Higher-order time domain boundary elements for elastodynamics: graded meshes and hp versions

Abstract

The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasi-optimal estimates for piecewise polynomial approximations of the Dirichlet trace of the solution and the traction. The results are applied to hp and graded versions of the time domain boundary element method for the weakly singular and the hypersingular integral equations. Numerical examples confirm the theoretical results for the Dirichlet and Neumann problems for screens and for poly- gonal domains in 2d. They exhibit the expected quasi-optimal convergence rates and the singular behavior of the solutions.

Appendices

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 [19, 29]. 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 Sect. 4. In the case of an open screen or line segment, $\partial \Gamma \ne \emptyset$, we first extend $\Gamma$ to a closed, orientable Lipschitz manifold $\widetilde{\Gamma }$. On $\Gamma$ we recall the usual Sobolev spaces of supported distributions:

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

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 $\omega \in \mathbb {C}\setminus \{0\}$:

$$\begin{aligned} ||u||_{s,\omega ,{\widetilde{\Gamma }}}=\left( \sum _{i=1}^p \int _{\mathbb {R}^n} (|\omega |^2+|\pmb {\xi }|^2)^s|{\mathcal {F}}\left\{ (\alpha _i u)\circ \varphi _i^{-1}\right\} (\pmb {\xi })|^2 d\pmb {\xi } \right) ^{\frac{1}{2}}\ . \end{aligned}$$

Here, ${\mathcal {F}}={\mathcal {F}}_{\varvec{x} \mapsto \pmb {\xi }}$ denotes the Fourier transform ${\mathcal {F}}\varphi (\pmb {\xi }) = \int e^{-i\varvec{x}\cdot \pmb {\xi }} \varphi (\varvec{x})\ d\varvec{x}$. Different $\omega \in \mathbb {C}\setminus \{0\}$ induce equivalent norms on $H^s(\Gamma )$, $\Vert u\Vert _{s,\omega ,\Gamma } = \inf _{v \in \widetilde{H}^s(\widetilde{\Gamma }\setminus \overline{\Gamma })} \ \Vert u+v\Vert _{s,\omega ,\widetilde{\Gamma }}$ and on $\widetilde{H}^s(\Gamma )$, $\Vert u\Vert _{s,\omega ,\Gamma , *} = \Vert e_+ u\Vert _{s,\omega ,\widetilde{\Gamma }}$. $e_+$ 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 $\Vert u\Vert _{s,\omega ,\Gamma , *}$ is stronger than $\Vert u\Vert _{s,\omega ,\Gamma }$.

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

$$\begin{aligned} H^r_\sigma (\mathbb {R}^+,{H}^s(\Gamma ))&=\{ u \in {\mathcal {D}}^{'}_{+}(H^s(\Gamma )): e^{-\sigma t} u \in {\mathcal {S}}^{'}_{+}(H^s(\Gamma )) \text { and } ||u||_{r,s,\Gamma }< \infty \}\ , \nonumber \\ H^r_\sigma (\mathbb {R}^+,\widetilde{H}^s({\Gamma }))&=\{ u \in {\mathcal {D}}^{'}_{+}(\widetilde{H}^s({\Gamma })): e^{-\sigma t} u \in {\mathcal {S}}^{'}_{+}(\widetilde{H}^s({\Gamma })) \text { and } ||u||_{r,s,\Gamma , *} < \infty \}\ . \end{aligned}$$

(85)

Here, ${\mathcal {D}}^{'}_{+}(E)$ 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 $E=\widetilde{H}^s({\Gamma })$. ${\mathcal {S}}^{'}_{+}(E)\subset {\mathcal {D}}^{'}_{+}(E)$ denotes the subspace of tempered distributions. The Sobolev spaces are endowed with the norms

$$\begin{aligned} \Vert u\Vert _{r,s}:=\Vert u\Vert _{r,s,\Gamma }&=\left( \int _{-\infty +i\sigma }^{+\infty +i\sigma }|\omega |^{2r}\ \Vert \hat{u}(\omega )\Vert ^2_{s,\omega ,\Gamma }\ d\omega \right) ^{\frac{1}{2}}\ ,\nonumber \\ \Vert u\Vert _{r,s,*} := \Vert u\Vert _{r,s,\Gamma ,*}&=\left( \int _{-\infty +i\sigma }^{+\infty +i\sigma }|\omega |^{2r}\ \Vert \hat{u}(\omega )\Vert ^2_{s,\omega ,\Gamma ,*}\ d\omega \right) ^{\frac{1}{2}}\,. \end{aligned}$$

(86)

They are Hilbert spaces. For $r=s=0$ they correspond to the weighted $L^2$-space with scalar product $\int _0^\infty e^{-2\sigma t} \int _\Gamma u \overline{v} d\Gamma _{\varvec{x}}\ dt$. Because $\Gamma$ is Lipschitz, these spaces are independent of the choice of $\alpha _i$ and $\varphi _i$ when $|s|\le 1$, as for standard Sobolev spaces.

We shall also use the norms $\Vert u\Vert _{r, s, (t_1, t_2]\times \Gamma }$ and $\Vert u\Vert _{r, s, (t_1, t_2]\times \Gamma ,*}$ for restrictions on the time interval $(t_1,t_2]$.

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

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 $r \in \mathbb {R}$:

$$\begin{aligned} {\mathcal {V}}&: H_\sigma ^{r+1}(\mathbb {R}^+, \widetilde{H}^{-\frac{1}{2}}(\Gamma )) \rightarrow H_\sigma ^{r}(\mathbb {R}^+, H^{\frac{1}{2}}(\Gamma )) \ , \ ,\\ {\mathcal {K}}'&: H_\sigma ^{r+1}(\mathbb {R}^+, \widetilde{H}^{-\frac{1}{2}}(\Gamma )) \rightarrow H_\sigma ^{r}(\mathbb {R}^+, H^{-\frac{1}{2}}(\Gamma ))\ ,\\ {\mathcal {K}}&: H_\sigma ^{r+1}(\mathbb {R}^+, \widetilde{H}^{\frac{1}{2}}(\Gamma )) \rightarrow H_\sigma ^{r}(\mathbb {R}^+, H^{\frac{1}{2}}(\Gamma )) \ , \\ {\mathcal {W}}&: H_\sigma ^{r+1}(\mathbb {R}^+, \widetilde{H}^{\frac{1}{2}}(\Gamma )) \rightarrow H_\sigma ^{r}(\mathbb {R}^+, H^{-\frac{1}{2}}(\Gamma ))\ . \end{aligned}$$

