in\mathbb{C}\setminus\\{0\\}$:

$||u||_{s,\omega,{\widetilde{\Gamma}}}=\left(\sum_{i=1}^{p}\int_{\mathbb{R}^{n}}(|\omega|^{2}+|\boldsymbol{\xi}|^{2})^{s}|\mathcal{F}\left\\{(\alpha_{i}u)\circ\varphi_{i}^{-1}\right\\}(\boldsymbol{\xi})|^{2}d\boldsymbol{\xi}\right)^{\frac{1}{2}}\
.$

Here, $\mathcal{F}=\mathcal{F}_{\mathbb{x}\mapsto\boldsymbol{\xi}}$ denotes
the Fourier transform $\mathcal{F}\varphi(\boldsymbol{\xi})=\int
e^{-i\mathbb{x}\cdot\boldsymbol{\xi}}\varphi(\mathbb{x})\ d\mathbb{x}$.
Different $\omega\in\mathbb{C}\setminus\\{0\\}$ induce equivalent norms on
$H^{s}(\Gamma)$,
$\|u\|_{s,\omega,\Gamma}=\inf_{v\in\widetilde{H}^{s}(\widetilde{\Gamma}\setminus\overline{\Gamma})}\
\|u+v\|_{s,\omega,\widetilde{\Gamma}}$ and on $\widetilde{H}^{s}(\Gamma)$,
$\|u\|_{s,\omega,\Gamma,\ast}=\|e_{+}u\|_{s,\omega,\widetilde{\Gamma}}$.
$e_{+}$ extends the distribution $u$ by $0$ from $\Gamma$ to
$\widetilde{\Gamma}$. When a specific $\omega$ is fixed, we write
$H^{s}_{\omega}(\Gamma)$ for $H^{s}(\Gamma)$, respectively
$\widetilde{H}^{s}_{\omega}(\Gamma)$ for $\widetilde{H}^{s}(\Gamma)$. The norm
$\|u\|_{s,\omega,\Gamma,\ast}$ is stronger than $\|u\|_{s,\omega,\Gamma}$.

We may now define a family of space-time anisotropic Sobolev spaces:

######  Definition A.1.

For $\sigma>0$ and $r,s\in\mathbb{R}$ define

$\displaystyle H^{r}_{\sigma}(\mathbb{R}^{+},{H}^{s}(\Gamma))$
$\displaystyle=\\{u\in\mathcal{D}^{{}^{\prime}}_{+}(H^{s}(\Gamma)):e^{-\sigma
t}u\in\mathcal{S}^{{}^{\prime}}_{+}(H^{s}(\Gamma))\textrm{ and
}||u||_{r,s,\Gamma}<\infty\\}\ ,$ $\displaystyle
H^{r}_{\sigma}(\mathbb{R}^{+},\widetilde{H}^{s}({\Gamma}))$
$\displaystyle=\\{u\in\mathcal{D}^{{}^{\prime}}_{+}(\widetilde{H}^{s}({\Gamma})):e^{-\sigma
t}u\in\mathcal{S}^{{}^{\prime}}_{+}(\widetilde{H}^{s}({\Gamma}))\textrm{ and
}||u||_{r,s,\Gamma,\ast}<\infty\\}\ .$ (85)

Here, $\mathcal{D}^{{}^{\prime}}_{+}(E)$ denotes the space of all
distributions on $\mathbb{R}$ with support in $[0,\infty)$, taking values in a
Hilbert space $E={H}^{s}({\Gamma})$, respectively
$E=\widetilde{H}^{s}({\Gamma})$.
$\mathcal{S}^{{}^{\prime}}_{+}(E)\subset\mathcal{D}^{{}^{\prime}}_{+}(E)$
denotes the subspace of tempered distributions. The Sobolev spaces are endowed
with the norms

$\displaystyle\|u\|_{r,s}:=\|u\|_{r,s,\Gamma}$
$\displaystyle=\left(\int_{-\infty+i\sigma}^{+\infty+i\sigma}|\omega|^{2r}\
\|\hat{u}(\omega)\|^{2}_{s,\omega,\Gamma}\ d\omega\right)^{\frac{1}{2}}\ ,$
$\displaystyle\|u\|_{r,s,\ast}:=\|u\|_{r,s,\Gamma,\ast}$
$\displaystyle=\left(\int_{-\infty+i\sigma}^{+\infty+i\sigma}|\omega|^{2r}\
\|\hat{u}(\omega)\|^{2}_{s,\omega,\Gamma,\ast}\
d\omega\right)^{\frac{1}{2}}\,.$ (86)

They are Hilbert spaces. For $r=s=0$ they correspond to the weighted
$L^{2}$-space with scalar product $\int_{0}^{\infty}e^{-2\sigma
t}\int_{\Gamma}u\overline{v}d\Gamma_{\mathbb{x}}\ dt$. Because $\Gamma$ is
Lipschitz, these spaces are independent of the choice of $\alpha_{i}$ and
$\varphi_{i}$ when $|s|\leq 1$, as for standard Sobolev spaces.
We shall also use the norms $\|u\|_{r,s,(t_{1},t_{2}]\times\Gamma}$ and
$\|u\|_{r,s,(t_{1},t_{2}]\times\Gamma,\ast}$ for restrictions on the time
interval $(t_{1},t_{2}]$.

Let now $\widetilde{\Gamma}=\partial\Omega^{\prime}$ the boundary of a
Lipschitz subset $\Omega\subset\mathbb{R}^{n}$ and
$\Gamma\subset\widetilde{\Gamma}$ open. Denote
$\Omega=\mathbb{R}^{n}\setminus\overline{\Omega^{\prime}}$.

We review the mapping properties for the weakly singular integral operator
$\mathcal{V}$ and the hypersingular operator $\mathcal{W}$.

######  Theorem A.2.

The single layer potential operator and the hypersingular operator are
continuous for $\sigma>0$ and $r\in\mathbb{R}$:

$\displaystyle\mathcal{V}$
$\displaystyle:H_{\sigma}^{r+1}(\mathbb{R}^{+},\widetilde{H}^{-\frac{1}{2}}(\Gamma))\to
H_{\sigma}^{r}(\mathbb{R}^{+},H^{\frac{1}{2}}(\Gamma))\ ,\ ,$
$\displaystyle\mathcal{K}^{\prime}$
$\displaystyle:H_{\sigma}^{r+1}(\mathbb{R}^{+},\widetilde{H}^{-\frac{1}{2}}(\Gamma))\to
H_{\sigma}^{r}(\mathbb{R}^{+},H^{-\frac{1}{2}}(\Gamma))\ ,$
$\displaystyle\mathcal{K}$
$\displaystyle:H_{\sigma}^{r+1}(\mathbb{R}^{+},\widetilde{H}^{\frac{1}{2}}(\Gamma))\to
H_{\sigma}^{r}(\mathbb{R}^{+},H^{\frac{1}{2}}(\Gamma))\ ,$
$\displaystyle\mathcal{W}$
$\displaystyle:H_{\sigma}^{r+1}(\mathbb{R}^{+},\widetilde{H}^{\frac{1}{2}}(\Gamma))\to
H_{\sigma}^{r}(\mathbb{R}^{+},H^{-\frac{1}{2}}(\Gamma))\ .$

This may be found in Theorem 3.1 in [13], see also [9] for $\mathcal{W}$ in
2d, with an analogous proof. See also [32] for a recent discussion of mapping
properties for the wave equation.

