exex∎ defndefn∎ remrem∎ *0.5in0.5in plain[]0 mystyle[] [0][][M. Capoferri and
I. Mann]Spectral asymptotics for linear elasticity: the case of mixed boundary
conditions0

# Spectral asymptotics for linear elasticity:
the case of mixed boundary conditions111MSC2020: Primary 35P20. Secondary
35Q74, 74J05.222Keywords:  elasticity, eigenvalue counting function, Dirichlet
conditions, free boundary conditions, mixed boundary conditions.

Matteo Capoferri  MC: Maxwell Institute for Mathematical Sciences and
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK;
<EMAIL_ADDRESS>https://mcapoferri.com.  0000-0001-6226-1407 Isabel Mann
IM: Department of Mathematics, University of York, York YO10 5DD, UK;
<EMAIL_ADDRESS>
###### Abstract

We establish two-term spectral asymptotics for the operator of linear
elasticity with mixed boundary conditions on a smooth compact Riemannian
manifold of arbitrary dimension. We illustrate our results by explicit
examples in dimension two and three, thus verifying our general formulae both
analytically and numerically.

###### Contents

  1. 1 Introduction
    1. 1.1 The operator of linear elasticity
    2. 1.2 Boundary value problems
    3. 1.3 Statement of the problem and main results
  2. 2 Proof of Theorem 1.8
    1. 2.1 A streamlined algorithm
    2. 2.2 Invariant subspaces
    3. 2.3 The proof
      1. 2.3.1 Computing $\mathrm{shift}_{P,\aleph}$: the two-dimensional case
      2. 2.3.2 Computing $\mathrm{shift}_{\perp,\aleph}$: normally polarised waves
      3. 2.3.3 Putting things together
  3. 3 Explicit examples
    1. 3.1 Two-dimensional examples
      1. 3.1.1 The disk
      2. 3.1.2 Flat cylinders
    2. 3.2 Three-dimensional examples
      1. 3.2.1 Flat cylinders
  4. Acknowledgements
  5. References

##  Appendix 1.§1. Introduction

The operator of linear elasticity is one of the fundamental operators of
mathematical physics, describing the deformation of an (isotropic) elastic
body. The main thrust of this paper is to derive an explicit formula for the
second asymptotic term (often called _second Weyl coefficient_) in the
expansion of the eigenvalue counting function for the operator of linear
elasticity with mixed boundary conditions on a smooth $d$-dimensional
Riemannian manifold with boundary. This paper complements the analysis
performed in [6], where the two cases of “pure” Dirichlet and free boundary
conditions were examined.

The structure of the paper as follows.

In subsections 1.1 and 1.2 we introduce setting and notation, before stating
the problem and our main results in subsection 1.3.

Section 2 is devoted to the proof of our main result, Theorem 1.8. The proof
comes in several steps: first we present a streamlined version of the
algorithm for the calculation of the second asymptotic term (subsection 2.1);
secondly, we reduce the problem at hand to the two-dimensional analogue plus a
much simpler $(d-2)$-dimensional problem by identifying appropriate invariant
subspaces (subsection 2.2); lastly, we prove our main result by implementing
the algorithm in each invariant subspace separately (subsection 2.3).

In section 3 we examine explicit examples in dimensions two and three. This is
an integral part of the paper which serves both as an illustration and a
verification of our results. Remarkably, for two- and three-dimensional flat
cylinders we write down the full spectrum of the operator of linear elasticity
with mixed boundary conditions explicitly, and compute the two-term spectral
asymptotics analytically.

### §1.1. The operator of linear elasticity

Let $(M,g)$ be a compact connected smooth Riemannian manifold of dimension
$d\geq 2$ with boundary $\partial M$. We denote by $\nabla$ the Levi-Civita
connection and by $\mathrm{Ric}$ the Ricci curvature tensor.

We define the operator of linear elasticity $\mathcal{L}$ acting on vector
fields $\mathbf{u}$ on $M$ as

$(\mathcal{L}\mathbf{u})^{\alpha}:=-\mu\left(\nabla_{\beta}\nabla^{\beta}u^{\alpha}+\operatorname{Ric}^{\alpha}{}_{\beta}u^{\beta}\right)-(\lambda+\mu)\nabla^{\alpha}\nabla_{\beta}u^{\beta}\,.$
(1.1)

Here and further on we adopt the Einstein summation convention over repeated
indices. The quantities $\lambda$ and $\mu$ are real constants known as _Lamé
parameters_ , assumed to satisfy the conditions

$\mu>0,\qquad d\lambda+2\mu>0\,,$ (1.2)

which guarantee strong convexity, see, e.g., [23, 1]. Furthermore, we assume
that the material density of the of the elastic medium $\rho_{\mathrm{mat}}$
differs from the Riemannian density $\sqrt{\det g}$ by a constant positive
factor.

The principal symbol $\mathcal{L}_{\mathrm{prin}}$ of $\mathcal{L}$
reads333Here and further on $\|\xi\|$ denotes the Riemannian norm of the
covector $\xi$.

$\left[\mathcal{L}_{\mathrm{prin}}\right]^{\alpha}{}_{\beta}(x,\xi)=\mu\|\xi\|^{2}\delta^{\alpha}{}_{\beta}+(\lambda+\mu)\xi^{\alpha}\xi_{\beta}\,,$
(1.3)

which, on account of (1.2), immediately implies that $\mathcal{L}$ is
elliptic. Indeed, the eigenvalues of $\mathcal{L}_{\mathrm{prin}}$ are

$\mu\|\xi\|^{2}\quad\text{(with multiplicity
$d-1$)},\qquad(\lambda+2\mu)\|\xi\|^{2}\quad\text{(with multiplicity $1$)}\,.$
(1.4)

Clearly, the operator $\mathcal{L}$ is formally self-adjoint with respect to
the $L^{2}$ inner product

$(\mathbf{u},\mathbf{v})_{L^{2}(M)}:=\int_{M}g_{\alpha\beta}\,u^{\alpha}v^{\beta}\,\sqrt{\det
g}\,\mathrm{d}x\,.$

### §1.2. Boundary value problems

Consider the potential energy of elastic deformation

$\mathcal{E}[\mathbf{u}]:=\frac{1}{2}\int_{M}\left(\lambda(\nabla_{\alpha}u^{\alpha})^{2}+\mu(\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{\alpha})\nabla^{\alpha}u^{\beta}\right)\sqrt{\det
g}\,\mathrm{d}x$ (1.5)

associated with the vector field of displacements $\mathbf{u}$. The quadratic
form $\mathcal{E}[\mathbf{u}]$ is nonnegative for $\mathbf{u}\in
H^{1}(\Omega)$ and strictly positive for $\mathbf{u}\in H^{1}_{0}(\Omega)$.
Observe that the structure of the quadratic functional (1.5) of linear
elasticity is the result of certain geometric assumptions, see [9, formula
(8.28)], as well as [8, Example 2.3 and formulae (2.5a), (2.5b) and (4.10e)].