This may be found in Theorem 3.1 in [12], see also [8] for ${\mathcal {W}}$ in 2d, with an analogous proof. See also [31] 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 $\Gamma \times \mathbb {R}^+$,

$$\begin{aligned} B_{D,\Gamma \times \mathbb {R}^+}(\pmb { \Phi },\pmb {\tilde{\Phi }}) := \int _{\mathbb {R}^+}\int _\Gamma {\mathcal {V}} \partial _t \pmb {\Phi }(t,\varvec{x})\ \pmb {\tilde{\Phi }}(t,\varvec{x})\ d\Gamma _{\varvec{x}} \, d_\sigma t \ , \end{aligned}$$

(87)

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

$$\begin{aligned} B_{N,\Gamma \times \mathbb {R}^+}(\pmb { \Psi },\pmb {\tilde{\Psi }}) := \int _{\mathbb {R}^+}\int _\Gamma {\mathcal {W}} \partial _t \pmb {\Psi }(t,\varvec{x})\ \pmb {\tilde{\Psi }}(t,\varvec{x})\ d\Gamma _{\varvec{x}} \, d_\sigma t \ , \end{aligned}$$

(88)

Proposition A.3

Let $\sigma >0$.

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

$$\begin{aligned} |B_{D,\Gamma \times \mathbb {R}^+}(\pmb { \Phi },\pmb {\tilde{\Phi }})| \lesssim \Vert \pmb { \Phi }\Vert _{1,-\frac{1}{2},\Gamma , *} \Vert \pmb {\tilde{\Phi }}\Vert _{1,-\frac{1}{2}, \Gamma ,*} \end{aligned}$$

(89)

and

$$\begin{aligned} \Vert \pmb { \Phi }\Vert _{0,-\frac{1}{2},\Gamma ,*}^2 \lesssim B_{D,\Gamma \times \mathbb {R}^+}(\pmb { \Phi },\pmb { \Phi }) . \end{aligned}$$

(90)

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

$$\begin{aligned} |B_{N,\Gamma \times \mathbb {R}^+}(\pmb { \Psi },\pmb {\tilde{\Psi }})| \lesssim \Vert \pmb { \Psi }\Vert _{1,\frac{1}{2},\Gamma , *} \Vert \pmb {\tilde{\Psi }}\Vert _{1,\frac{1}{2}, \Gamma ,*} \end{aligned}$$

(91)

and

$$\begin{aligned} \Vert \pmb { \Psi }\Vert _{0,\frac{1}{2},\Gamma ,*}^2 \lesssim B_{N,\Gamma \times \mathbb {R}^+}(\pmb { \Psi },\pmb { \Psi }) . \end{aligned}$$

(92)

Proof

The inequalities (89) and (91) follow from the mapping properties in Theorem A.2. The coercivity (92) was shown in [7, 8] in 2d, and the proof holds verbatim in any dimension.

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

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

(93)

We assume $\textrm{Im}\ (\omega ) \ge \sigma >0$. 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

$$\begin{aligned} B_{D,\omega }([{{\textbf {p}}}],\overline{\pmb {q}}) = \langle -i\omega {\mathcal {V}}_{\omega } [{{\textbf {p}}}],\overline{\pmb {q}} \rangle _{\widetilde{\Gamma }}= \langle -i\omega {{\textbf {g}}},\overline{\pmb {q}}\rangle _{\widetilde{\Gamma }} \end{aligned}$$

(94)

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

It involves the single layer operator ${\mathcal {V}}_{\omega }$ obtained from ${\mathcal {V}}$ by Fourier transformation. Using Green’s formula as in [8], Thm 3.1, we have

$$\begin{aligned} \int _{\Omega '\cup \Omega } \left( \overline{\sigma ({{\textbf {u}}})} : \varepsilon ({{\textbf {u}}})-\rho \omega ^2\vert {{\textbf {u}}}\vert ^2\right) d {{\textbf {x}}}=\int _{\widetilde{\Gamma }} {{\textbf {u}}} \cdot \overline{[\sigma ({{\textbf {u}}}){{\textbf {n}}}]} d\widetilde{\Gamma }\equiv \langle {\mathcal {V}}_{\omega } [{{\textbf {p}}}],\overline{[{{\textbf {p}}}]} \rangle _{\widetilde{\Gamma }}. \end{aligned}$$

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

$$\begin{aligned} \textrm{Re}\, i\overline{\omega } \langle {\mathcal {V}}_{\omega } [{{\textbf {p}}}],\overline{[{{\textbf {p}}}]} \rangle _{\widetilde{\Gamma }}&= \textrm{Re} \left( i\overline{\omega } \int _{\Omega '\cup \Omega } \overline{\sigma ({{\textbf {u}}})}: \varepsilon ({{\textbf {u}}})d {{\textbf {x}}}\right) + \textrm{Re} \left( -i\omega \int _{\Omega '\cup \Omega } \rho \vert \omega \vert ^2\vert {{\textbf {u}}}\vert ^2 d {{\textbf {x}}}\right) \nonumber \\&=2 \textrm{Im} (\omega ) E_{\omega }\geqslant 0, \end{aligned}$$

(95)

with

$$\begin{aligned} E_{\omega }=\frac{1}{2}\int _{\Omega '\cup \Omega } \left( \overline{\sigma ({{\textbf {u}}})}: \varepsilon ({{\textbf {u}}})+ \rho \vert \omega \vert ^2\vert {{\textbf {u}}}\vert ^2 \right) d {{\textbf {x}}}\ . \end{aligned}$$

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

$$\begin{aligned} E_{\omega }\geqslant C_{\sigma }\Vert {{\textbf {u}}} \Vert ^2_{1,\omega ,\Omega '\cup \Omega } \end{aligned}$$

(96)

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

$$\begin{aligned} \vert \langle -i\omega {\mathcal {V}}_{\omega } [{{\textbf {p}}}],\overline{[{{\textbf {p}}}]} \rangle _{\widetilde{\Gamma }}\vert \geqslant \widetilde{C}_{\sigma }\Vert {{\textbf {u}}} \Vert ^2_{1,\omega ,\Omega '\cup \Omega }\ . \end{aligned}$$