For convenience of the reader we recall basic properties of the bilinear form
for the Dirichlet problem in the infinite space-time cylinder
$\Gamma\times\mathbb{R}^{+}$,

$\displaystyle
B_{D,\Gamma\times\mathbb{R}^{+}}(\boldsymbol{\Phi},\boldsymbol{\tilde{\Phi}}):=\int_{\mathbb{R}^{+}}\int_{\Gamma}\mathcal{V}\partial_{t}\boldsymbol{\Phi}(t,\mathbb{x})\
\boldsymbol{\tilde{\Phi}}(t,\mathbb{x})\ d\Gamma_{\mathbb{x}}\,d_{\sigma}t\ ,$
(87)

where $d_{\sigma}t=e^{-2\sigma t}dt$, as well as the corresponding bilinear
form for the Neumann problem,

$\displaystyle
B_{N,\Gamma\times\mathbb{R}^{+}}(\boldsymbol{\Psi},\boldsymbol{\tilde{\Psi}}):=\int_{\mathbb{R}^{+}}\int_{\Gamma}\mathcal{W}\partial_{t}\boldsymbol{\Psi}(t,\mathbb{x})\
\boldsymbol{\tilde{\Psi}}(t,\mathbb{x})\ d\Gamma_{\mathbb{x}}\,d_{\sigma}t\ ,$
(88)

######  Proposition A.3.

Let $\sigma>0$.
a) For every $\boldsymbol{\Phi},\boldsymbol{\tilde{\Phi}}\in
H^{1}_{\sigma}(\mathbb{R}^{+},H^{-\frac{1}{2}}(\Gamma))^{n}$ there holds:

$|B_{D,\Gamma\times\mathbb{R}^{+}}(\boldsymbol{\Phi},\boldsymbol{\tilde{\Phi}})|\lesssim\|\boldsymbol{\Phi}\|_{1,-\frac{1}{2},\Gamma,\ast}\|\boldsymbol{\tilde{\Phi}}\|_{1,-\frac{1}{2},\Gamma,\ast}$
(89)

and

$\|\boldsymbol{\Phi}\|_{0,-\frac{1}{2},\Gamma,\ast}^{2}\lesssim
B_{D,\Gamma\times\mathbb{R}^{+}}(\boldsymbol{\Phi},\boldsymbol{\Phi}).$ (90)

b) For every $\boldsymbol{\Psi},\boldsymbol{\tilde{\Psi}}\in
H^{1}_{\sigma}(\mathbb{R}^{+},H^{\frac{1}{2}}(\Gamma))^{n}$ there holds:

$|B_{N,\Gamma\times\mathbb{R}^{+}}(\boldsymbol{\Psi},\boldsymbol{\tilde{\Psi}})|\lesssim\|\boldsymbol{\Psi}\|_{1,-\frac{1}{2},\Gamma,\ast}\|\boldsymbol{\tilde{\Psi}}\|_{1,\frac{1}{2},\Gamma,\ast}$
(91)

and

$\|\boldsymbol{\Psi}\|_{0,\frac{1}{2},\Gamma,\ast}^{2}\lesssim
B_{N,\Gamma\times\mathbb{R}^{+}}(\boldsymbol{\Psi},\boldsymbol{\Psi}).$ (92)

###### Proof.

The inequalities (89) and (91) follow from the mapping properties in Theorem
A.2. The coercivity (92) was shown in [8, 9] in 2d, and the proof holds
verbatim in any dimension.

To show (90), we consider the elastic problem in the frequency domain:

$\left\\{\begin{array}[]{l}(\lambda+\mu)\nabla(\nabla\cdot\textbf{u})+\mu\Delta\textbf{u}+\rho\omega^{2}\textbf{u}=\textrm{div}\>\sigma(\textbf{u})+\rho\omega^{2}\textbf{u}=0,\quad\textbf{x}\in\Omega^{\prime}\cup\Omega\\\
\textbf{u}=\textbf{g},\quad\textbf{x}\in\widetilde{\Gamma}.\end{array}\right..$
(93)

We assume $\mathrm{Im}\ (\omega)\geq\sigma>0$. The energetic weak formulation
for the single layer equation for the traction
$[\textbf{p}]=[\sigma(\textbf{u})\textbf{n}]$ in frequency domain is given by
(using Parseval’s identity):
Find $[\textbf{p}]\in H^{-\frac{1}{2}}_{\omega}(\widetilde{\Gamma})^{n}$ such
that

$B_{D,\omega}([\textbf{p}],\overline{\boldsymbol{q}})=\langle-i\omega\mathcal{V}_{\omega}[\textbf{p}],\overline{\boldsymbol{q}}\rangle_{\widetilde{\Gamma}}=\langle-i\omega\textbf{g},\overline{\boldsymbol{q}}\rangle_{\widetilde{\Gamma}}$
(94)

for all $\boldsymbol{q}\in H^{-\frac{1}{2}}_{\omega}(\widetilde{\Gamma})^{n}$.

It involves the single layer operator $\mathcal{V}_{\omega}$ obtained from
$\mathcal{V}$ by Fourier transformation. Using Green’s formula as in [9], Thm
3.1, we have

$\int_{\Omega^{\prime}\cup\Omega}\left(\overline{\sigma(\textbf{u})}:\varepsilon(\textbf{u})-\rho\omega^{2}|\textbf{u}|^{2}\right)d\textbf{x}=\int_{\widetilde{\Gamma}}\textbf{u}\cdot\overline{[\sigma(\textbf{u})\textbf{n}]}d\widetilde{\Gamma}\equiv\langle\mathcal{V}_{\omega}[\textbf{p}],\overline{[\textbf{p}]}\rangle_{\widetilde{\Gamma}}.$

Now note that
$|\langle-i\omega\mathcal{V}_{\omega}[\textbf{p}],\overline{[\textbf{p}]}\rangle_{\widetilde{\Gamma}}|\geqslant\mathrm{Re}\>i\overline{\omega}\langle\mathcal{V}_{\omega}[\textbf{p}],\overline{[\textbf{p}]}\rangle_{\widetilde{\Gamma}}$
and

$\displaystyle\mathrm{Re}\>i\overline{\omega}\langle\mathcal{V}_{\omega}[\textbf{p}],\overline{[\textbf{p}]}\rangle_{\widetilde{\Gamma}}$
$\displaystyle=\mathrm{Re}\left(i\overline{\omega}\int_{\Omega^{\prime}\cup\Omega}\overline{\sigma(\textbf{u})}:\varepsilon(\textbf{u})d\textbf{x}\right)+\mathrm{Re}\left(-i\omega\int_{\Omega^{\prime}\cup\Omega}\rho|\omega|^{2}|\textbf{u}|^{2}d\textbf{x}\right)$
$\displaystyle=2\mathrm{Im}(\omega)E_{\omega}\geqslant 0,$ (95)

with

$E_{\omega}=\frac{1}{2}\int_{\Omega^{\prime}\cup\Omega}\left(\overline{\sigma(\textbf{u})}:\varepsilon(\textbf{u})+\rho|\omega|^{2}|\textbf{u}|^{2}\right)d\textbf{x}\
.$

Physically, $E_{\omega}$ is the energy of the displacement u, and it satisfies

$E_{\omega}\geqslant
C_{\sigma}\|\textbf{u}\|^{2}_{1,\omega,\Omega^{\prime}\cup\Omega}$ (96)

for a positive constant $C_{\sigma}$. From (95) and (96) we deduce that

$|\langle-i\omega\mathcal{V}_{\omega}[\textbf{p}],\overline{[\textbf{p}]}\rangle_{\widetilde{\Gamma}}|\geqslant\widetilde{C}_{\sigma}\|\textbf{u}\|^{2}_{1,\omega,\Omega^{\prime}\cup\Omega}\
.$

From the trace theorem there exists a positive constant $C_{trace}$ such that

$2C_{trace}\|\textbf{u}\|^{2}_{1,\omega,\Omega^{\prime}\cup\Omega}\geqslant
2\|\textbf{p}|_{\widetilde{\Gamma}_{+}}\|_{-1/2,\omega,\widetilde{\Gamma}}^{2}+2\|\textbf{p}|_{\widetilde{\Gamma}_{-}}\|_{-1/2,\omega,\widetilde{\Gamma}}^{2}\geqslant\|[\textbf{p}]\|_{-1/2,\omega,\widetilde{\Gamma}}^{2}.$

Coercivity in the frequency domain follows:

$|\langle-i\omega\mathcal{V}_{\omega}[\textbf{p}],\overline{[\textbf{p}]}\rangle_{\widetilde{\Gamma}}|\geqslant\frac{\widetilde{C}_{\sigma}}{2C_{trace}}\|[\textbf{p}]\|_{-1/2,\omega,\widetilde{\Gamma}}^{2}.$
(97)

To show (90), it remains to translate the coercivity (97) from the frequency
domain into the time domain. Integrating (94) in $\omega$ and using the
Parseval identity, noting $\mathcal{F}_{\omega\rightarrow
t}^{-1}\left(\widehat{\varphi}(\omega+i\sigma)\right)=\varphi(t)e^{-\sigma
t}$, we get the identity

$\int_{\mathbb{R}+i\omega_{I}^{0}}\int_{\widetilde{\Gamma}}-i\omega\mathcal{V}_{\omega}\widehat{\boldsymbol{\Phi}}\cdot\overline{\widehat{\boldsymbol{\Phi}}}d\widetilde{\Gamma}d\omega=\int_{0}^{+\infty}\int_{\widetilde{\Gamma}}e^{-2\sigma
t}\frac{\partial}{\partial
t}\left(\mathcal{V}\boldsymbol{\Phi}\right)\cdot\boldsymbol{\Phi}d\widetilde{\Gamma}dt=B_{D}\left(\boldsymbol{\Phi},\boldsymbol{\Phi}\right).$