Performing integration by parts in (1.5) one obtains the Green identity for
the elasticity operator

$2\,\mathcal{E}[\mathbf{u}]=(\mathbf{u},\mathcal{L}\mathbf{u})_{L^{2}(M)}+(\mathbf{u},\mathcal{T}\mathbf{u})_{L^{2}(\partial
M)}\,,$ (1.6)

where $\mathcal{T}$ is the boundary traction operator defined as

$(\mathcal{T}\mathbf{u})^{\alpha}:=\lambda
n^{\alpha}\nabla_{\beta}u^{\beta}+\mu\left(n^{\beta}\nabla_{\beta}u^{\alpha}+n_{\beta}\nabla^{\alpha}u^{\beta}\right).$

Here $\mathbf{n}$ is the exterior unit normal vector to the boundary $\partial
M$.

Examination of (1.6) supplies appropriate boundary conditions for
$\mathcal{L}$. In the current paper, we will be concerned with the following
four sets of boundary conditions.

  * •

Dirichlet boundary conditions:

$\left.\mathbf{u}\right|_{\partial M}=0.$ (1.7)

  * •

Free boundary conditions:

$\left.\mathcal{T}\mathbf{u}\right|_{\partial M}=0.$ (1.8)

  * •

Dirichlet-free (DF) boundary conditions:

$\left.\left[\mathbf{u}-\left(g_{\alpha\beta}\,n^{\alpha}u^{\beta}\right)\,\mathbf{n}\right]\right|_{\partial
M}=0\,,\qquad\left.g_{\alpha\beta}\,n^{\alpha}(\mathcal{T}\mathbf{u})^{\beta}\right|_{\partial
M}=0\,.$ (1.9)

  * •

Free-Dirichlet (FD) boundary conditions:

$\left.\left[\mathcal{T}\mathbf{u}-\left(g_{\alpha\beta}\,n^{\alpha}(\mathcal{T}u)^{\beta}\right)\,\mathbf{n}\right]\right|_{\partial
M}=0\,,\qquad\left.g_{\alpha\beta}\,n^{\alpha}u^{\beta}\right|_{\partial
M}=0\,.$ (1.10)

We refer to the boundary conditions DF and FD as _mixed boundary conditions_.
The former (DF) corresponds to Dirichlet boundary conditions being imposed
tangentially to the boundary and free conditions imposed in the normal
direction to the boundary; the latter (FD) corresponds to free boundary
conditions being imposed tangentially to the boundary and Dirichlet conditions
imposed in the normal direction to the boundary.

The boundary conditions (1.7)–(1.10) are of Shapiro–Lopatinski type [20] for
$\mathcal{L}$, hence the corresponding boundary value problems are elliptic.
This leads to the following four eigenvalue problems for $\mathcal{L}$.

  1. 1.

The Dirichlet problem (Dir). The _Dirichlet eigenvalue problem_ consists in
seeking $\mathbf{u}\in H^{1}(\Omega)$, $\mathbf{u}\neq 0$, and
$\Lambda\in\mathbb{R}$ such that

$\mathcal{L}\mathbf{u}=\Lambda\mathbf{u}$ (1.11)

subject to the boundary conditions (1.7). The problem (1.11), (1.7) has
discrete spectrum, consisting of discrete eigenvalues

$(0<)\Lambda_{1}^{\mathrm{Dir}}\leq\Lambda_{2}^{\mathrm{Dir}}\leq\dots$

enumerated with account of multiplicity and accumulating to $+\infty$.

  2. 2.

The free boundary problem (free). The _free boundary eigenvalue problem_
consists in seeking $\mathbf{u}\in H^{1}(\Omega)$, $\mathbf{u}\neq 0$, and
$\Lambda\in\mathbb{R}$ satisfying (1.11), subject to the boundary conditions
(1.8). The problem (1.11), (1.8) has discrete spectrum, consisting of discrete
eigenvalues

$(0\leq)\Lambda_{1}^{\mathrm{free}}\leq\Lambda_{2}^{\mathrm{free}}\leq\dots$

enumerated with account of multiplicity and accumulating to $+\infty$.

  3. 3.

The Dirichlet-free problem (DF). The _Dirichlet-free eigenvalue problem_
consists in seeking $\mathbf{u}\in H^{1}(\Omega)$, $\mathbf{u}\neq 0$, and
$\Lambda\in\mathbb{R}$ satisfying (1.11), subject to the boundary conditions
(1.9). The problem (1.11), (1.9) has discrete spectrum, consisting of discrete
eigenvalues

$(0\leq)\Lambda_{1}^{\mathrm{DF}}\leq\Lambda_{2}^{\mathrm{DF}}\leq\dots$

enumerated with account of multiplicity and accumulating to $+\infty$.

  4. 4.

The free-Dirichlet problem (FD). The _free-Dirichlet eigenvalue problem_
consists in seeking $\mathbf{u}\in H^{1}(\Omega)$, $\mathbf{u}\neq 0$, and
$\Lambda\in\mathbb{R}$ satisfying (1.11), subject to the boundary conditions
(1.10). The problem (1.11), (1.10) has discrete spectrum, consisting of
discrete eigenvalues

$(0\leq)\Lambda_{1}^{\mathrm{FD}}\leq\Lambda_{2}^{\mathrm{FD}}\leq\dots$

enumerated with account of multiplicity and accumulating to $+\infty$.

######  Remark 1.1.

The problems $\mathrm{Dir}$, $\mathrm{free}$, DF and FD also admit a minmax
formulation. We refer the interested reader to [22] for details.

Let us briefly elaborate on the physical meaning of the above eigenvalue
problems. The spectral parameter $\Lambda$ appearing in (1.11) has the
following interpretation

$\Lambda=\frac{\rho_{\mathrm{mat}}}{\sqrt{\det g}}\,\omega^{2}\,,$

where $\omega$ is the angular natural frequency of oscillation of the elastic
medium. The boundary conditions $\mathrm{Dir}$ (1.7) describe a body whose
boundary is “clamped”, i.e., completely prevented from moving, whereas the
conditions $\mathrm{free}$ (1.8) describe the opposite situation, in which the
boundary is free to oscillate without restrictions. Mixed DF boundary
conditions describe a body that is allowed to deform in the direction normal
to the boundary, but is prevented from deforming in the directions tangential
to the boundary. Similarly, mixed FD boundary conditions describe a body that
is allowed to “slide” along its boundary, but is prevented from deforming in
the direction normal to the boundary. Clearly, mixed boundary conditions are
physically meaningful and describe realistic scenarios relevant for
applications — see also [22] for further discussions in this respect.