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

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

Coercivity in the frequency domain follows:

$$\begin{aligned} \vert \langle -i\omega {\mathcal {V}}_{\omega } [{{\textbf {p}}}],\overline{[{{\textbf {p}}}]} \rangle _{\widetilde{\Gamma }}\vert \geqslant \frac{\widetilde{C}_{\sigma }}{2C_{trace}} \Vert [{{\textbf {p}}}] \Vert _{-1/2,\omega ,\widetilde{\Gamma }}^2. \end{aligned}$$

(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

$$\begin{aligned} \int _{\mathbb {R}+i\omega _I^0}\int _{\widetilde{\Gamma }}-i\omega {\mathcal {V}}_{\omega } \widehat{\pmb {\Phi }}\cdot \overline{\widehat{\pmb {\Phi }}}d\widetilde{\Gamma }d\omega =\int _0^{+\infty }\int _{\widetilde{\Gamma }}e^{-2\sigma t}\frac{\partial }{\partial t}\left( {\mathcal {V}}\pmb {\Phi }\right) \cdot \pmb {\Phi } d\widetilde{\Gamma }dt=B_D\left( \pmb {\Phi },\pmb {\Phi } \right) . \end{aligned}$$

We now use (97):

$$\begin{aligned} \textrm{Re}\, B_D\left( \pmb {\Phi },\pmb {\Phi } \right) = \int _{\mathbb {R}+i\sigma }\textrm{Re}\, i\overline{\omega } \langle {\mathcal {V}}_{\omega } \pmb {{\widehat{\Phi }}},\overline{\pmb {{\widehat{\Phi }}}} \rangle _{\widetilde{\Gamma }} \geqslant \frac{\widetilde{C}_{\sigma }}{2C_{trace}} \int _{\mathbb {R}+i \sigma } \Vert \pmb {{\widehat{\Phi }}} \Vert _{-1/2,\omega ,\widetilde{\Gamma }}^2 d\omega \ . \end{aligned}$$

Therefore

$$\begin{aligned} |B_D\left( \pmb {\Phi },\pmb {\Phi } \right) | \ge \frac{\widetilde{C}_{\omega _I}}{2C_{trace}} \Vert \pmb {\Phi }\Vert _{0,-1/2,\widetilde{\Gamma }}^2\ . \end{aligned}$$

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

Appendix B

In the following, let us describe the approach by Matyukevich and Plamenevskiǐ from [40] 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 [23], 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 [40], 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 $m \ge 2$ and $\mathbb {K} \subset \mathbb {R}^m$ an open cone. We use local coordinates $\varvec{x} = (\varvec{y},z)$ in the wedge $\mathbb {D}$.

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

$$\begin{aligned} {\mathcal {L}}(D_{\varvec{x}},D_{t})\varvec{u}(\varvec{x},t)&= \varvec{f}(\varvec{x},t),&(\varvec{x},t) \in \mathbb {D} \times \mathbb {R} \ , \end{aligned}$$

(98)

$$\begin{aligned} \varvec{u}(\varvec{x},t)&= \varvec{g}{(\varvec{x},t)},&(\varvec{x},t) \in \partial \mathbb {D} \times \mathbb {R}\ . \end{aligned}$$

(99)

with the matrix differential operator $({\mathcal {L}}(D_{\varvec{x}},D_{t})\varvec{u}(\varvec{x},t))_p=\partial _t^2\varvec{u}(\varvec{x},t) - \sum _{k,l,q=0}^{{n}} \partial _{k} a^{kl}_{pq}(\varvec{x})\partial _l u_q(\varvec{x},t)$, $p=1, \dots , {n}$.

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

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

(100)

We denote by ${\mathcal {A}}_D(\tau ) = {\mathcal {L}}(D_{\varvec{x}},\tau )$ the closure of this operator in $L^{2}(\mathbb {D})$. We first note a well-posedness result, Theorem 4.1.2 in [40].

Proposition B.1

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

$$\begin{aligned} \gamma ^{2} \int _{\mathbb {D} } (|\tau |^{2}|\varvec{v}(\varvec{x},\tau )|^{2} + |D_{\varvec{x}}\varvec{v}(\varvec{x},\tau )|^{2})d\varvec{x} \le c \int _{\mathbb {D} } |\hat{\varvec{f}}(\varvec{x},\tau )|^{2}d\varvec{x}\ . \end{aligned}$$

(101)

Proof

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

$$\begin{aligned} B^\tau _D(\varvec{u},\varvec{v}) = \int _{\mathbb {D} } \sum _{i,j,k,l} C_{kl}^{ij}(\varvec{x}) \partial _k u_{i}(\varvec{x}) \partial _l \overline{v_{j}(\varvec{x})} d\varvec{x} - \tau ^{2} \int _{\mathbb {D}} \varvec{u}(\varvec{x})\cdot \overline{\varvec{v}(\varvec{x})} d\varvec{x}\ , \end{aligned}$$

where $C_{kl}^{ij}$ denotes the Hooke tensor from Sect. 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 [40]: If $\tau ^2 \in \mathbb {C}\setminus \overline{\mathbb {R}_+}$, then there exists $\delta = \delta (\tau )>0$ such that $|B^\tau _D(\varvec{u},\varvec{u})|\ge \delta \Vert \varvec{u}; H^1(\mathbb {D})\Vert ^2$.

The assertion then follows by applying the Lax-Milgram theorem.$\square$

1.1 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 $\mathbb {K}$: $r = |\varvec{y}|$, $\pmb {\omega } = \frac{\varvec{y}}{|\varvec{y}|}$. We first assume that $\varvec{v}$ solves the non-homogeneous Dirichlet problem with parameters $\tau \in \mathbb {R}- i \gamma$ and $\xi \in \mathbb {R}$,

$$\begin{aligned} {\mathcal {L}}(D_{\varvec{y}}, \xi , \tau )\varvec{v}(\varvec{y}, \xi , \tau )&= \hat{\varvec{f}}(\varvec{y},\xi ,\tau ),\quad \varvec{y} \in \mathbb {K} \end{aligned}$$

(102)

$$\begin{aligned} \varvec{v}(\varvec{y},\xi , \tau )&=\hat{\varvec{g}}(\varvec{y},\xi ,\tau ), \quad \varvec{y} \in \partial \mathbb {K}\ . \end{aligned}$$

(103)