We now use (97):

$\mathrm{Re}\>B_{D}\left(\boldsymbol{\Phi},\boldsymbol{\Phi}\right)=\int_{\mathbb{R}+i\sigma}\mathrm{Re}\>i\overline{\omega}\langle\mathcal{V}_{\omega}\boldsymbol{\widehat{\Phi}},\overline{\boldsymbol{\widehat{\Phi}}}\rangle_{\widetilde{\Gamma}}\geqslant\frac{\widetilde{C}_{\sigma}}{2C_{trace}}\int_{\mathbb{R}+i\sigma}\|\boldsymbol{\widehat{\Phi}}\|_{-1/2,\omega,\widetilde{\Gamma}}^{2}d\omega\
.$

Therefore

$|B_{D}\left(\boldsymbol{\Phi},\boldsymbol{\Phi}\right)|\geq\frac{\widetilde{C}_{\omega_{I}}}{2C_{trace}}\|\boldsymbol{\Phi}\|_{0,-1/2,\widetilde{\Gamma}}^{2}\
.$

Proposition A.3 follows by restricting to distributions supported in
$\Gamma\subset\widetilde{\Gamma}$. ∎

## Appendix B

In the following, let us describe the approach by Matyukevich and Plamenevskiǐ
from [41] to prove the asymptotic expansion of the solution to the
elastodynamic Dirichlet problem (4) - (6) in a neighborhood of a non-smooth
boundary point of the domain. For ease of reference to the work of
Plamenevskiǐ and coauthors, as well as [24], this section adopts some of the
notation from the analysis community, rather than the notation commonly found
in numerical works. In particular, the $\sigma>0$ from other sections in the
article is here called $\gamma$, singular exponents $\lambda_{\ell}$ are
denoted by $i{\lambda_{\ell}}$, and the definition of the Fourier transform
and its inverse are interchanged. However, note that the dimensions $n$ and
$m$ are interchanged compared to the specific reference [41], but they agree
with the main body of this paper.

In the following we consider two model geometries, wedge and corner, to
describe the local behavior of solutions to this and more general systems near
non-smooth boundary points of the domain. They are of the form
$\mathbb{D}=\mathbb{K}\times\mathbb{R}^{n-m}\subset\mathbb{R}^{n}$, with
$m\geq 2$ and $\mathbb{K}\subset\mathbb{R}^{m}$ an open cone. We use local
coordinates $\mathbb{x}=(\mathbb{y},z)$ in the wedge $\mathbb{D}$.

For $n\geq 2$ we consider the elastodynamic problem (4) - (6) in the space-
time cylinder $\mathbb{D}\times\mathbb{R}$, written abstractly in the form:

$\displaystyle\mathcal{L}(D_{\mathbb{x}},D_{t})\mathbb{u}(\mathbb{x},t)$
$\displaystyle=\mathbb{f}(\mathbb{x},t),$
$\displaystyle(\mathbb{x},t)\in\mathbb{D}\times\mathbb{R}\ ,$ (98)
$\displaystyle\mathbb{u}(\mathbb{x},t)$
$\displaystyle=\mathbb{g}{(\mathbb{x},t)},$
$\displaystyle(\mathbb{x},t)\in\partial\mathbb{D}\times\mathbb{R}\ .$ (99)

with the matrix differential operator
$(\mathcal{L}(D_{\mathbb{x}},D_{t})\mathbb{u}(\mathbb{x},t))_{p}=\partial_{t}^{2}\mathbb{u}(\mathbb{x},t)-\sum_{k,l,q=0}^{{n}}\partial_{k}a^{kl}_{pq}(\mathbb{x})\partial_{l}u_{q}(\mathbb{x},t)$,
$p=1,\dots,{n}$.

Applying the Fourier transform $\mathcal{F}_{{t}\mapsto\tau}$ leads to a
parameter-dependent elliptic problem, with $\tau=\sigma-i\gamma$, $\gamma>0$,
$\sigma\in\mathbb{R}$:

$\mathcal{L}(D_{\mathbb{x}},\tau)\mathbb{v}(\mathbb{x},\tau)=\hat{\mathbb{f}}(\mathbb{x},\tau),\
\
\mathbb{x}\in\mathbb{D},\quad\mathbb{v}(\mathbb{x},\tau)=0,\quad\mathbb{x}\in\partial\mathbb{D}\
.$ (100)

We denote by $\mathcal{A}_{D}(\tau)=\mathcal{L}(D_{\mathbb{x}},\tau)$ the
closure of this operator in $L^{2}(\mathbb{D})$. We first note a well-
posedness result, Theorem 4.1.2 in [41].

######  Proposition B.1.

For every $\hat{\mathbb{f}}\in L^{2}(\mathbb{D})$ and $\tau=\sigma-i\gamma$,
$\sigma\in\mathbb{R}$, $\gamma>0$, there exists a unique solution $\mathbb{v}$
of (100). Further, there exists a constant $c>0$ independent of $\tau$ and
$\hat{\mathbb{f}}$ such that

$\gamma^{2}\int_{\mathbb{D}}(|\tau|^{2}|\mathbb{v}(\mathbb{x},\tau)|^{2}+|D_{\mathbb{x}}\mathbb{v}(\mathbb{x},\tau)|^{2})d\mathbb{x}\leq
c\int_{\mathbb{D}}|\hat{\mathbb{f}}(\mathbb{x},\tau)|^{2}d\mathbb{x}\ .$ (101)

###### Proof.

On the standard Sobolev space $H^{1}_{0}(\mathbb{D})$ we define the
sesquilinear form

$B^{\tau}_{D}(\mathbb{u},\mathbb{v})=\int_{\mathbb{D}}\sum_{i,j,k,l}C_{kl}^{ij}(\mathbb{x})\partial_{k}u_{i}(\mathbb{x})\partial_{l}\overline{v_{j}(\mathbb{x})}d\mathbb{x}-\tau^{2}\int_{\mathbb{D}}\mathbb{u}(\mathbb{x})\cdot\overline{\mathbb{v}(\mathbb{x})}d\mathbb{x}\
,$

where $C_{kl}^{ij}$ denotes the Hooke tensor from Section 2. A key property of
$B^{\tau}_{D}$ is the Korn inequality, which estimates $B^{\tau}_{D}$ in terms
of the norm of $H^{1}(\mathbb{D})$; see Proposition 4.1.3 in [41]: If
$\tau^{2}\in\mathbb{C}\setminus\overline{\mathbb{R}_{+}}$, then there exists
$\delta=\delta(\tau)>0$ such that
$|B^{\tau}_{D}(\mathbb{u},\mathbb{u})|\geq\delta\|\mathbb{u};H^{1}(\mathbb{D})\|^{2}$.
The assertion then follows by applying the Lax-Milgram theorem.∎

###  B.1 Solution of parameter-dependent Dirichlet problem in a cone

For a finer analysis one performs a Fourier transform
$\mathcal{F}_{z\mapsto\xi}$ in the variable $z$ in (98), (99) and introduces
polar coordinates in $\mathbb{K}$: $r=|\mathbb{y}|$,
$\boldsymbol{\omega}=\frac{\mathbb{y}}{|\mathbb{y}|}$. We first assume that
$\mathbb{v}$ solves the non-homogeneous Dirichlet problem with parameters
$\tau\in\mathbb{R}-i\gamma$ and $\xi\in\mathbb{R}$,

$\displaystyle\mathcal{L}(D_{\mathbb{y}},\xi,\tau)\mathbb{v}(\mathbb{y},\xi,\tau)$
$\displaystyle=\hat{\mathbb{f}}(\mathbb{y},\xi,\tau),\quad\mathbb{y}\in\mathbb{K}$
(102) $\displaystyle\mathbb{v}(\mathbb{y},\xi,\tau)$
$\displaystyle=\hat{\mathbb{g}}(\mathbb{y},\xi,\tau),\quad\mathbb{y}\in\partial\mathbb{K}\
.$ (103)

For simplicity, we first consider the homogeneous Dirichlet problem,
corresponding to $\mathbb{g}=0$. The corresponding statements for nonzero
Dirichlet data $\mathbb{g}$ can be deduced from the general results for a
wedge in Subsection B.2.