### §1.3. Statement of the problem and main results

Consider, for each set of boundary conditions
$\aleph\in\\{\mathrm{Dir},\mathrm{free},\mathrm{DF},\mathrm{FD}\\}$, the
corresponding _eigenvalue counting function_
$N_{\aleph}:\mathbb{R}\to\mathbb{N}$ defined as

$N_{\aleph}(\Lambda):=\\#\left\\{k\ |\
\Lambda_{k}^{\aleph}<\Lambda\right\\}\,.$ (1.12)

Clearly, the function (1.12) is monotonically non-decreasing in $\Lambda$ and
vanishes identically for $\Lambda\leq\Lambda_{1}^{\aleph}$.

The study of the asymptotic behaviour of eigenvalue counting functions of the
type (1.12) as $\Lambda\to+\infty$ for (semibounded) elliptic operators is a
well established area of mathematics, pioneered by Lord Rayleigh’s _The theory
of sound_ [25] in 1877. What started as an investigation prompted by practical
questions from physics soon attracted the interest of pure mathematicians, as
people realised that the coefficients in these expansions contain geometric
invariants (see, e.g., [21, Chapter 6]). We refer the reader to [26, 3, 19]
for historical overviews of the development of the subject.

Before stating our main result, let us summarise, without proof, some known
facts concerning (1.12). In what follows $(M,g)$ satisfies the conditions from
subsection 1.1.

######  Proposition 1.2.

We have

$N_{\aleph}(\Lambda)=a\operatorname{Vol}_{d}(M)\Lambda^{d/2}+o\left(\Lambda^{d/2}\right)\quad\text{as}\quad\Lambda\to+\infty,$
(1.13)

where

$a=\frac{1}{(4\pi)^{d/2}\Gamma\left(1+\frac{d}{2}\right)}\left(\frac{d-1}{\mu^{d/2}}+\frac{1}{(\lambda+2\mu)^{d/2}}\right)$
(1.14)

is the _Weyl constant_ for linear elasticity, $\operatorname{Vol}_{d}(M)$ is
the Riemannian volume of $M$, and $\Gamma$ is the gamma function.

The one-term asymptotic expansion (1.13) is often referred to as _Weyl law_.
Note that in the special case $d=3$ formula (1.13) was already established,
indirectly and on the basis of physical arguments, by P. Debye in 1912 [13]. A
rigorous mathematical proof was provided shortly afterwards by H. Weyl [29].
It is worth emphasising that the coefficient $a$ is independent of the choice
of boundary conditions.

Let $\mathcal{A}$ be an elliptic semibounded differential operator of even
order $2m$ acting between sections of Hermitian $C^{\infty}$ vector bundles of
dimension $N$ over a smooth $d$-dimensional manifold $M$ with boundary,
supplemented by differential boundary conditions $\mathcal{B}$ satisfying the
(parabolic version of the) Shapiro–Lopatniski conditions [2]. Then it is known
[15, Theorem 2.6.1] that the trace of the Green kernel $G(x,y,t)$ for the
boundary value problem

$\begin{cases}\left(\frac{\partial}{\partial
t}+\mathcal{A}\right)\mathbf{u}=0&\text{in }M,\\\
\mathcal{B}\mathbf{u}=0&\text{on }\partial M,\end{cases}$

admits a complete asymptotic expansion

$\mathcal{Z}_{\mathcal{B}}(t):=\int_{M}\operatorname{tr}G(x,x,t)\,\operatorname{dVol}_{M}\sim\widetilde{c}_{d-1}\,t^{-\frac{d}{2m}}+\widetilde{c}_{d-2}\,t^{-\frac{d}{2m}+\frac{1}{2m}}+\dots+\widetilde{c}_{d-k}t^{-\frac{d}{2m}+\frac{n-1}{2m}}+\dots{\\\
}\text{as}\quad t\to 0^{+},$ (1.15)

where $\mathrm{tr}$ stands for matrix trace. Moreover, all coefficients in
(1.15) are locally determined [15, §2]. We refer the reader to [17], [16,
§4.2] and references therein for further details and generalisations.

Formula (1.15) allows us to define Weyl coefficients for elliptic operators on
manifolds with boundary.

######  Definition 1.3.

For $1\leq n\leq d$, we define the _$n$ -th Weyl coefficient_ for the elliptic
boundary value problem $(\mathcal{A},\mathcal{B})$ to be the number

$c_{d-n}:=\frac{\widetilde{c}_{d-n}}{\Gamma\left(\frac{d-n+1}{2m}\right)}\,,$
(1.16)

where the $\widetilde{c}_{d-n}$ is the coefficient of
$t^{-\frac{d}{2m}+\frac{n-1}{2m}}$ in the expansion (1.15) and $\Gamma$ is the
gamma function.

######  Remark 1.4.

Note that Definition 1.3 agrees with the definition of Weyl coefficients in
[4, 7, 10, 5, 11] for (pseudo)differential operators on compact manifolds
without boundary, where Weyl coefficients are defined to be the coefficients
appearing in the complete asymptotic expansion for the mollified derivative of
the counting function. Whilst such a complete asymptotic expansion always
exists when $\partial M=\emptyset$ (see, e.g., [18]), we are unaware of a
similar result for manifolds _with_ boundary. This is why in the latter case
defining Weyl coefficients is somewhat more delicate.

We should also point out that the standard convention in the literature is to
call Weyl coefficients the constants appearing in the asymptotic expansion of
the mollified counting function, as opposed to its derivative. The two
definitions are, effectively, the same up to integrating factors; as a matter
of convenience and consistency with previous papers by the first author, we
will stick here with Definition 1.3.

######  Fact 1.5.

Suppose that the eigenvalue counting function $N(\Lambda)$ of the elliptic
eigenvalue problem

$\begin{cases}\mathcal{A}\mathbf{u}=\Lambda\mathbf{u}&\text{in }M,\\\
\mathcal{B}\mathbf{u}=0&\text{on }\partial M\end{cases}$

admits a $j$-term asymptotic expansion

$N(\Lambda)=C_{d}\Lambda^{\frac{d}{2m}}+C_{d-1}\Lambda^{\frac{d-1}{2m}}+\dots+C_{d-j+1}\Lambda^{\frac{d-j+1}{2m}}+o(\Lambda^{\frac{d-j+1}{2m}})\quad\text{as}\quad\Lambda\to+\infty$
(1.17)

for some $1\leq j\leq d$. Then

$C_{d-n+1}=\frac{2m}{d-n+1}c_{d-n}\qquad\text{for}\quad 1\leq n\leq j\,.$
(1.18)

######  Remark 1.6.

We should like to emphasise that Weyl coefficients (1.16) are defined — and
can be computed — _irrespective_ of whether an asymptotic expansion for the
eigenvalue counting function of the form (1.17) exists.