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

Proposition B.2

(Theorem 6.2.5, [40]) Let $\tau \in \mathbb {R}-i\gamma$ with $\gamma >0$. For all $\hat{\varvec{f}} \in L^2(\mathbb {K})$, There exists a unique, strong solution $\varvec{v}$ of (102), (103), and

$$\begin{aligned} \gamma ^2(p^2 \Vert \varvec{v};L^2(\mathbb {K})\Vert ^2 + \Vert D_{\varvec{x}} \varvec{v};L^2(\mathbb {K})\Vert ^2) \le c \Vert \hat{\varvec{f}};L^2(\mathbb {K})\Vert ^2 \ . \end{aligned}$$

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

Define the weighted Sobolev norms

$$\begin{aligned} \Vert v; H^{s}_\beta (\mathbb {K}) \Vert&= \left( \sum _{|\alpha | \le s} \int _{\mathbb {K}} r^{2(\beta + |\alpha | - s)} |D_{\varvec{x}}^\alpha v|^2 \right) ^{\frac{1}{2}} d\varvec{x}\ , \end{aligned}$$

(104)

$$\begin{aligned} \Vert v; H^{s}_\beta (\mathbb {K},p) \Vert&= \left( \sum _{k=0}^{s} p^{2k} \Vert v;H^{s-k}_\beta (\mathbb {K}) \Vert ^{2} \right) ^{\frac{1}{2}} \ . \end{aligned}$$

(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 $\chi _\tau (\varvec{x}) = \chi (|\tau | \varvec{y})$. From Proposition B.2 one obtains with $p =\sqrt{|\xi |^2+|\tau |^2}$, and c independent of $\xi$, $\tau$,

$$\begin{aligned}{} & {} \gamma ^2\Vert \varvec{v}; H^1_\beta (\mathbb {K}, p)\Vert ^2 + \Vert \chi _\tau \varvec{v}; H^2_\beta (\mathbb {K},p)\Vert ^2 \nonumber \\{} & {} \quad \le c \left\{ \Vert {\mathcal {L}}(D_{\varvec{y}},\xi ,\tau )\varvec{v}; H^0_\beta (\mathbb {K})\Vert ^2+ \frac{p^{2(1-\beta )}}{\gamma ^2}\Vert {\mathcal {L}}(D_{\varvec{y}},\xi ,\tau )\varvec{v}; L^2(\mathbb {K})\Vert \right\} . \end{aligned}$$

(106)

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

$$\begin{aligned} {\mathcal {A}}_D(\lambda )\pmb {\varphi } = \left\{ r^{2-i\lambda } {\mathcal {L}}(D_{\varvec{y}},0,0) r^{i\lambda }\pmb {\varphi },\pmb {\varphi }|_{\partial \Xi }\right\} \end{aligned}$$

(107)

defines a map

$$\begin{aligned} {\mathcal {A}}_D(\lambda ) : H^2(\Xi ) \rightarrow L^2(\Xi ) \times H^{3/2}(\partial \Xi )\ , \end{aligned}$$

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 ${\mathcal {L}}(D_{\varvec{x}},D_{t})\varvec{v} = \partial _t ^{2}\varvec{v} + A(D_{\varvec{x}})\varvec{v}$ with

$$\begin{aligned} A(D_{\varvec{x}}) = A(D_{\varvec{y}}, D_z) = D_{k} A^{kl} D_{l}\ , \end{aligned}$$

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

$$\begin{aligned} \left\{ r^{2-i\lambda } A(D_{\varvec{y}},0) r^{i\lambda }\pmb {\varphi },\pmb {\varphi }|_{\partial \Xi }\right\} \ . \end{aligned}$$

(108)

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

$$\begin{aligned} \varvec{u}_\ell (\varvec{y}) = r^{i\lambda _\ell } \sum _{q=0}^k \frac{1}{q!} (i \ln (r))^q \pmb {\varphi }_\ell ^{(k-q)}(\pmb {\omega }) \end{aligned}$$

(109)

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

$$\begin{aligned} {\mathcal {L}}(D_{\varvec{y}}, 0, 0)\varvec{u}(\varvec{y})&= 0,\quad \varvec{y} \in \mathbb {K}\ , \end{aligned}$$

(110)

$$\begin{aligned} \varvec{u}(\varvec{y})&=0, \quad \varvec{y} \in \partial \mathbb {K}\ . \end{aligned}$$

(111)

Here, $\{\pmb {\varphi }_\ell ^{(0)}, \dots , \pmb {\varphi }_\ell ^{(k)}\}$ is a Jordan chain to $\lambda _\ell$, consisting of an eigenvector $\pmb {\varphi }_\ell ^{(k)}$ and generalized eigenvectors $\pmb {\varphi }_\ell ^{(0)}, \dots , \pmb {\varphi }_\ell ^{(k-1)}$. Let $\kappa _1 \ge \kappa _2 \ge \dots \ge \kappa _J$ denote the partial multiplicities of the $\lambda _\ell$ , and let $\{\pmb {\varphi }_\ell ^{(0,j)}, \dots , \pmb {\varphi }_\ell ^{(\kappa _j-1,j)} : j=1, \dots , J\}$ be a canonical system of Jordan chains. The functions

$$\begin{aligned} \varvec{u}^{(k,j)}_\ell (\varvec{y}) = r^{i \lambda _\ell } \sum _{q=0}^k \frac{1}{q!} (i \ln (r))^q \pmb {\varphi }_\ell ^{(k-q,j)}(\pmb {\omega }), \end{aligned}$$

(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 Sect. 3.1 for a discussion of the eigenvalues and eigenfunctions in the case of a polygon, Sect. 3.2 for an edge, and Section 3.3 for a circular cone.

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

The dual vector functions

$$\begin{aligned} \varvec{v}^{(k,j)}_\ell (\varvec{y}) = r^{i \overline{\lambda _\ell } - (m-2)} \sum _{q=0}^k \frac{1}{q!} (i \ln (r))^q \pmb {\psi }^{(k-q,j)}_\ell (\pmb {\omega }), \end{aligned}$$

(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 [40]), respectively [44].