######  Proposition B.2 (Theorem 6.2.5, [41]).

Let $\tau\in\mathbb{R}-i\gamma$ with $\gamma>0$. For all $\hat{\mathbb{f}}\in
L^{2}(\mathbb{K})$, There exists a unique, strong solution $\mathbb{v}$ of
(102), (103), and

$\gamma^{2}(p^{2}\|\mathbb{v};L^{2}(\mathbb{K})\|^{2}+\|D_{\mathbb{x}}\mathbb{v};L^{2}(\mathbb{K})\|^{2})\leq
c\|\hat{\mathbb{f}};L^{2}(\mathbb{K})\|^{2}\ .$

Here $p=\sqrt{|\xi|^{2}+|\tau|^{2}}$, and $c$ is independent of $\xi$, $\tau$.

Define the weighted Sobolev norms

$\displaystyle\|v;H^{s}_{\beta}(\mathbb{K})\|$
$\displaystyle=\left(\sum_{|\alpha|\leq
s}\int_{\mathbb{K}}r^{2(\beta+|\alpha|-s)}|D_{\mathbb{x}}^{\alpha}v|^{2}\right)^{\frac{1}{2}}d\mathbb{x}\
,$ (104) $\displaystyle\|v;H^{s}_{\beta}(\mathbb{K},p)\|$
$\displaystyle=\left(\sum_{k=0}^{s}p^{2k}\|v;H^{s-k}_{\beta}(\mathbb{K})\|^{2}\right)^{\frac{1}{2}}\
.$ (105)

Let $\chi\in C^{\infty}_{0}(\mathbb{K})$ be a cut-off function which is $=1$
in a neighborhood of the vertex of the cone $\mathbb{K}$, and
$\chi_{\tau}(\mathbb{x})=\chi(|\tau|\mathbb{y})$. From Proposition B.2 one
obtains with $p=\sqrt{|\xi|^{2}+|\tau|^{2}}$, and $c$ independent of $\xi$,
$\tau$,

$\gamma^{2}\|\mathbb{v};H^{1}_{\beta}(\mathbb{K},p)\|^{2}+\|\chi_{\tau}\mathbb{v};H^{2}_{\beta}(\mathbb{K},p)\|^{2}\leq
c\left\\{\|\mathcal{L}(D_{\mathbb{y}},\xi,\tau)\mathbb{v};H^{0}_{\beta}(\mathbb{K})\|^{2}+\frac{p^{2(1-\beta)}}{\gamma^{2}}\|\mathcal{L}(D_{\mathbb{y}},\xi,\tau)\mathbb{v};L^{2}(\mathbb{K})\|\right\\}.$
(106)

Set $\Xi=\mathbb{K}\cap S^{{m}-1}$. For every $\lambda\in\mathbb{C}$ the
pencil

$\mathcal{A}_{D}(\lambda)\boldsymbol{\varphi}=\left\\{r^{2-i\lambda}\mathcal{L}(D_{\mathbb{y}},0,0)r^{i\lambda}\boldsymbol{\varphi},\boldsymbol{\varphi}|_{\partial\Xi}\right\\}$
(107)

defines a map

$\mathcal{A}_{D}(\lambda):H^{2}(\Xi)\to L^{2}(\Xi)\times H^{3/2}(\partial\Xi)\
,$

which is an isomorphism except for a discrete set of eigenvalues
$\\{\lambda_{\ell}\\}$.

For the elastodynamic equation $\mathcal{L}$ has constant coefficients and is
of the form
$\mathcal{L}(D_{\mathbb{x}},D_{t})\mathbb{v}=\partial_{t}^{2}\mathbb{v}+A(D_{\mathbb{x}})\mathbb{v}$
with

$A(D_{\mathbb{x}})=A(D_{\mathbb{y}},D_{z})=D_{k}A^{kl}D_{l}\ ,$

where each of the $A^{kl}$ is a constant matrix $A^{kl}=(a^{kl}_{ij})_{i,j}$.
The operator pencil is then given by

$\left\\{r^{2-i\lambda}A(D_{\mathbb{y}},0)r^{i\lambda}\boldsymbol{\varphi},\boldsymbol{\varphi}|_{\partial\Xi}\right\\}\
.$ (108)

We assume that the strip $\\{\lambda\in\mathbb{C}:m-3\leq 2\mathrm{Im}\
\lambda\leq m-2\\}$ does not intersect the spectrum of $\mathcal{A}_{D}$. For
an eigenvalue $\lambda_{\ell}$ of $\mathcal{A}_{D}$ we take a power-like
solution

$\mathbb{u}_{\ell}(\mathbb{y})=r^{i\lambda_{\ell}}\sum_{q=0}^{k}\frac{1}{q!}(i\ln(r))^{q}\boldsymbol{\varphi}_{\ell}^{(k-q)}(\boldsymbol{\omega})$
(109)

of the homogeneous Dirichlet problem with $\tau=0$, $\xi=0$:

$\displaystyle\mathcal{L}(D_{\mathbb{y}},0,0)\mathbb{u}(\mathbb{y})$
$\displaystyle=0,\quad\mathbb{y}\in\mathbb{K}\ ,$ (110)
$\displaystyle\mathbb{u}(\mathbb{y})$
$\displaystyle=0,\quad\mathbb{y}\in\partial\mathbb{K}\ .$ (111)

Here,
$\\{\boldsymbol{\varphi}_{\ell}^{(0)},\dots,\boldsymbol{\varphi}_{\ell}^{(k)}\\}$
is a Jordan chain to $\lambda_{\ell}$, consisting of an eigenvector
$\boldsymbol{\varphi}_{\ell}^{(k)}$ and generalized eigenvectors
$\boldsymbol{\varphi}_{\ell}^{(0)},\dots,\boldsymbol{\varphi}_{\ell}^{(k-1)}$.
Let $\kappa_{1}\geq\kappa_{2}\geq\dots\geq\kappa_{J}$ denote the partial
multiplicities of the $\lambda_{\ell}$ , and let
$\\{\boldsymbol{\varphi}_{\ell}^{(0,j)},\dots,\boldsymbol{\varphi}_{\ell}^{(\kappa_{j}-1,j)}:j=1,\dots,J\\}$
be a canonical system of Jordan chains. The functions

$\mathbb{u}^{(k,j)}_{\ell}(\mathbb{y})=r^{i\lambda_{\ell}}\sum_{q=0}^{k}\frac{1}{q!}(i\ln(r))^{q}\boldsymbol{\varphi}_{\ell}^{(k-q,j)}(\boldsymbol{\omega}),$
(112)

where $k=0,\dots,\kappa_{j}-1$ and $j=1,\dots,J$, constitute a basis in the
space of power-like solutions corresponding to $\lambda_{\ell}$.

######  Remark B.3.

In special geometries the spectral problem for $\mathcal{A}_{D}$ admits an
explicit solution. See Section 3.1 for a discussion of the eigenvalues and
eigenfunctions in the case of a polygon, Section 3.2 for an edge, and Section
3.3 for a circular cone.

Let $\mathbb{V}_{\ell}^{(k,j)}$ be the infinite series of dual functions
satisfying the homogeneous equations (110), (111), and let
$\mathbb{V}_{\ell,{M}}^{(k,j)}$ be its truncation after ${M}$ terms.

The dual vector functions

$\mathbb{v}^{(k,j)}_{\ell}(\mathbb{y})=r^{i\overline{\lambda_{\ell}}-(m-2)}\sum_{q=0}^{k}\frac{1}{q!}(i\ln(r))^{q}\boldsymbol{\psi}^{(k-q,j)}_{\ell}(\boldsymbol{\omega}),$
(113)

form a basis in the space of power-like solutions to (110), (111) that
correspond to the eigenvalue $\overline{\lambda_{\ell}}+i(m-2)$. The bases
match under specific orthogonality and normalization conditions (see, for
example, (114) in [41]), respectively [45].

Denote by $\\{\mathbb{u}_{\ell}^{k,j}\\}$, $\\{\mathbb{v}_{\ell}^{k,j}\\}$ the
matched bases of power-like solutions of (110), (111). Next we consider the
homogeneous problem with parameters $\tau\in\mathbb{R}-i\gamma$ and
$\xi\in\mathbb{R}^{{n-m}}$, corresponding to (102), (103),

$\displaystyle\mathcal{L}(D_{\mathbb{y}},\xi,\tau)\mathbb{v}(\mathbb{y},\xi,\tau)$
$\displaystyle=0,\quad\mathbb{y}\in\mathbb{K}$ (114)
$\displaystyle\mathbb{v}(\mathbb{y},\xi,\tau)$
$\displaystyle=0,\quad\mathbb{y}\in\partial\mathbb{K}\ .$ (115)

Substituting $\mathbb{u}_{\ell}^{(k,j)}$ in (114), (115), we construct the
formal series

$\mathbb{U}_{\ell}^{(k,j)}(\mathbb{y},\xi,\tau)=\sum_{q=0}^{\infty}r^{i\lambda_{\ell}+q}\mathbb{P}{(k,j)}_{q}(\boldsymbol{\omega},\xi,\tau,\ln(r))$
(116)

satisfying (114), (115). Here $\mathbb{P}{(k,j)}_{q}$ are polynomials in
$\xi,\tau,\ln(r)$, with coefficients smoothly depending on
$\boldsymbol{\omega}\in\Xi$. Replacing $\\{\mathbb{u}_{\ell}^{k,j}\\}$ by
$\\{\mathbb{v}_{\ell}^{k,j}\\}$, we obtain the formal series

$\mathbb{V}_{\ell}^{(k,j)}(\mathbb{y},\xi,\tau)=\sum_{q=0}^{\infty}r^{i(\overline{\lambda_{\ell}}+i{(m-n-2)})+q}\mathbb{Q}{(k,j)}_{q}(\boldsymbol{\omega},\xi,\tau,\ln(r)),$
(117)

satisfying (114), (115). The functions $\mathbb{Q}{(k,j)}_{q}$ again obey
analogous properties to $\mathbb{P}{(k,j)}_{q}$.