Of course, when $\mathcal{A}=\mathcal{L}$ and
$\mathcal{B}=\mathcal{B}_{\aleph}$, $\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}$,
formula (1.15) specialises to read

$\mathcal{Z}_{\aleph}(t)=\sum_{k=1}^{\infty}e^{-\Lambda_{k}^{\aleph}t}=\Gamma\left(\frac{d}{2}+1\right)\,a\operatorname{Vol}_{d}(M)t^{-\frac{d}{2}}+c_{d-2}^{\aleph}\,t^{-\frac{d-1}{2}}+o(t^{-\frac{d-1}{2}})\,\quad\text{as}\quad
t\to 0^{+},$

where the first asymptotic term has been written explicitly in terms of (1.14)
— cf. (1.13) and (1.18).

Whilst the existence of a one-term asymptotic expansion (Weyl’s law) is always
guaranteed — see Proposition 1.2 — the validity of a two-term expansion of the
form (1.17) for $\mathcal{L}$ with boundary conditions $\mathcal{B}_{\aleph}$,
$\aleph\in\\{\mathrm{Dir},\mathrm{free},\mathrm{DF},\mathrm{FD}\\}$, is still
an open problem. Nevertheless, such an expansion is known to exist under
additional dynamical assumptions on certain _branching Hamiltonian billiards_
on the cotangent bundle $T^{*}M$. We recall the result below, referring the
reader to [28] for additional details and precise statements.

######  Theorem 1.7.

Suppose that $(M,g)$ is such that the corresponding billiards is neither
absolutely periodic nor dead-end. Then

$N_{\aleph}(\Lambda)=a\operatorname{Vol}_{d}(M)\Lambda^{d/2}+C_{d-1}^{\aleph}\Lambda^{(d-1)/2}+o\left(\Lambda^{(d-1)/2}\right)\quad\text{as}\quad\Lambda\to+\infty\,,$
(1.19)

for any set of boundary conditions
$\aleph\in\\{\mathrm{Dir},\mathrm{free},\mathrm{DF},\mathrm{FD}\\}$.

Theorem 1.7 is a special case of [27, Theorem 6.1], which is applicable here
because the eigenvalues (1.4) of $\mathcal{L}_{\mathrm{prin}}$ have constant
multiplicity as functions of $(x,\xi)\in T^{*}M\setminus\\{0\\}$.

We are now ready to state our main result.

######  Theorem 1.8.

Let $(M,g)$ be a smooth compact connected $d$-dimensional Riemannian manifold
with boundary $\partial M$, $d\geq 2$. The second Weyl coefficient for the
elliptic boundary value problem $(\mathcal{L},\mathcal{B}_{\aleph})$,
$\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}$, is given by

$c^{\aleph}_{d-2}=\frac{d-1}{2}C^{\aleph}_{d-1}=\frac{d-1}{2}b_{\aleph}\operatorname{Vol}_{d-1}(\partial
M)\,,$ (1.20)

where

$b_{\aleph}:=@\frac{1}{2^{d+1}\pi^{\frac{d-1}{2}}\Gamma\left(\frac{d+1}{2}\right)}\left(\frac{d-3}{\mu^{\frac{d-1}{2}}}+\frac{1}{(\lambda+2\mu)^{\frac{d-1}{2}}}\right)\quad\text{with}\quad@=\begin{cases}-&\text{for}\
\,\aleph=\mathrm{DF},\\\ +&\text{for}\ \,\aleph=\mathrm{FD}.\end{cases}$
(1.21)

The above theorem, whose proof will be given in Section 2, warrants a number
of remarks.

  1. (i)

Formulae for $b_{\aleph}$ in the case of ”pure” boundary conditions
$\aleph\in\\{\mathrm{Dir},\mathrm{free}\\}$ were obtained in [6, Theorem
1.8]444To ease the comparison, note that formulae (1.27) and (1.28) in [6,
Theorem 1.8] are expressed in terms of the auxiliary quantity
$\alpha:=\frac{\mu}{\lambda+2\mu}\in\left(0,\frac{d}{2(d-1)}\right)$..

  2. (ii)

Remarkably, formula (1.21) is very simple and elegant. This is noteworthy and
in general not the case for boundary value problems for linear elasticity, in
that one would expect the Lamé parameters $\lambda$ and $\mu$ to mix up in a
rather complicated way in the second Weyl coefficient, owing to boundary
conditions mixing longitudinal and transverse waves. For instance, the
expressions for $b_{\mathrm{Dir}}$ and $b_{\mathrm{free}}$ contain integrals
of inverse trigonometric functions depending on
$\alpha:=\frac{\mu}{\lambda+2\mu}$ in a nontrivial fashion, see [6, formulae
(1.27) and (1.28)]. The underlying reason for (1.21) being so simple is that
mixed boundary conditions $\mathrm{DF}$ and $\mathrm{FD}$, unlike
$\mathrm{Dir}$ and $\mathrm{free}$, _do not_ mix up components of the vector
fields they act upon when one switches to the associated one-dimensional
spectral problem — see Section 2.1, and formula (2.3) in particular.

  3. (iii)

Note that

$b_{FD}<0<b_{DF}$ (1.22)

for $d=2$, whereas

$b_{DF}<0<b_{FD}$ (1.23)

for $d\geq 3$.

  4. (iv)

We should like to emphasise that computing the second Weyl coefficient for
_systems_ of PDEs is not easy. Indeed, the subject area of two-term
asymptotics for elliptic systems has experienced a troubled development, up
until the last decade; we refer the reader to [12, Section 11] for a
historical overview.

Before moving on to the proof of our main theorem, let us recall a well known
fact which will bring about some simplifications in subsequent sections.

######  Fact 1.9.

The first two Weyl coefficients do not feel the geometry of $M$ or of its
boundary $\partial M$. Therefore, it suffices to determine these coefficients
in the case where $M$ is a smooth domain in $\mathbb{R}^{d}$ equipped with the
Euclidean metric.

##  Appendix 2.§2. Proof of Theorem 1.8

This section is concerned with the proof of Theorem 1.8. We will break the
proof, somewhat long and technical, into several steps.

On account of Fact 1.9, in the remainder of this section we will assume that
$M$ is a smooth domain in $\mathbb{R}^{d}$ and that $g$ is the Euclidean
metric, $g_{\alpha\beta}=\delta_{\alpha\beta}$.

### §2.1. A streamlined algorithm

In order to prove Theorem 1.8 we will rely on a constructive algorithm for the
second Weyl coefficient due to Vassiliev [27]. The original results from [27]
(see also [26]) apply, strictly speaking, to scalar operators. A roadmap for
the generalisation to systems was given in [27, § 6], whereas a detailed
exposition of the algorithm for systems was recently provided in [6]. For the
convenience of the reader, we present below a streamlined version of the
latter algorithm, adapted to the special case of the operator of linear
elasticity (1.1).