Denote by $\{\varvec{u}_\ell ^{k,j}\}$, $\{\varvec{v}_\ell ^{k,j}\}$ 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),

$$\begin{aligned} {\mathcal {L}}(D_{\varvec{y}}, \xi , \tau )\varvec{v}(\varvec{y}, \xi , \tau )&= 0,\quad \varvec{y} \in \mathbb {K} \end{aligned}$$

(114)

$$\begin{aligned} \varvec{v}(\varvec{y},\xi , \tau )&=0, \quad \varvec{y} \in \partial \mathbb {K}\ . \end{aligned}$$

(115)

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

$$\begin{aligned} \varvec{U}_\ell ^{(k,j)}(\varvec{y},\xi ,\tau ) = \sum _{q=0}^\infty r^{i\lambda _\ell + q} \varvec{P}{(k,j)}_q(\pmb {\omega },\xi ,\tau ,\ln (r)) \end{aligned}$$

(116)

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

$$\begin{aligned} \varvec{V}_\ell ^{(k,j)}(\varvec{y},\xi ,\tau ) = \sum _{q=0}^\infty r^{i(\overline{\lambda _\ell }+i{(m-n-2)}) + q} \varvec{Q}{(k,j)}_q(\pmb {\omega },\xi ,\tau ,\ln (r)), \end{aligned}$$

(117)

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

In reference [40] the formal series $\varvec{U}_\ell ^{(k,j)}$, $\varvec{V}_\ell ^{(k,j)}$ are constructed for these bases.

Consider now (102), (103) with $\chi \varvec{v}\in H^2_\beta (\mathbb {K})$, $\hat{\varvec{f}} \in H^0_\beta (\mathbb {K}) \cap H^0_\gamma (\mathbb {K})$, 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 $\{\lambda \in \mathbb {C} : \textrm{Im} \ \lambda = \gamma + \frac{m}{2}-2\}$ does not intersect the spectrum of the pencil ${\mathcal {A}}_D$, then we have

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

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

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:

$$\begin{aligned} \Vert \varvec{v}; DH_\beta (\mathbb {K}, \xi , \tau ) \Vert = \left( \gamma ^2 \Vert \varvec{v}; H^1_\beta (\mathbb {K},p)\Vert ^2 + \Vert \chi _p \varvec{v}; H^2_\beta (\mathbb {K},p)\Vert ^2 \right) ^{\frac{1}{2}}\ ,\\ \Vert \hat{\varvec{f}}; RH_\beta (\mathbb {K}, \xi , \tau ) \Vert = \left( \Vert \hat{\varvec{f}}; H^0_\beta (\mathbb {K})\Vert ^2 + p^{2(1-\beta )} \gamma ^{-2}\Vert \hat{\varvec{f}}; L^2(\mathbb {K})\Vert ^2 \right) ^{\frac{1}{2}} \ , \end{aligned}$$

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 ${\mathcal {L}}(D_{\varvec{y}},\xi ,\tau )$ from Problem (102), (103), defines a continuous map ${\mathcal {L}}(D_{\varvec{y}},\xi ,\tau ): DH_\beta (\mathbb {K}, \xi , \tau ) \rightarrow RH_\beta (\mathbb {K}, \xi , \tau )$.

In [40], Matyukevich and Plamenevskiǐ investigate the dependence of properties of ${\mathcal {L}}(D_{\varvec{y}},\xi ,\tau )$ on $\beta$. Let $1>\beta _1>\beta _2> \dots$ be numbers in $(-\infty ,1]$ such that every line $\{\lambda \in \mathbb {C} : \textrm{Im}\ \lambda = \beta _r+\frac{m}{2}-2\}$ 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, [40]) Suppose that $\beta \in (\beta _1,1]$, $\gamma >0$ and $\hat{\varvec{f}} \in RH_\beta (\mathbb {K}, \xi , \tau )$. Then (102), (103) with right hand side $\hat{\varvec{f}}$ admits a unique solution $\varvec{v}$ satisfying

$$\begin{aligned} \Vert \varvec{v}; DH_\beta (\mathbb {K}, \xi , \tau ) \Vert \le c \Vert \hat{\varvec{f}}; RH_\beta (\mathbb {K}, \xi , \tau ) \Vert , \end{aligned}$$

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

Theorem B.5

(Proposition 6.4.1, [40]) Suppose $\gamma >0$, $\beta \in (\beta _{r+1}, \beta _r)$, $0<\beta _r-\beta <1$, $\hat{\varvec{f}} \in RH_\beta (\mathbb {K}, \xi , \tau )$ and

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

for all $\varvec{w}_{\ell }^{(k,j)}$ corresponding to eigenvalues of ${\mathcal {A}}_D$ in the strip $\{\textrm{Im} \ \lambda \in (\beta _{r+1}+\frac{m}{2}-2, \beta _1+\frac{m}{2}-2)\}$. Then the solution $\varvec{v}$ of (102), (103), admits the representation

$$\begin{aligned} \varvec{v}(\varvec{y},\xi ,\tau ) = \chi (p\varvec{y}) \sum _{\ell } \sum _{k,j}c^{(k,j)}_\ell (\xi ,\tau ) \varvec{u}_\ell ^{(k,j)}(\varvec{y}) + \varvec{v}_0(\varvec{y},\xi ,\tau )\ . \end{aligned}$$

Here the outer summmation over $\ell$ sums over all eigenvalues $\lambda _\ell$ of the pencil with $\textrm{Im}\ \lambda = \beta _r+\frac{m}{2}-2$, while the inner summation sums over a basis $\{\varvec{u}_\ell ^{(k,j)}\}$ of power-like solutions as in (109) corresponding to $\lambda _\ell$. The remainder $\varvec{v}_0$ belongs to $DH_\beta (\mathbb {K}, \xi , \tau )$.

There holds

$$\begin{aligned} c_\ell ^{(k,j)}(\xi ,\tau ) = p^{i\lambda _\ell } \sum _q \frac{1}{q!} (i \ln (p))^q d_\ell ^{(k+q,j)}(\xi ,\tau )\ , \end{aligned}$$

with

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

Moreover there holds

$$\begin{aligned}{} & {} \Vert \varvec{v}_0; DH_\beta (\mathbb {K}, \xi , \tau ) \Vert \le c \Vert \hat{\varvec{f}}; RH_\beta (\mathbb {K}, \xi , \tau ) \Vert , \\{} & {} |d_\ell ^{(k,j)}(\xi ,\tau )|\le c p^{\beta +\frac{m}{2}-2}\Vert \hat{\varvec{f}}; RH_\beta (\mathbb {K}, \xi , \tau ) \Vert , \end{aligned}$$

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

1.2 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 $\mathbb {D}$,

$$\begin{aligned} {\mathcal {L}}(\varvec{x},D_{\varvec{x}},\tau )\varvec{v}(\varvec{x},\tau ) = \hat{\varvec{f}}(\varvec{x},\tau ), \quad \varvec{x} \in \mathbb {D}, \end{aligned}$$