In reference [41] the formal series $\mathbb{U}_{\ell}^{(k,j)}$,
$\mathbb{V}_{\ell}^{(k,j)}$ are constructed for these bases.

Consider now (102), (103) with $\chi\mathbb{v}\in H^{2}_{\beta}(\mathbb{K})$,
$\hat{\mathbb{f}}\in H^{0}_{\beta}(\mathbb{K})\cap
H^{0}_{\gamma}(\mathbb{K})$, for $\gamma<\beta$. As above, $\chi\in
C^{\infty}_{0}(\mathbb{K})$ denotes a cut-off function which is $=1$ in a
neighborhood of the vertex of the cone $\mathbb{K}$. If the line
$\\{\lambda\in\mathbb{C}:\mathrm{Im}\ \lambda=\gamma+\frac{m}{2}-2\\}$ does
not intersect the spectrum of the pencil $\mathcal{A}_{D}$, then we have

$\mathbb{v}=\chi\sum c_{\ell}^{(k,j)}\mathbb{U}^{(k,j)}_{\ell,{M}}+\mathbb{h}\
,$

where the remainder $\mathbb{h}$ is such that $\chi\mathbb{h}\in
H^{2}_{\gamma}(\mathbb{K})$. Here $\mathbb{U}^{(k,j)}_{\ell,{M}}$ is the
partial sum of the series $\mathbb{U}^{(k,j)}_{\ell}$ containing M terms such
that $\chi r^{i\lambda_{\ell}+({M}+1)}\mathbb{P}_{{M}+1}^{(k,j)}\in
H^{2}_{\gamma}(\mathbb{K})$. The asymptotic formula for $\mathbb{v}$ involves
the summands corresponding to the eigenvalues of the pencil in the strip
$\\{\lambda\in\mathbb{C}:\mathrm{Im}\
\lambda\in(\gamma+\frac{m}{2}-2,\beta+\frac{m}{2}-2)\\}$, so that
$\chi\mathbb{U}^{(k,j)}_{\ell,{M}}\in H^{2}_{\beta}(\mathbb{K})$ and
$\chi\mathbb{U}^{(k,j)}_{\ell,{M}}\not\in H^{2}_{\gamma}(\mathbb{K})$

To state the main result for the expansion of the parameter-dependent problem
near the vertex of the cone $\mathbb{K}$, we introduce the following function
spaces:

$\displaystyle\|\mathbb{v};DH_{\beta}(\mathbb{K},\xi,\tau)\|=\left(\gamma^{2}\|\mathbb{v};H^{1}_{\beta}(\mathbb{K},p)\|^{2}+\|\chi_{p}\mathbb{v};H^{2}_{\beta}(\mathbb{K},p)\|^{2}\right)^{\frac{1}{2}}\
,$
$\displaystyle\|\hat{\mathbb{f}};RH_{\beta}(\mathbb{K},\xi,\tau)\|=\left(\|\hat{\mathbb{f}};H^{0}_{\beta}(\mathbb{K})\|^{2}+p^{2(1-\beta)}\gamma^{-2}\|\hat{\mathbb{f}};L^{2}(\mathbb{K})\|^{2}\right)^{\frac{1}{2}}\
,$

where $p=\sqrt{|\xi|^{2}+|\tau|^{2}}$ and $\tau=\sigma-i\gamma$
($\sigma\in\mathbb{R}$, $\gamma>0$). By Proposition B.2 and (106), the
operator $\mathcal{L}(D_{\mathbb{y}},\xi,\tau)$ from Problem (102), (103),
defines a continuous map
$\mathcal{L}(D_{\mathbb{y}},\xi,\tau):DH_{\beta}(\mathbb{K},\xi,\tau)\to
RH_{\beta}(\mathbb{K},\xi,\tau)$.

In [41], Matyukevich and Plamenevskiǐ investigate the dependence of properties
of $\mathcal{L}(D_{\mathbb{y}},\xi,\tau)$ on $\beta$. Let
$1>\beta_{1}>\beta_{2}>\dots$ be numbers in $(-\infty,1]$ such that every line
$\\{\lambda\in\mathbb{C}:\mathrm{Im}\ \lambda=\beta_{r}+\frac{m}{2}-2\\}$
contains at least one eigenvalue of the pencil $\mathcal{A}_{D}$.

Matyukevich and Plamenevskiǐ obtain the following results:

######  Theorem B.4 (Theorem 6.3.5, [41]).

Suppose that $\beta\in(\beta_{1},1]$, $\gamma>0$ and $\hat{\mathbb{f}}\in
RH_{\beta}(\mathbb{K},\xi,\tau)$. Then (102), (103) with right hand side
$\hat{\mathbb{f}}$ admits a unique solution $\mathbb{v}$ satisfying

$\|\mathbb{v};DH_{\beta}(\mathbb{K},\xi,\tau)\|\leq
c\|\hat{\mathbb{f}};RH_{\beta}(\mathbb{K},\xi,\tau)\|,$

where $c$ is independent of $(\xi,\tau)$.

######  Theorem B.5 (Proposition 6.4.1, [41]).

Suppose $\gamma>0$, $\beta\in(\beta_{r+1},\beta_{r})$, $0<\beta_{r}-\beta<1$,
$\hat{\mathbb{f}}\in RH_{\beta}(\mathbb{K},\xi,\tau)$ and

$(\hat{\mathbb{f}},\mathbb{w}_{\ell}^{(k,j)}(\cdot,\xi,\overline{\tau}))_{L^{2}(\mathbb{K})}=0$

for all $\mathbb{w}_{\ell}^{(k,j)}$ corresponding to eigenvalues of
$\mathcal{A}_{D}$ in the strip $\\{\mathrm{Im}\
\lambda\in(\beta_{r+1}+\frac{m}{2}-2,\beta_{1}+\frac{m}{2}-2)\\}$. Then the
solution $\mathbb{v}$ of (102), (103), admits the representation

$\mathbb{v}(\mathbb{y},\xi,\tau)=\chi(p\mathbb{y})\sum_{\ell}\sum_{k,j}c^{(k,j)}_{\ell}(\xi,\tau)\mathbb{u}_{\ell}^{(k,j)}(\mathbb{y})+\mathbb{v}_{0}(\mathbb{y},\xi,\tau)\
.$

Here the outer summmation over $\ell$ sums over all eigenvalues
$\lambda_{\ell}$ of the pencil with $\mathrm{Im}\
\lambda=\beta_{r}+\frac{m}{2}-2$, while the inner summation sums over a basis
$\\{\mathbb{u}_{\ell}^{(k,j)}\\}$ of power-like solutions as in (109)
corresponding to $\lambda_{\ell}$. The remainder $\mathbb{v}_{0}$ belongs to
$DH_{\beta}(\mathbb{K},\xi,\tau)$.

There holds

$c_{\ell}^{(k,j)}(\xi,\tau)=p^{i\lambda_{\ell}}\sum_{q}\frac{1}{q!}(i\ln(p))^{q}d_{\ell}^{(k+q,j)}(\xi,\tau)\
,$

with

$d_{\ell}^{(k,j)}(\xi,\tau)=p^{-2}\left(\hat{\mathbb{f}}(p^{-1}\cdot,\xi,\tau),\mathbb{w}_{\ell}^{(k,j)}(\cdot,\xi/p,\bar{\tau}/p)\right)_{L^{2}(\mathbb{K})}\
.$

Moreover there holds

$\|\mathbb{v}_{0};DH_{\beta}(\mathbb{K},\xi,\tau)\|\leq
c\|\hat{\mathbb{f}};RH_{\beta}(\mathbb{K},\xi,\tau)\|,\ \ \
|d_{\ell}^{(k,j)}(\xi,\tau)|\leq
cp^{\beta+\frac{m}{2}-2}\|\hat{\mathbb{f}};RH_{\beta}(\mathbb{K},\xi,\tau)\|,$

with a constant $c$ independent of $\xi$, $\tau$ and $\hat{\mathbb{f}}$.