In a neighbourhood of the boundary $\partial M$ we introduce local coordinates
$x=(x^{\prime},z)$, with $x^{\prime}\in\mathbb{R}^{d-1}$ and
$z:=\operatorname{dist}(x,\partial M)$ for $x\in\operatorname{Int}(M)$, so
that $\partial M=\\{z=0\\}$ and $z>0$ inside of $M$. Similarly, we adopt the
notation $\xi=(\xi^{\prime},\zeta)\in\mathbb{R}^{d-1}\times\mathbb{R}$ and
$\mathbf{u}=(\mathbf{u}^{\prime},u_{d})$. Furthermore, we denote by
$\mathcal{B}_{\aleph}$ the differential operators implementing the boundary
conditions (1.9) and (1.10) for $\aleph=\mathrm{DF}$ and $\aleph=\mathrm{FD}$,
respectively.

Consider the _one-dimensional spectral problem_

$\mathcal{L}^{\prime}\mathbf{u}(z)=\Lambda\mathbf{u}(z),\qquad\left.\mathcal{B}_{\aleph}^{\prime}\mathbf{u}\right|_{z=0}=0\,,\qquad\aleph\in\\{\mathrm{DF},\mathrm{FD}\\},$
(2.1)

where $\mathcal{L}^{\prime}$ and $\mathcal{B}_{\aleph}^{\prime}$ are the
ordinary differential operators acting on vector functions
$\mathbf{u}=\mathbf{u}(z)$ defined in accordance with

$\mathcal{L}^{\prime}\mathbf{u}:=\mu\left(|\xi^{\prime}|^{2}-\frac{\mathrm{d}^{2}}{\mathrm{d}z^{2}}\right)\mathbf{u}-(\lambda+2\mu)\begin{pmatrix}i\xi^{\prime}\\\
\frac{\mathrm{d}}{\mathrm{d}z}\end{pmatrix}\left(i\xi^{\prime}\cdot\mathbf{u}^{\prime}+\frac{\mathrm{d}u_{d}}{\mathrm{d}z}\right)$
(2.2)

and

$\left.\mathcal{B}^{\prime}_{\mathrm{DF}}\mathbf{u}(z)\right|_{z=0}=\begin{pmatrix}\mathbf{u}^{\prime}(0)\\\
-(\lambda+2\mu)\frac{\mathrm{d}u_{d}}{\mathrm{d}z}(0)\end{pmatrix},\qquad\left.\mathcal{B}^{\prime}_{\mathrm{FD}}\mathbf{u}(z)\right|_{z=0}=\begin{pmatrix}-\mu\frac{\mathrm{d}\mathbf{u}^{\prime}}{\mathrm{d}z}(0)\\\
u_{d}(0)\end{pmatrix}.$ (2.3)

The operators (2.2) and (2.3) are obtained from $\mathcal{L}$ and
$\mathcal{B}_{\aleph}$, $\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}$, by replacing
partial derivatives along the boundary with $i$ times the corresponding
component of momentum, $\partial_{x^{\prime}}\mapsto i\xi^{\prime}$.
Furthermore, in the second component of
$\mathcal{B}^{\prime}_{\mathrm{DF}}\mathbf{u}$ we dropped terms proportional
to $\mathbf{u}^{\prime}(0)$, whereas in the first component of
$\mathcal{B}^{\prime}_{\mathrm{FD}}\mathbf{u}$ we dropped terms proportional
to $u_{d}(0)$.

Suppose $\xi^{\prime}\neq 0$.

Step 1: Thresholds and continuous spectrum. The principal symbol
$\mathcal{L}^{\prime}_{\mathrm{prin}}(\zeta)$ of $\mathcal{L}^{\prime}$
(recall (1.3)) has eigenvalues

$h_{1}(\zeta)=\mu(|\xi^{\prime}|^{2}+\zeta^{2})\quad\text{(with multiplicity
$d-1$)},\quad
h_{2}(\zeta)=(\lambda+2\mu)(|\xi^{\prime}|^{2}+\zeta^{2})\quad\text{(with
multiplicity $1$)}\,.$ (2.4)

We define the _thresholds_ of the continuous spectrum as the nonnegative real
numbers $\Lambda_{*}$ such that the equation

$h_{k}(\zeta)=\Lambda_{*}$

has a multiple real root for either $k=1$ or $k=2$. Formula (2.4) immediately
implies that we have two thresholds for (2.1):

$\Lambda_{*}^{(1)}=\mu|\xi^{\prime}|^{2},\qquad\Lambda_{*}^{(2)}=(\lambda+2\mu)|\xi^{\prime}|^{2}\,.$
(2.5)

Observe that, due to (1.2), we have $\Lambda_{*}^{(1)}<\Lambda_{*}^{(2)}$.

Formula (2.5) then implies that the problem (2.1) has continuous spectrum
$[\Lambda_{*}^{(1)},+\infty)$ — see, e.g., [26, Appendix A]. Furthermore, the
thresholds partition the continuous spectrum into two zones
$I^{(1)}:=(\Lambda_{*}^{(1)},\Lambda_{*}^{(2)})$ and
$I^{(2)}:=(\Lambda_{*}^{(2)},+\infty)$, where the continuous spectrum has
multiplicity $d-1$ and $d$, respectively.

Step 2: Eigenfunctions of the continuous spectrum. Let
$\mathbf{v}_{k}(\zeta)$, $k=1,\dots,d-1$, be orthonormalised eigenvectors of
$\mathcal{L}^{\prime}_{\mathrm{prin}}$ corresponding to the eigenvalue
$h_{1}(\zeta)$, and let

$\mathbf{v}_{d}(\zeta):=\frac{1}{\sqrt{|\xi^{\prime}|^{2}+\zeta^{2}}}\begin{pmatrix}\xi^{\prime}\\\
\zeta\end{pmatrix}=\frac{1}{|\xi|}\xi$ (2.6)

be the normalised eigenvector of $\mathcal{L}^{\prime}_{\mathrm{prin}}$
corresponding to the eigenvalue $h_{2}(\zeta)$. Let

$\zeta_{1}^{\pm}(\Lambda):=\pm\sqrt{\frac{\Lambda}{\mu}-|\xi^{\prime}|^{2}}\quad\text{and}\quad\zeta_{2}^{\pm}(\Lambda):=\pm\sqrt{\frac{\Lambda}{\lambda+2\mu}-|\xi^{\prime}|^{2}}$

for $\Lambda\geq\mu|\xi^{\prime}|^{2}$ and
$\Lambda\geq(\lambda+2\mu)|\xi^{\prime}|^{2}$, respectively. Note that the
quantities $\zeta_{k}^{\pm}(\Lambda)$ are solutions of
$h_{k}(\zeta)-\Lambda=0$. Then, in view of elementary theory of matrix
ordinary differential equations, we seek _eigenfunctions of the continuous
spectrum_ (or _generalised eigenfunctions_) for (2.1) in the form