(118)

$$\begin{aligned} \varvec{v}(\varvec{x},\tau ) = \hat{\varvec{g}}(\varvec{x},\tau ), \quad \varvec{x} \in \partial \mathbb {D}\ , \end{aligned}$$

(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 $\varvec{x}=(\varvec{y},z)$ and introduces polar coordinates in $\mathbb {K}$: $r = |\varvec{y}|$, $\pmb {\omega } = \frac{\varvec{y}}{|\varvec{y}|}$. Define

$$\begin{aligned} \Vert u; H^{s}_\beta (\mathbb {D}) \Vert = \left( \sum _{|\alpha | \le s} \int _{\mathbb {D}} r^{{2(}\beta + |\alpha | - s{)}} |D_{\varvec{x}}^\alpha u|^2 \right) ^{\frac{1}{2}}\ , \end{aligned}$$

(120)

$$\begin{aligned} \Vert u; H^{s}_\beta (\mathbb {D},p) \Vert = \left( \sum _{k=0}^{s} p^{2k} \Vert u;H^{s-k}_\beta (\mathbb {D}) \Vert ^{2} \right) ^{\frac{1}{2}} \ . \end{aligned}$$

(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, [40]) Suppose that the wedge $\mathbb {D}$ is admissible in the sense of [40], $\{ \hat{\varvec{f}},\hat{\varvec{g}} \} \in L^{2}(\mathbb {D}) \times H^{1}(\partial \mathbb {D})$ and $\tau = \sigma - i \gamma$, $\sigma \in \mathbb {R}$, $\gamma > 0$. Then there exists a unique strong solution $\varvec{v}$ of (118) and (119). Furthermore, there exists a constant $c>0$ independent of $\tau$ such that

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

Higher regularity has been obtained by Matyukevich and Plamenevskiǐ in the spaces $H^{s}_\beta (\mathbb {D})$. Following [40] we only state the result for homogeneous boundary conditions.

Proposition B.7

(Proposition 5.1.1, [40]) Let $\beta \le 1$. Assume $\textrm{Im}\ \lambda = \beta + \frac{m}{2}-2$ does not intersect the spectrum of ${\mathcal {A}}_{D}$. Then for $\varvec{v} \in H^{2}_{\beta }(\mathbb {D},1) \cap H^{1}_{\beta =0}(\mathbb {D})$ with ${\mathcal {L}}(D_{\varvec{x}},0)\varvec{v} \in L^{2}(\mathbb {D})$ there holds

$$\begin{aligned}&\Vert \chi _\tau \varvec{v}, H^2_\beta (\mathbb {D}, |\tau |)\Vert ^2 + \gamma ^2 \Vert \varvec{v}, H^1_{\beta }(\mathbb {D}, |\tau |)\Vert ^2 \nonumber \\ {}&\qquad \le c \left\{ \Vert {\mathcal {L}}(D_{\varvec{x}},\tau )\varvec{v}, H^0_\beta (\mathbb {D})\Vert ^2 + |\tau |^{2(1-\beta )} \gamma ^{-2} \Vert {\mathcal {L}}(D_{\varvec{x}},\tau )\varvec{v}, L^2(\mathbb {D})\Vert ^2 \right\} \ , \end{aligned}$$

(122)

where $\chi _\tau (\varvec{x}) = \chi (|\tau | \varvec{y})$ 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 $\varvec{v}$, $\tau = \sigma -i\gamma$, $\sigma \in \mathbb {R}$, $\gamma >0$.

A corresponding result for the wave equation with inhomogeneous boundary conditions has been considered in [46], Formula (7), but we omit the more involved statement.

The proof in [40] is based on three steps: (i) estimates far from the edge, (ii) estimates near the edge, (iii) the global a priori estimate (101).

1.3 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 $\varvec{x}=(\varvec{y},z) \in \mathbb {D}$ and parameter $q>0$:

$$\begin{aligned} \Vert w; H^s_\beta ({\mathcal {Q}})\Vert = \left( \sum _{|\alpha | \le s} \int _{\mathbb {R}} \int _{\mathbb {D}}r^{2(\beta -s+|\alpha |)} |D_{\varvec{x},t}^\alpha w(\varvec{x},t)|^2 \ d\varvec{x} \ dt\right) ^{1/2} \ , \\ \Vert w; H^s_\beta ({\mathcal {Q}},q)\Vert = \left( \sum _{k=0}^{s} q^{2k} \Vert w; H^{s-k}_\beta (Q)\Vert ^2\right) ^{1/2}\ . \end{aligned}$$

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

$$\begin{aligned} \Vert w; V^s_\beta ({\mathcal {Q}},\gamma )\Vert = \Vert w^\gamma ; H^s_\beta ({\mathcal {Q}},\gamma )\Vert \ . \end{aligned}$$

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 $(\varvec{f},\varvec{g}) \in V^0_0({\mathcal {Q}},\gamma ) \times V^{3/2}_0(\partial {\mathcal {Q}},\gamma )$, and let $\varvec{v}$ be a strong solution to (118), (119) in $\mathbb {D}$ with right hand side $(\hat{\varvec{f}}, \hat{\varvec{g}})$. Then $\varvec{u}(\varvec{y},z,t) = {\mathcal {F}}^{-1}_{{(\xi ,\tau ) \rightarrow (z,t)}} \varvec{v}(\varvec{y},\xi ,\tau )$ is called a strong solution of (98), (99).

Proposition B.6 implies that for any $(\varvec{f},\varvec{g}) \in V^0_0({\mathcal {Q}},\gamma ) \times V^{3/2}_0(\partial {\mathcal {Q}},\gamma )$ with $\gamma >0$ the problem (98), (99) admits a unique strong solution and

$$\begin{aligned}{} & {} \gamma ^2 \Vert \varvec{{u}}; V^1_0({\mathcal {Q}},\gamma )\Vert ^2 +\gamma \Vert \varvec{p}(\varvec{{u}}), V^0_\gamma (\partial \mathbb {D},\gamma ) \Vert ^2\\{} & {} \quad \le c\left( \Vert \varvec{f}; V^0_0({\mathcal {Q}},\gamma )\Vert ^2+\gamma \Vert \varvec{g}; V^{3/2}_0(\partial {\mathcal {Q}},\gamma )\Vert ^2\right) \ , \end{aligned}$$

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

$$\begin{aligned} X \varvec{u}(\varvec{y},z,t) = {\mathcal {F}}^{-1}_{(\xi ,\tau ) \rightarrow (z,t)} \chi (p\varvec{y}) {\mathcal {F}}_{(z',t') \rightarrow (\xi ,\tau )} \varvec{u}(\varvec{y},z',t') \end{aligned}$$