###  B.2 Solution of a parameter-dependent Dirichlet problem in a wedge

By means of an inverse Fourier transform $\mathcal{F}_{\xi\mapsto z}^{-1}$ in
the dual edge variable $\xi$, we obtain results for the general Dirichlet
problem in the wedge $\mathbb{D}$,

$\displaystyle\mathcal{L}(\mathbb{x},D_{\mathbb{x}},\tau)\mathbb{v}(\mathbb{x},\tau)=\hat{\mathbb{f}}(\mathbb{x},\tau),\quad\mathbb{x}\in\mathbb{D},$
(118)
$\displaystyle\mathbb{v}(\mathbb{x},\tau)=\hat{\mathbb{g}}(\mathbb{x},\tau),\quad\mathbb{x}\in\partial\mathbb{D}\
,$ (119)

the problem in the frequency domain corresponding to (98), (99).

The regularity of the solutions is described in the following weighted Sobolev
spaces on $\mathbb{D}=\mathbb{K}\times\mathbb{R}^{n-m}$. In $\mathbb{D}$, one
uses the coordinates $\mathbb{x}=(\mathbb{y},z)$ and introduces polar
coordinates in $\mathbb{K}$: $r=|\mathbb{y}|$,
$\boldsymbol{\omega}=\frac{\mathbb{y}}{|\mathbb{y}|}$. Define

$\displaystyle\|u;H^{s}_{\beta}(\mathbb{D})\|=\left(\sum_{|\alpha|\leq
s}\int_{\mathbb{D}}r^{{2(}\beta+|\alpha|-s{)}}|D_{\mathbb{x}}^{\alpha}u|^{2}\right)^{\frac{1}{2}}\
,$ (120)
$\displaystyle\|u;H^{s}_{\beta}(\mathbb{D},p)\|=\left(\sum_{k=0}^{s}p^{2k}\|u;H^{s-k}_{\beta}(\mathbb{D})\|^{2}\right)^{\frac{1}{2}}\
.$ (121)

Corresponding spaces $H^{s}_{\beta}(\partial\mathbb{D})$ and
$H^{s}_{\beta}(\partial\mathbb{D},p)$ on $\partial\mathbb{D}$ are obtained as
trace spaces for $H^{s}_{\beta}(\mathbb{D})$, respectively
$H^{s}_{\beta}(\mathbb{D},p)$.

The basic existence result is given by:

######  Proposition B.6 (Theorem 4.2.2, [41]).

Suppose that the wedge $\mathbb{D}$ is admissible in the sense of [41],
$\\{\hat{\mathbb{f}},\hat{\mathbb{g}}\\}\in L^{2}(\mathbb{D})\times
H^{1}(\partial\mathbb{D})$ and $\tau=\sigma-i\gamma$, $\sigma\in\mathbb{R}$,
$\gamma>0$. Then there exists a unique strong solution $\mathbb{v}$ of (118)
and (119). Furthermore, there exists a constant $c>0$ independent of $\tau$
such that

$\gamma^{2}\|\mathbb{v},H^{1}(\mathbb{D},|\tau|)\|^{2}+\gamma\|\mathbb{p}(\mathbb{v}),L^{2}(\partial\mathbb{D})\|^{2}\leq
c\left(\|\hat{\mathbb{f}}\|^{2}_{L^{2}(\mathbb{D})}+\gamma\|\hat{\mathbb{g}},H^{1}(\partial\mathbb{D},|\tau|)\|^{2}\right)\
.$

Higher regularity has been obtained by Matyukevich and Plamenevskiǐ in the
spaces $H^{s}_{\beta}(\mathbb{D})$. Following [41] we only state the result
for homogeneous boundary conditions.

######  Proposition B.7 (Proposition 5.1.1, [41]).

Let $\beta\leq 1$. Assume $\mathrm{Im}\ \lambda=\beta+\frac{m}{2}-2$ does not
intersect the spectrum of $\mathcal{A}_{D}$. Then for $\mathbb{v}\in
H^{2}_{\beta}(\mathbb{D},1)\cap H^{1}_{\beta=0}(\mathbb{D})$ with
$\mathcal{L}(D_{\mathbb{x}},0)\mathbb{v}\in L^{2}(\mathbb{D})$ there holds

$\displaystyle\|\chi_{\tau}\mathbb{v},H^{2}_{\beta}(\mathbb{D},|\tau|)\|^{2}+\gamma^{2}\|\mathbb{v},H^{1}_{\beta}(\mathbb{D},|\tau|)\|^{2}$
(122) $\displaystyle\qquad\leq
c\left\\{\|\mathcal{L}(D_{\mathbb{x}},\tau)\mathbb{v},H^{0}_{\beta}(\mathbb{D})\|^{2}+|\tau|^{2(1-\beta)}\gamma^{-2}\|\mathcal{L}(D_{\mathbb{x}},\tau)\mathbb{v},L^{2}(\mathbb{D})\|^{2}\right\\}\
,$

where $\chi_{\tau}(\mathbb{x})=\chi(|\tau|\mathbb{y})$ for some $\chi\in
C^{\infty}_{0}(\overline{\mathbb{K}})$ which is $=1$ in a neighborhood of the
vertex of the cone $\mathbb{K}$. The constant $c$ is independent of
$\mathbb{v}$, $\tau=\sigma-i\gamma$, $\sigma\in\mathbb{R}$, $\gamma>0$.

A corresponding result for the wave equation with inhomogeneous boundary
conditions has been considered in [51], Formula (7), but we omit the more
involved statement.

The proof in [41] is based on three steps: (i) estimates far from the edge,
(ii) estimates near the edge, (iii) the global a priori estimate (101).

###  B.3 Solution of a time-dependent problem in a wedge; non-homogeneous
boundary conditions

We now present results for the time-dependent system (98), (99), with constant
coefficients, obtained from the frequency-domain results via the inverse
Fourier transform. They are stated in terms of the following weighted function
spaces in the space-time cylinder $\mathcal{Q}=\mathbb{D}\times\mathbb{R}$,
with coordinates $\mathbb{x}=(\mathbb{y},z)\in\mathbb{D}$ and parameter $q>0$:

$\|w;H^{s}_{\beta}(\mathcal{Q})\|=\left(\sum_{|\alpha|\leq
s}\int_{\mathbb{R}}\int_{\mathbb{D}}r^{2(\beta-s+|\alpha|)}|D_{\mathbb{x},t}^{\alpha}w(\mathbb{x},t)|^{2}\
d\mathbb{x}\ dt\right)^{1/2}\ ,$
$\|w;H^{s}_{\beta}(\mathcal{Q},q)\|=\left(\sum_{k=0}^{s}q^{2k}\|w;H^{s-k}_{\beta}(Q)\|^{2}\right)^{1/2}\
.$

If $\gamma>0$, we set $w^{\gamma}(\mathbb{x},t):=\exp(-\gamma
t)w(\mathbb{x},t)$ and define

$\|w;V^{s}_{\beta}(\mathcal{Q},\gamma)\|=\|w^{\gamma};H^{s}_{\beta}(\mathcal{Q},\gamma)\|\
.$

The corresponding spaces on the boundary $\partial\mathcal{Q}$ are defined as
the trace spaces of $H^{s}_{\beta}(Q)$, respectively
$V^{s}_{\beta}(\mathcal{Q},\gamma)$.

######  Definition B.8.

Assume $(\mathbb{f},\mathbb{g})\in V^{0}_{0}(\mathcal{Q},\gamma)\times
V^{3/2}_{0}(\partial\mathcal{Q},\gamma)$, and let $\mathbb{v}$ be a strong
solution to (118), (119) in $\mathbb{D}$ with right hand side
$(\hat{\mathbb{f}},\hat{\mathbb{g}})$. Then
$\mathbb{u}(\mathbb{y},z,t)=\mathcal{F}^{-1}_{{(\xi,\tau)\to(z,t)}}\mathbb{v}(\mathbb{y},\xi,\tau)$
is called a strong solution of (98), (99).