$\mathbf{u}(z;\Lambda)=\frac{1}{\sqrt{4\pi\mu}\left|\zeta_{1}^{+}(\Lambda)\right|^{1/2}}\sum_{j=1}^{d-1}\left(c_{j}^{+}\mathbf{v}_{j}(\zeta_{1}^{+}(\Lambda))e^{i\zeta_{1}^{+}(\Lambda)z}+c_{j}^{-}\mathbf{v}_{j}(\zeta_{1}^{-}(\Lambda))e^{i\zeta_{1}^{-}(\Lambda)z}\right)\\\
+C\,\mathbf{v}_{d}\left(i\sqrt{|\xi^{\prime}|^{2}-\frac{\Lambda}{\lambda+2\mu}}\right)e^{-\sqrt{|\xi^{\prime}|^{2}-\frac{\Lambda}{\lambda+2\mu}}z}$
(2.7)

for $\Lambda\in I^{(1)}$, and in the form

$\mathbf{u}(z;\Lambda)=\frac{1}{\sqrt{4\pi\mu}\left|\zeta_{1}^{+}(\Lambda)\right|^{1/2}}\sum_{j=1}^{d-1}\left(c_{j}^{+}\mathbf{v}_{j}(\zeta_{1}^{+}(\Lambda))e^{i\zeta_{1}^{+}(\Lambda)z}+c_{j}^{-}\mathbf{v}_{j}(\zeta_{1}^{-}(\Lambda))e^{i\zeta_{1}^{-}(\Lambda)z}\right)\\\
+\frac{1}{\sqrt{4\pi(\lambda+2\mu)}\left|\zeta_{2}^{+}(\Lambda)\right|^{1/2}}\left(c_{d}^{+}\mathbf{v}_{d}(\zeta_{2}^{+}(\Lambda))e^{i\zeta_{2}^{+}(\Lambda)z}+c_{d}^{-}\mathbf{v}_{j}(\zeta_{2}^{-}(\Lambda))e^{i\zeta_{1}^{-}(\Lambda)z}\right)$
(2.8)

for $\Lambda\in I^{(2)}$. The complex numbers $c_{j}^{\pm}$ in (2.7) and (2.8)
are called _incoming_ (-) and _outgoing_ (+) complex wave amplitudes, and are
assumed not to be all zero.

Step 3: The scattering matrix. By imposing that (2.7) and (2.8) satisfy the
boundary conditions, one can express the coefficients $c_{j}^{+}$ in terms of
the coefficients $c_{j}^{-}$. This defines the _scattering matrices_
$S^{(k)}(\Lambda)$, $k=1,2$, via the identities

$\begin{pmatrix}c_{1}^{+}\\\ \vdots\\\
c_{d-1}^{+}\end{pmatrix}=S^{(1)}(\Lambda)\begin{pmatrix}c_{1}^{-}\\\ \vdots\\\
c_{d-1}^{-}\end{pmatrix}\qquad\text{for}\ \Lambda\in I^{(1)}$

and

$\begin{pmatrix}c_{1}^{+}\\\ \vdots\\\
c_{d}^{+}\end{pmatrix}=S^{(2)}(\Lambda)\begin{pmatrix}c_{1}^{-}\\\ \vdots\\\
c_{d}^{-}\end{pmatrix}\qquad\text{for}\ \Lambda\in I^{(2)}.$

The matrix $S^{(1)}(\Lambda)$ (resp. $S^{(2)}(\Lambda)$) is a
$(d-1)\times(d-1)$ (resp. $d\times d$) unitary matrix. The way in which the
$c_{j}^{\pm}$ are arranged into a $(d-1)$-dimensional (resp. $d$-dimensional)
vector is unimportant and will not affect the quantities computed in the next
steps.

Step 4: The phase shift. Compute the _phase shift_ , defined as

$\varphi_{\aleph}(\Lambda;\xi^{\prime}):=\begin{cases}0&\text{for}\
\Lambda\leq\Lambda_{1}^{*}\\\ \arg\det
S^{k}(\Lambda)+\mathfrak{s}^{(k)}&\text{for}\ \Lambda\in I^{(k)}\\\
\end{cases}$ (2.9)

where the branch of the multivalued function $\arg$ are chosen in such a way
that $\varphi(\Lambda)$ is continuous in each interval $I^{(k)}$, and the
shifts $\mathfrak{s}^{(k)}$ are constants determined by the requirement that
the jump of the phase shift at the thresholds satisfy

$\frac{1}{\pi}\lim_{\epsilon\to
0^{+}}\left(\varphi(\Lambda_{*}^{(k)}+\epsilon)-\varphi(\Lambda_{*}^{(k)}-\epsilon)\right)=j_{*}^{(k)}-\frac{m_{k}}{2},\qquad
k\in\\{1,2\\}\,.$ (2.10)

Here $m_{k}$ is the multiplicity of the eigenvalue $h_{k}$ and $j_{*}^{(k)}$
is the number of linearly independent vectors $\mathbf{v}$ such that

$\mathbf{v}e^{i\zeta_{+}(\Lambda_{*}^{(k)})z}+\mathbf{f}(z)$ (2.11)

is a solution of the one-dimensional problem (2.1), with $\mathbf{f}(z)=o(1)$
as $z\to+\infty$. The threshold $\Lambda_{*}^{(k)}$ is called _rigid_ if
$j_{*}^{(k)}=0$ and _soft_ if $j_{*}^{(k)}=m_{k}$

Step 5: The one-dimensional counting function. Compute the _one-dimensional
counting function_ , defined as

$N_{\aleph,\mathrm{1D}}(\Lambda;\xi^{\prime}):=\\#\\{\text{eigenvalues of
\eqref{one-dimensional spectral problem} strictly smaller than}\
\Lambda\\}\,.$ (2.12)

Step 6: The spectral shift function. Compute the _spectral shift function_ ,
defined as

$\mathrm{shift}_{\aleph}(\Lambda;\xi^{\prime}):=\frac{1}{2\pi}\varphi_{\aleph}(\Lambda;\xi^{\prime})+N_{\aleph,\mathrm{1D}}(\Lambda;\xi^{\prime}).$
(2.13)

Then we have the following.