(123)

and

$$\begin{aligned} \Lambda ^\mu \varvec{u}(\varvec{y},z,t) = {\mathcal {F}}^{-1}_{\tau \rightarrow t} |\tau |^\mu {\mathcal {F}}_{t'\rightarrow \tau } \varvec{u}(\varvec{y},z,t')\ . \end{aligned}$$

(124)

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

$$\begin{aligned} \Vert \varvec{v}; DV_{\beta }({\mathcal {Q}},\gamma )\Vert&= \left( \gamma ^2 \Vert \varvec{v}; V^1_\beta ({\mathcal {Q}},\gamma )\Vert ^2 + \Vert X\varvec{v}; V_\beta ^2({\mathcal {Q}},\gamma )\Vert ^2+ \gamma \Vert \partial _\nu \varvec{v}; V^0_\beta (\partial {\mathcal {Q}},\gamma ) \Vert ^2\right) ^{1/2}\ , \end{aligned}$$

(125)

$$\begin{aligned} \Vert \varvec{f}; RV_{\beta }({\mathcal {Q}},\gamma )\Vert&= \left( \Vert \varvec{f}; V^0_\beta ({\mathcal {Q}},\gamma )\Vert ^2 + \gamma ^{-2}\Vert \Lambda ^{1-\beta }\varvec{f}; V^0_0({\mathcal {Q}},\gamma )\Vert ^2\right) ^{1/2}\ . \end{aligned}$$

(126)

$$\begin{aligned} \Vert (\varvec{f},\varvec{g}); {\mathcal {R}}V_{\beta }({\mathcal {Q}},\gamma )\Vert&= \left( \Vert \varvec{f}; RV_\beta ({\mathcal {Q}},\gamma )\Vert ^2 + \Vert X\varvec{g}; V_\beta ^{3/2}(\partial {\mathcal {Q}},\gamma )\Vert ^2+ \gamma \Vert \varvec{g}; V^1_0(\partial {\mathcal {Q}},\gamma ) \Vert ^2\right. \nonumber \\ {}&\qquad \left. + \gamma ^{-1} \Vert \Lambda ^{1-\beta }\varvec{g}; V^1_0(\partial {\mathcal {Q}},\gamma ) \Vert ^2\right) ^{1/2}\ . \end{aligned}$$

(127)

More generally, one may introduce for $q \in \mathbb {N}_0$

$$\begin{aligned}{} & {} \Vert \varvec{f}; RV_{\beta ,q}({\mathcal {Q}},\gamma )\Vert \\{} & {} \quad = \left( \sum _{j=0}^q \gamma ^{-2j}\Vert \Lambda ^{j}\varvec{f}; V^{q-j}_{\beta +q-j}({\mathcal {Q}},\gamma )\Vert ^2 + \gamma ^{-2(1+q)}\Vert \Lambda ^{1-\beta +q}\varvec{f}; V^0_0({\mathcal {Q}},\gamma )\Vert ^2\right) ^{1/2}\ , \end{aligned}$$

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, [40], for $\mathbf{{g}}=0$ and $q=0$. It may be extended to inhomogeneous boundary conditions and $q>0$ using the arguments in [33].

Theorem B.9

Suppose $q \in \mathbb {N}_0$, $\gamma >0$ and $(\varvec{f},\varvec{g}) \in {\mathcal {R}}V_{\beta ,q}({\mathcal {Q}},\gamma )$. a) If $\beta \in (\beta _1,1)$, the strong solution $\varvec{u}$ to (98), (99) belongs to $DV_{\beta ,q}({\mathcal {Q}},\gamma )$ and there exists $c>0$ independent of $\gamma$ such that

$$\begin{aligned} \Vert \varvec{u}; DV_{\beta ,q}({\mathcal {Q}},\gamma )\Vert \le c \Vert (\varvec{f},\varvec{g}); {\mathcal {R}}V_{\beta ,q}({\mathcal {Q}},\gamma )\Vert \ . \end{aligned}$$

b) If $\beta \in (\beta _{r+1},\beta _r)$, then there exists a solution $\varvec{u}$ 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 $\varvec{w}_\ell ^{k,j}$ corresponding to eigenvalues $\lambda _\ell$ of ${\mathcal {A}}_D$ with $\textrm{Im}\ \lambda \in [\beta _r + \frac{m}{2}-2, \beta _1 + \frac{m}{2}-2]$,

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

(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, [40], for $\mathbf{{g}}=0$ and $q=0$. The extension to inhomogeneous boundary data $\mathbf{{g}}$ follows as in Sect. 3: choose an extension $\widetilde{\textbf{g}}$ in the domain with Dirichlet trace $\mathbf{{g}}$. Theorem 7.5, [40] then assures an asymptotic expansion of the function $\textbf{U} = \textbf{u}-\widetilde{\textbf{g}}$, which satisfies homogeneous boundary conditions. The expansion of $\textbf{u} = \textbf{U} + \widetilde{\textbf{g}}$ then follows.

Theorem B.10

Suppose $\gamma >0$ and $(\varvec{f},\varvec{g}) \in {\mathcal {R}}V_{\beta ,q}({\mathcal {Q}},\gamma )$ 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 $\varvec{w}_\ell ^{k,j}$ corresponding to eigenvalues $\lambda _\ell$ of ${\mathcal {A}}_D$ with $\textrm{Im}\ \lambda \in [\beta _r + \frac{m}{2}-2, \beta _1 + \frac{m}{2}-2]$ the relation (128) holds. Then the solution $\varvec{u}$ to (98), (99) admits an asymptotic expansion

$$\begin{aligned} \varvec{u}(\varvec{y},z,t) = \sum _\ell \sum _{k,j} (X \tilde{c}_\ell ^{k,j})(\varvec{y},z,t) \varvec{u}_\ell ^{k,j}(\varvec{y}) + \varvec{u}_0(\varvec{y},z,t)\ , \end{aligned}$$

(129)

with $\varvec{u}_0 \in DV_{\beta ,q}({\mathcal {Q}},\gamma )$. Here the first sum is over all eigenvalues $\lambda _\ell$ with $\textrm{Im}\ \lambda =\beta _r + \frac{m}{2}-2$, while the second sum is over all generalized eigenfunctions $\varvec{u}_\ell ^{k,j}$ corresponding to $\lambda _\ell$. The coefficients $\tilde{c}_\ell ^{k,j}(z,t)$ are defined by