Proposition B.6 implies that for any $(\mathbb{f},\mathbb{g})\in
V^{0}_{0}(\mathcal{Q},\gamma)\times V^{3/2}_{0}(\partial\mathcal{Q},\gamma)$
with $\gamma>0$ the problem (98), (99) admits a unique strong solution and

$\gamma^{2}\|\mathbb{{u}};V^{1}_{0}(\mathcal{Q},\gamma)\|^{2}+\gamma\|\mathbb{p}(\mathbb{{u}}),V^{0}_{\gamma}(\partial\mathbb{D},\gamma)\|^{2}\leq
c\left(\|\mathbb{f};V^{0}_{0}(\mathcal{Q},\gamma)\|^{2}+\gamma\|\mathbb{g};V^{3/2}_{0}(\partial\mathcal{Q},\gamma)\|^{2}\right)\
,$

for a constant $c>0$ independent of $\gamma$.

Let $\chi\in C^{\infty}(\overline{\mathbb{K}})$ be a cut-off function which is
identically $1$ in a neighborhood of the conical point $0$. Define

$X\mathbb{u}(\mathbb{y},z,t)=\mathcal{F}^{-1}_{(\xi,\tau)\to(z,t)}\chi(p\mathbb{y})\mathcal{F}_{(z^{\prime},t^{\prime})\to(\xi,\tau)}\mathbb{u}(\mathbb{y},z^{\prime},t^{\prime})$
(123)

and

$\Lambda^{\mu}\mathbb{u}(\mathbb{y},z,t)=\mathcal{F}^{-1}_{\tau\to
t}|\tau|^{\mu}\mathcal{F}_{t^{\prime}\to\tau}\mathbb{u}(\mathbb{y},z,t^{\prime})\
.$ (124)

Higher regularity theorems involve the following norms in $Q$: For
$\beta\in\mathbb{R}$ and $\gamma>0$

$\displaystyle\|\mathbb{v};DV_{\beta}(\mathcal{Q},\gamma)\|$
$\displaystyle=\left(\gamma^{2}\|\mathbb{v};V^{1}_{\beta}(\mathcal{Q},\gamma)\|^{2}+\|X\mathbb{v};V_{\beta}^{2}(\mathcal{Q},\gamma)\|^{2}+\gamma\|\partial_{\nu}\mathbb{v};V^{0}_{\beta}(\partial\mathcal{Q},\gamma)\|^{2}\right)^{1/2}\
,$ (125) $\displaystyle\|\mathbb{f};RV_{\beta}(\mathcal{Q},\gamma)\|$
$\displaystyle=\left(\|\mathbb{f};V^{0}_{\beta}(\mathcal{Q},\gamma)\|^{2}+\gamma^{-2}\|\Lambda^{1-\beta}\mathbb{f};V^{0}_{0}(\mathcal{Q},\gamma)\|^{2}\right)^{1/2}\
.$ (126)
$\displaystyle\|(\mathbb{f},\mathbb{g});\mathcal{R}V_{\beta}(\mathcal{Q},\gamma)\|$
$\displaystyle=\left(\|\mathbb{f};RV_{\beta}(\mathcal{Q},\gamma)\|^{2}+\|X\mathbb{g};V_{\beta}^{3/2}(\partial\mathcal{Q},\gamma)\|^{2}+\gamma\|\mathbb{g};V^{1}_{0}(\partial\mathcal{Q},\gamma)\|^{2}\right.$
$\displaystyle\qquad\left.+\gamma^{-1}\|\Lambda^{1-\beta}\mathbb{g};V^{1}_{0}(\partial\mathcal{Q},\gamma)\|^{2}\right)^{1/2}\
.$ (127)

More generally, one may introduce for $q\in\mathbb{N}_{0}$

$\|\mathbb{f};RV_{\beta,q}(\mathcal{Q},\gamma)\|=\left(\sum_{j=0}^{q}\gamma^{-2j}\|\Lambda^{j}\mathbb{f};V^{q-j}_{\beta+q-j}(\mathcal{Q},\gamma)\|^{2}+\gamma^{-2(1+q)}\|\Lambda^{1-\beta+q}\mathbb{f};V^{0}_{0}(\mathcal{Q},\gamma)\|^{2}\right)^{1/2}\
,$

and similarly $\mathcal{R}V_{\beta,q}(\mathcal{Q},\gamma)$ and
$DV_{\beta,q}(\mathcal{Q},\gamma)$.

The following result may then be found in Theorem 7.4, [41], for ${\bf{g}}=0$
and $q=0$. It may be extended to inhomogeneous boundary conditions and $q>0$
using the arguments in [35].

######  Theorem B.9.

Suppose $q\in\mathbb{N}_{0}$, $\gamma>0$ and
$(\mathbb{f},\mathbb{g})\in\mathcal{R}V_{\beta,q}(\mathcal{Q},\gamma)$. a) If
$\beta\in(\beta_{1},1)$, the strong solution $\mathbb{u}$ to (98), (99)
belongs to $DV_{\beta,q}(\mathcal{Q},\gamma)$ and there exists $c>0$
independent of $\gamma$ such that

$\|\mathbb{u};DV_{\beta,q}(\mathcal{Q},\gamma)\|\leq
c\|(\mathbb{f},\mathbb{g});\mathcal{R}V_{\beta,q}(\mathcal{Q},\gamma)\|\ .$

b) If $\beta\in(\beta_{r+1},\beta_{r})$, then there exists a solution
$\mathbb{u}$ to (98), (99) if and only if for all $\xi\in\mathbb{R}^{n-m}$,
for all $\tau\in\mathbb{R}-i\gamma$ and for all $\mathbb{w}_{\ell}^{k,j}$
corresponding to eigenvalues $\lambda_{\ell}$ of $\mathcal{A}_{D}$ with
$\mathrm{Im}\ \lambda\in[\beta_{r}+\frac{m}{2}-2,\beta_{1}+\frac{m}{2}-2]$,

$(\hat{\mathbb{f}}(\cdot,\xi,\tau),\mathbb{w}_{\ell}^{k,j}(\cdot,\xi,\overline{\tau}))_{L^{2}(\mathbb{K})}+(\hat{\mathbb{g}}(\cdot,\xi,\tau),{\boldsymbol{p}}(\mathbb{w}_{\ell}^{k,j})(\cdot,\xi,\overline{\tau}))_{L^{2}(\partial\mathbb{K})}=0\
.$ (128)

We can now state the main result of this section, which gives the asymptotics
of the time-dependent problem in a neighborhood of the edge. It may be found
in Theorem 7.5, [41], for ${\bf{g}}=0$ and $q=0$. The extension to
inhomogeneous boundary data ${\bf{g}}$ follows as in Section 3: choose an
extension $\widetilde{{\bf g}}$ in the domain with Dirichlet trace ${\bf{g}}$.
Theorem 7.5, [41] then assures an asymptotic expansion of the function ${\bf
U}={\bf u}-\widetilde{{\bf g}}$, which satisfies homogeneous boundary
conditions. The expansion of ${\bf u}={\bf U}+\widetilde{{\bf g}}$ then
follows.

######  Theorem B.10.

Suppose $\gamma>0$ and
$(\mathbb{f},\mathbb{g})\in\mathcal{R}V_{\beta,q}(\mathcal{Q},\gamma)$ for
$\beta\in(\beta_{r+1},\beta_{r})$ with $0<\beta_{r}-\beta<1$. Assume that for
all $\xi\in\mathbb{R}^{n-m}$, for all $\tau\in\mathbb{R}-i\gamma$ and for all
$\mathbb{w}_{\ell}^{k,j}$ corresponding to eigenvalues $\lambda_{\ell}$ of
$\mathcal{A}_{D}$ with $\mathrm{Im}\
\lambda\in[\beta_{r}+\frac{m}{2}-2,\beta_{1}+\frac{m}{2}-2]$ the relation
(128) holds. Then the solution $\mathbb{u}$ to (98), (99) admits an asymptotic
expansion

$\mathbb{u}(\mathbb{y},z,t)=\sum_{\ell}\sum_{k,j}(X\tilde{c}_{\ell}^{k,j})(\mathbb{y},z,t)\mathbb{u}_{\ell}^{k,j}(\mathbb{y})+\mathbb{u}_{0}(\mathbb{y},z,t)\
,$ (129)

with $\mathbb{u}_{0}\in DV_{\beta,q}(\mathcal{Q},\gamma)$. Here the first sum
is over all eigenvalues $\lambda_{\ell}$ with $\mathrm{Im}\
\lambda=\beta_{r}+\frac{m}{2}-2$, while the second sum is over all generalized
eigenfunctions $\mathbb{u}_{\ell}^{k,j}$ corresponding to $\lambda_{\ell}$.
The coefficients $\tilde{c}_{\ell}^{k,j}(z,t)$ are defined by