######  Theorem 2.1.

The second Weyl coefficient555Recall that the second Weyl coefficient and the
second coefficient in the asymptotic expansion for the eigenvalue counting
function (1.19) are related in accordance with (1.20) — see also (1.18). is
given by

$c_{d-2}^{\aleph}=\frac{d-1}{2(2\pi)^{d-1}}\int_{T^{*}\partial
M}\mathrm{shift}_{\aleph}(1;\xi^{\prime})\,\mathrm{d}x^{\prime}\,\mathrm{d}\xi^{\prime}=\frac{d-1}{2}\frac{\mathrm{Vol}_{d-1}(\partial
M)}{(2\pi)^{d-1}}\int_{\mathbb{R}^{d-1}}\mathrm{shift}_{\aleph}(1;\xi^{\prime})\,\mathrm{d}\xi^{\prime}\,.$
(2.14)

Theorem 2.1 is a specialisation to the case at hand of [28, Theorem 2] (once
the latter has been extended to systems).

### §2.2. Invariant subspaces

Implementing the algorithm from subsection 2.1 as written for an arbitrary
dimension $d$ is quite tricky. Rather than applying our algorithm to (1.11),
(1.9) and (1.11), (1.10) directly, we shall first simplify the problem by
decomposing the general $d$-dimensional problem into a two-dimensional
analogue plus a much simpler $(d-2)$-dimensional problem. The arguments in
this subsection can be traced back, in essence, to observations by
Dupuis–Mazo–Onsager [14], see also [6, Section 3]. The key idea is to
decompose elastic waves into two polarised components: one polarised in the
plane of propagation and the other normally to it.

To this end, suppose we have fixed $\xi^{\prime}\in\mathbb{R}^{d-1}$,
$\xi^{\prime}\neq 0$, and define

$P:=\mathrm{span}\left\\{\frac{1}{|\xi^{\prime}|}\begin{pmatrix}\xi^{\prime}\\\
0\end{pmatrix},\begin{pmatrix}0^{\prime}\\\
1\end{pmatrix}\right\\}\subset\mathbb{R}^{d}\,.$

Let $P^{\perp}$ be the orthogonal complement of $P$ in $\mathbb{R}^{d}$. One
can easily check the following facts.

######  Fact 2.2.

  1. (i)

For every $\zeta\in\mathbb{R}$ the eigenvector $\mathbf{v}_{d}(\zeta)$ (2.6)
is an element of $P$.

  2. (ii)

For every $\zeta\in\mathbb{R}$ the orthogonal subspaces $P$ and $P^{\perp}$
are invariant subspaces of
$\mathcal{L}_{\mathrm{prin}}(\xi^{\prime},\zeta)=\mathcal{L}^{\prime}_{\mathrm{prin}}(\zeta)$
— recall (1.3).

  3. (iii)

The restriction
$\left.\mathcal{L}^{\prime}_{\mathrm{prin}}\right|_{P^{\perp}}$ of
$\mathcal{L}^{\prime}_{\mathrm{prin}}$ to $P^{\perp}$ has one eigenvalue,
$\mu|\xi|^{2}$, of multiplicity $d-2$.

  4. (iv)

The restriction $\left.\mathcal{L}^{\prime}_{\mathrm{prin}}\right|_{P}$ of
$\mathcal{L}^{\prime}_{\mathrm{prin}}$ to $P$ has two eigenvalues,
$\mu|\xi|^{2}$ and $(\lambda+2\mu)|\xi|^{2}$, each of multiplicity $1$.

The decomposition $\mathbb{R}^{d}=P\oplus P^{\perp}$ induces a corresponding
decomposition at the level of vector fields. Let

$\mathbf{P}:=\\{\mathbf{u}\in C^{\infty}[0,+\infty)\ |\ \mathbf{u}(z)\in
P\quad\forall z\in[0,+\infty)\\}\,,$ (2.15)
$\mathbf{P}^{\perp}:=\\{\mathbf{u}\in C^{\infty}[0,+\infty)\ |\
\mathbf{u}(z)\in P^{\perp}\quad\forall z\in[0,+\infty)\\}\,.$ (2.16)

######  Proposition 2.3.

The vector spaces (2.15) and (2.16) are invariant subspaces for the operator
(2.2), compatible with mixed boundary conditions $\mathrm{DF}$ and
$\mathrm{FD}$. Namely,

$\mathcal{L}^{\prime}\mathbf{P}\subset\mathbf{P},\qquad\mathcal{L}^{\prime}\mathbf{P}^{\perp}\subset\mathbf{P}^{\perp},$
(2.17)

and

$\left.\mathcal{B}_{\aleph}^{\prime}\mathbf{P}\right|_{z=0}\subset
P,\qquad\left.\mathcal{B}_{\aleph}^{\prime}\mathbf{P}\right|_{z=0}\subset
P^{\perp},\qquad\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}\,.$ (2.18)

###### Proof.

A generic element of $\mathbf{P}$ reads

$\mathbf{u}_{\parallel}=\frac{1}{\|\xi^{\prime}\|}\begin{pmatrix}\xi^{\prime}\\\
0\end{pmatrix}f_{1}(z)+\begin{pmatrix}0^{\prime}\\\
1\end{pmatrix}f_{2}(z)\,,\qquad f_{1},f_{2}\in C^{\infty}[0,+\infty)$ (2.19)

whereas a generic element of $\mathbf{P}^{\perp}$ reads

$\mathbf{u}_{\perp}(z)=\sum_{j=1}^{d-2}\begin{pmatrix}\psi_{j}\\\
0\end{pmatrix}f_{j}(z),\qquad f_{j}\in C^{\infty}[0,+\infty),$ (2.20)

where the $\psi_{j}$’s, $j=1,\dots,d-2$, are linearly independent columns in
$\mathbb{R}^{d-1}$ orthogonal to $\xi^{\prime}$.

Formula (2.17) follows from [6, Lemma 3.1(a)].

Substituting (2.19) and (2.20) into (2.3) we obtain

$\left.\left(\mathcal{B}^{\prime}_{DF}\mathbf{u}_{\parallel}\right)\right|_{z=0}=\frac{1}{\|\xi^{\prime}\|}\begin{pmatrix}\xi^{\prime}\\\
0\end{pmatrix}f_{1}(0)-(\lambda+2\mu)\begin{pmatrix}0^{\prime}\\\
1\end{pmatrix}\frac{\mathrm{d}f_{2}}{\mathrm{d}z}(0)\,,$ (2.21)
$\left.\left(\mathcal{B}^{\prime}_{DF}\mathbf{u}_{\perp}\right)\right|_{z=0}=\sum_{j=1}^{d-2}\begin{pmatrix}\psi_{j}\\\
0\end{pmatrix}\,f_{j}(0)$ (2.22)

for $\aleph=\mathrm{DF}$ and

$\left.\left(\mathcal{B}^{\prime}_{FD}\mathbf{u}_{\parallel}\right)\right|_{z=0}=-\mu\frac{1}{\|\xi^{\prime}\|}\begin{pmatrix}\xi^{\prime}\\\
0\end{pmatrix}\frac{\mathrm{d}f_{1}}{\mathrm{d}z}(0)+\begin{pmatrix}0^{\prime}\\\
1\end{pmatrix}f_{2}(0)\,,$ (2.23)
$\left.\left(\mathcal{B}^{\prime}_{FD}\mathbf{u}_{\perp}\right)\right|_{z=0}=-\mu\sum_{j=1}^{d-2}\begin{pmatrix}\psi_{j}\\\
0\end{pmatrix}\frac{\mathrm{d}f_{j}}{\mathrm{d}z}(0)$ (2.24)

for $\aleph=\mathrm{FD}$. Formulae (2.21)–(2.24) imply that mixed boundary
conditions preserve our invariant subspaces, so that (2.18) holds. This
concludes the proof. ∎