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

where

$$\begin{aligned} c_\ell ^{k,j} = p^{i\lambda _\ell } \sum _q \frac{1}{q!} (i \ln p)^q d_\ell ^{(k+q,j)}(\xi ,\tau ), \end{aligned}$$

(130)

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

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

Moreover, the following estimates hold: $\Vert e^{-\gamma t} \tilde{d}_\ell ; H^{2-\frac{m}{2}-\beta }(\mathbb {R}^{n-m+1})\Vert \le c \Vert (\varvec{f},\varvec{g}); {\mathcal {R}}V_{\beta ,q}({\mathcal {Q}},\gamma )\Vert$ and $\Vert \varvec{u}_0; DV_{\beta ,q}({\mathcal {Q}},\gamma )\Vert \le c \Vert (\varvec{f},\varvec{g}); {\mathcal {R}}V_{\beta ,q}({\mathcal {Q}},\gamma )\Vert$.

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 [34, 40]. The boundary condition affects the corresponding stencil ${\mathcal {A}}_N$ and consequently its eigenvalues $i \lambda _\ell$ and singular functions $\varvec{w}_\ell ^{k,j}$.

Appendix C

We recall certain auxiliary results from [21], which are used in the proofs of Theorem 5.3 and Theorem 5.7.

Lemma C.1

( [21], 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 $\tilde{u}\in H^r_\sigma (\mathbb {R}^+,\widetilde{H}^s(\Gamma ))$, $u\in H^r_\sigma (\mathbb {R}^+,H^s(\Gamma ))$,

$$\begin{aligned} \sum \limits _{j=1}^N \Vert u\Vert ^2_{r,s,\Gamma _j} \le \Vert u\Vert ^2_{r,s,\Gamma } \ ,\qquad \Vert \tilde{u}\Vert ^2_{r,s,\Gamma , *} \le \sum \limits _{j=1}^N \Vert \tilde{u}\Vert ^2_{r,s,\Gamma _j, *}\ . \end{aligned}$$

(131)

Lemma C.2

( [21], Lemma 8) Let $I_j = [0, h_j]$, $r\in \mathbb {R}$, $0\le s_j\le 1$, $f_2\in \widetilde{H}^{-s_2}(I_2)$, $f_1 \in \widetilde{H}^r(\mathbb {R}^+, H^{-s_1}(I_1)$. Then there holds

$$\begin{aligned} \Vert f_1(t,x) f_2(y) \Vert _{r, -s_1 - s_2, I_1\times I_2, *} \le \Vert f_1\Vert _{r, -s_1, I_1, *} \Vert f_2\Vert _{\tilde{H}^{-s_2}(I_2)} \ . \end{aligned}$$

A similar result holds in the positive Sobolev norms:

Lemma C.3

( [21], Lemma 9) Let $I_j = [0, h_j],\; 0\le s\le 1,\; f_2\in \widetilde{H}^{s}(I_2)$, $f_1 \in \widetilde{H}^s(\mathbb {R}^+, H^{s}(I_1)$. Then there holds

$$\begin{aligned} \Vert f_1(t,x) f_2(y) \Vert _{r, s, I_1\times I_2, *} \le \Vert f_1\Vert _{r, s, I_1, *} \Vert f_2\Vert _{\tilde{H}^{s}(I_2)}\ . \end{aligned}$$

Lemma C.4

( [21], Lemma 10) Let $0 \le r \le \rho \le q+1$, $-1\le s\le 0$, $R=[0,h_1]\times [0,h_2],\; u\in H^{\rho }([0,\Delta t), H^1(R))$, $\Pi _t^q u$ 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 $p = \Pi _t^q \Pi ^0_{x,y} u$ we have

$$\begin{aligned}&\Vert u-p\Vert _{r,s,R,*}\lesssim (\Delta t)^{\rho -r}{\max \{h_1,h_2,\Delta t \}^{-s}}\Vert \partial _t^\rho u\Vert _{L^2([0,\Delta t)\times R)} \nonumber \\&\quad + \max \{h_1,h_2,\Delta t \}^{-s}\left( h_1 \Vert u_x\Vert _{L^2([0,\Delta t) \times R)} + h_2 \Vert u_y\Vert _{L^2({[0,\Delta t) \times }R)} \right) . \end{aligned}$$

(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

$$\begin{aligned}{} & {} \Vert u-p \Vert _{r,s,R,*}\lesssim (\Delta t)^{\rho -r}{\max \{h_1,h_2,\Delta t \}^{-s}}\Vert \partial _t^\rho u\Vert _{L^2([0,\Delta t)\times R)} \nonumber \\{} & {} \quad +\left( h_1^{1-s}\Vert u_x\Vert _{{L^2}([0,\Delta t) \times R)} + h_2^{1-s}\Vert u_y\Vert _{{L^2}([0,\Delta t) \times R)} \right) . \end{aligned}$$

Lemma C.5

( [21], 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 $r\in \mathbb {R}$

$$\begin{aligned} \Vert u-p\Vert _{r,0,[0,\Delta t)\times Q}&\lesssim \max \{h_1, \Delta t\}^2\Vert u_{xx}\Vert _{r,0,[0,\Delta t)\times Q} \nonumber \\&\quad + \max \{h_2, \Delta t\}^2\Vert u_{yy}\Vert _{r,0,[0,\Delta t)\times Q}\nonumber \\&\qquad + ( \max \{h_1, \Delta t\}^2+ \max \{h_2, \Delta t\}^2 )\Vert u_{tt}\Vert _{r,0,[0,\Delta t)\times Q} \nonumber \\ {}&\qquad + \max \{h_1, \Delta t\}^2 \max \{h_2, \Delta t\}\Vert u_{xxy}\Vert _{r,0,[0,\Delta t)\times Q} \end{aligned}$$

(133)

$$\begin{aligned} \Vert (u-p)_x\Vert _{r,0,[0,\Delta t)\times Q}&\lesssim \max \{h_1, \Delta t\}\Vert u_{xx}\Vert _{r,0,[0,\Delta t)\times Q} \nonumber \\&\qquad + \max \{h_1, \Delta t\}\Vert u_{xt}\Vert _{r,0,[0,\Delta t)\times Q} \nonumber \\ {}&\qquad + \max \{h_2, \Delta t\}^2\Vert u_{xyy}\Vert _{r,0,[0,\Delta t)\times Q}. \end{aligned}$$

(134)