$\tilde{c}_{\ell}^{k,j}=\mathcal{F}^{-1}_{(\xi,\tau)\to(z,t)}c_{\ell}^{k,j},$

where

$c_{\ell}^{k,j}=p^{i\lambda_{\ell}}\sum_{q}\frac{1}{q!}(i\ln
p)^{q}d_{\ell}^{(k+q,j)}(\xi,\tau),$ (130)

and, with $p=\sqrt{|\xi|^{2}+|\tau|^{2}}$ and $\mathbb{w}_{\ell}^{k,j}$ as in
Theorem B.9,

$d_{\ell}^{(k+q,j)}(\xi,\tau)=p^{-2}(\hat{\mathbb{f}}(p^{-1}\cdot,\xi,\tau),\mathbb{w}_{\ell}^{k,j}(\cdot,\xi/p,\overline{\tau}/p))_{L^{2}(\mathbb{K})}+p^{-1}(\hat{\mathbb{g}}(p^{-1}\cdot,\xi,\tau),{\boldsymbol{p}}(\mathbb{w}_{\ell}^{k,j})(p^{-1}\cdot,\xi/p,\overline{\tau}/p))_{L^{2}(\partial\mathbb{K})}.$

Moreover, the following estimates hold: $\|e^{-\gamma
t}\tilde{d}_{\ell};H^{2-\frac{m}{2}-\beta}(\mathbb{R}^{n-m+1})\|\leq
c\|(\mathbb{f},\mathbb{g});\mathcal{R}V_{\beta,q}(\mathcal{Q},\gamma)\|$
and $\|\mathbb{u}_{0};DV_{\beta,q}(\mathcal{Q},\gamma)\|\leq
c\|(\mathbb{f},\mathbb{g});\mathcal{R}V_{\beta,q}(\mathcal{Q},\gamma)\|$.

The explicit formulas show that for $f$ smooth in time also the coefficients
$d_{\ell}$ will be smooth in time.

Analogous results for the Neumann problem may be obtained in a similar way,
see [36, 41]. The boundary condition affects the corresponding stencil
$\mathcal{A}_{N}$ and consequently its eigenvalues $i\lambda_{\ell}$ and
singular functions $\mathbb{w}_{\ell}^{k,j}$.

## Appendix C

We recall certain auxiliary results from [22], which are used in the proofs of
Theorem 5.3 and Theorem 5.7.

######  Lemma C.1 ([22], Lemma 3).

Let $\Gamma,\,\Gamma_{j}\;(j=1,\dots,N)$ be Lipschitz domains with
$\overline{\Gamma}=\bigcup\limits_{j=1}^{N}\overline{\Gamma}_{j}$,
$s\in[-1,1]$ and $r\in\mathbb{R}$. Then for all $\tilde{u}\in
H^{r}_{\sigma}(\mathbb{R}^{+},\widetilde{H}^{s}(\Gamma))$, $u\in
H^{r}_{\sigma}(\mathbb{R}^{+},H^{s}(\Gamma))$,

$\displaystyle\sum\limits_{j=1}^{N}\|u\|^{2}_{r,s,\Gamma_{j}}\leq\|u\|^{2}_{r,s,\Gamma}\
,\qquad\|\tilde{u}\|^{2}_{r,s,\Gamma,\ast}\leq\sum\limits_{j=1}^{N}\|\tilde{u}\|^{2}_{r,s,\Gamma_{j},\ast}\
.$ (131)

######  Lemma C.2 ([22], Lemma 8).

Let $I_{j}=[0,h_{j}]$, $r\in\mathbb{R}$, $0\leq s_{j}\leq 1$,
$f_{2}\in\widetilde{H}^{-s_{2}}(I_{2})$,
$f_{1}\in\widetilde{H}^{r}(\mathbb{R}^{+},H^{-s_{1}}(I_{1})$. Then there holds

$\|f_{1}(t,x)f_{2}(y)\|_{r,-s_{1}-s_{2},I_{1}\times
I_{2},\ast}\leq\|f_{1}\|_{r,-s_{1},I_{1},\ast}\|f_{2}\|_{\tilde{H}^{-s_{2}}(I_{2})}\
.$

A similar result holds in the positive Sobolev norms:

######  Lemma C.3 ([22], Lemma 9).

Let $I_{j}=[0,h_{j}],\;0\leq s\leq 1,\;f_{2}\in\widetilde{H}^{s}(I_{2})$,
$f_{1}\in\widetilde{H}^{s}(\mathbb{R}^{+},H^{s}(I_{1})$. Then there holds

$\|f_{1}(t,x)f_{2}(y)\|_{r,s,I_{1}\times
I_{2},\ast}\leq\|f_{1}\|_{r,s,I_{1},\ast}\|f_{2}\|_{\tilde{H}^{s}(I_{2})}\ .$

######  Lemma C.4 ([22], Lemma 10).

Let $0\leq r\leq\rho\leq q+1$, $-1\leq s\leq 0$,
$R=[0,h_{1}]\times[0,h_{2}],\;u\in H^{\rho}([0,\Delta t),H^{1}(R))$,
$\Pi_{t}^{q}u$ the orthogonal projection onto piecewise polynomials in $t$ of
order $q$, $\Pi_{x,y}^{0}u=\frac{1}{h_{1}h_{2}}\int\limits_{R}u(t,x,y)dy\,dx$.
Then for $p=\Pi_{t}^{q}\Pi_{x,y}u$ we have

$\displaystyle\|u-p\|_{r,s,R,\ast}$ $\displaystyle\lesssim(\Delta
t)^{\rho-r}{\max\\{h_{1},h_{2},\Delta
t\\}^{-s}}\|\partial_{t}^{\rho}u\|_{L^{2}([0,\Delta t)\times R)}$
$\displaystyle\qquad+\max\\{h_{1},h_{2},\Delta
t\\}^{-s}\left(h_{1}\|u_{x}\|_{L^{2}([0,\Delta t)\times
R)}+h_{2}\|u_{y}\|_{L^{2}({[0,\Delta t)\times}R)}\right).$ (132)

If $u(t,x,y)=u_{1}(t,x)u_{2}(y),\;u_{1}\in H^{{\rho}}([0,\Delta
t),H^{1}([0,h_{1}])),\;u_{2}\in H^{1}([0,h_{2}])$ then

$\|u-p\|_{r,s,R,\ast}\lesssim(\Delta t)^{\rho-r}{\max\\{h_{1},h_{2},\Delta
t\\}^{-s}}\|\partial_{t}^{\rho}u\|_{L^{2}([0,\Delta t)\times
R)}+\left(h_{1}^{1-s}\|u_{x}\|_{{L^{2}}([0,\Delta t)\times
R)}+h_{2}^{1-s}\|u_{y}\|_{{L^{2}}([0,\Delta t)\times R)}\right).$

######  Lemma C.5 ([22], Lemma 11).

Let $Q=[0,h_{1}]\times[0,h_{2}],u\in H^{3}([0,\Delta t)\times Q),p$ the
bilinear interpolant of $u$ at the vertices of $Q$. Then there holds for
$r\in\mathbb{R}$

$\displaystyle\|u-p\|_{r,0,[0,\Delta t)\times Q}$
$\displaystyle\lesssim\max\\{h_{1},\Delta t\\}^{2}\|u_{xx}\|_{r,0,[0,\Delta
t)\times Q}+\max\\{h_{2},\Delta t\\}^{2}\|u_{yy}\|_{r,0,[0,\Delta t)\times Q}$
$\displaystyle\qquad+(\max\\{h_{1},\Delta t\\}^{2}+\max\\{h_{2},\Delta
t\\}^{2})\|u_{tt}\|_{r,0,[0,\Delta t)\times Q}$
$\displaystyle\qquad+\max\\{h_{1},\Delta t\\}^{2}\max\\{h_{2},\Delta
t\\}\|u_{xxy}\|_{r,0,[0,\Delta t)\times Q}$ (133)
$\displaystyle\|(u-p)_{x}\|_{r,0,[0,\Delta t)\times Q}$
$\displaystyle\lesssim\max\\{h_{1},\Delta t\\}\|u_{xx}\|_{r,0,[0,\Delta
t)\times Q}+\max\\{h_{1},\Delta t\\}\|u_{xt}\|_{r,0,[0,\Delta t)\times Q}$
$\displaystyle\qquad+\max\\{h_{2},\Delta t\\}^{2}\|u_{xyy}\|_{r,0,[0,\Delta
t)\times Q}$ (134)

## Acknowledgements

This research was supported through the “Oberwolfach Research Fellows” program
in 2020.

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