######  Remark 2.4.

The crucial property established by Proposition 2.3 is expressed by formula
(2.18). An analogue of Proposition 2.3 for “pure” Dirichlet and free boundary
conditions was proved in [6]. That this extends to a mixture of the two is not
clear _a priori_ , because both the operator and boundary conditions mix up
components in a nontrivial fashion. Indeed, if one, say, imposes different
boundary conditions in different directions _along_ the boundary, the
statement of the proposition is false.

Let $\Pi_{P}$ be the orthogonal projection in $\mathbb{R}^{d}$ onto $P$, and
let us define

$\mathcal{L}^{\prime}_{P,\aleph}:=\left.\mathcal{L}^{\prime}_{\aleph}\right|_{\Pi_{P}D(\mathcal{L}^{\prime}_{\aleph})}\qquad\text{and}\qquad\mathcal{L}^{\prime}_{\perp,\aleph}:=\left.\mathcal{L}^{\prime}_{\aleph}\right|_{(I-\Pi_{P})D(\mathcal{L}^{\prime}_{\aleph})}$
(2.25)

to be the restriction of the operator $\mathcal{L}^{\prime}$ with boundary
conditions $\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}$ to the invariant subspaces
of its domain $D(\mathcal{L}^{\prime}_{\aleph})$ induced by (2.15) and (2.16)
by combining Proposition 2.3 with a standard density argument. It then follows
that operator $\mathcal{L}^{\prime}_{\aleph}$ decomposes as

$\mathcal{L}^{\prime}_{\aleph}=\mathcal{L}^{\prime}_{P,\aleph}\oplus\mathcal{L}^{\prime}_{\perp,\aleph},$
(2.26)

so that, by the Spectral Theorm, we have

$\mathrm{shift}_{\aleph}=\mathrm{shift}_{P,\aleph}+\mathrm{shift}_{\perp,\aleph}\,.$
(2.27)

In other words, the spectral shift function for the problem (2.1) can be
obtained by computing the spectral shift functions for
$\mathcal{L}^{\prime}_{P,\aleph}$ and $\mathcal{L}^{\prime}_{\perp,\aleph}$
separately, and adding up the results in the end.

By examining the structure of our equations, it is not hard to see that
$\mathrm{shift}_{P,\aleph}$ coincides with the the spectral shift function for
the problem (2.1) in the special case $d=2$ (we will revisit this point more
formally in subsection 2.3). Therefore, in view of Theorem 2.1 and formula
(2.27), the decomposition (2.26) reduces the problem at hand to computing

  1. (i)

the spectral shift function for (2.1) in two dimensions and

  2. (ii)

the spectral shift function of the restriction of our operator to normally
polarised vector fields in arbitrary dimension $d>2$.

### §2.3. The proof

We are now ready to prove Theorem 1.8.

Due to rotational symmetry, we observe that the spectral shift function will
only depend on $\xi^{\prime}$ via its norm $|\xi^{\prime}|$. Therefore, it
suffices to implement our algorithm and determine the spectral shift function
in the special case

$\overline{\xi^{\prime}}=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
1\end{pmatrix}\in\mathbb{R}^{d-1}\,.$ (2.28)

The general case can then be recovered by rescaling the spectral parameter as

$\Lambda\mapsto\frac{\Lambda}{|\xi^{\prime}|^{2}}$ (2.29)

at the very end.

In the next two subsections we will assume that $\xi^{\prime}$ has been chosen
in accordance with (2.28).

####  2.3.1 Computing $\mathrm{shift}_{P,\aleph}$: the two-dimensional case

On account of (2.28), the domain of $\mathcal{L}^{\prime}_{P,\aleph}$ is
comprised of vector functions of the form

$\begin{pmatrix}0\\\ \vdots\\\ 0\\\ f_{1}(z)\\\ f_{2}(z)\end{pmatrix}\,.$
(2.30)

Furthermore, $\mathcal{L}^{\prime}_{P,\aleph}$ acts on (2.30) as the one-
dimensional operator associated with the full elasticity operator in two
spatial dimensions. More precisely, let $\mathcal{L}^{\prime}_{2,\aleph}$ be
the one-dimensional operator (2.1) associated with the operator (1.1) for
$d=2$ and boundary conditions $\mathcal{B}_{\aleph}$. Then we have

$\mathcal{L}^{\prime}_{P,\aleph}\begin{pmatrix}0\\\ \vdots\\\ 0\\\ f_{1}(z)\\\
f_{2}(z)\end{pmatrix}=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
\mathcal{L}^{\prime}_{2,\aleph}\begin{pmatrix}f_{1}(z)\\\
f_{2}(z)\end{pmatrix}\end{pmatrix}.$

See also Fact 2.2(iv). This implies

$\mathrm{shift}_{P,\aleph}=\mathrm{shift}_{2,\aleph}.$ (2.31)

The goal of this subsection is then to prove the following.

######  Proposition 2.5.

We have666Observe that in two dimensions formula (2.28) reads
$\overline{\xi^{\prime}}=1$.

$\mathrm{shift}_{2,\aleph}(\Lambda;1)=@\frac{1}{4}\mathbb{1}_{(\mu,\lambda+2\mu)}(\Lambda)\quad\text{where}\quad@=\begin{cases}+&\text{for
}\aleph=\mathrm{DF}\\\ -&\text{for }\aleph=\mathrm{FD}\end{cases}$ (2.32)

and $\mathbb{1}_{A}$ denotes the characteristic function of the set $A$.

In order to prove Proposition 2.5 let us implement the algorithm from
subsection 2.1. The associated one-dimensional spectral problem (2.1) has
continuous spectrum $[\mu,+\infty)$, with thresholds

$\Lambda_{*}^{(1)}=\mu\,,\qquad\Lambda_{*}^{(2)}=(\lambda+2\mu)\,.$

The latter partition the continuous spectrum into two intervals
$I^{(1)}=(\mu,\lambda+2\mu)$ and $I^{(2)}=(\lambda+2\mu,+\infty)$ of
multiplicity $1$ and $2$, respectively.

The eigenfunctions of the continuous spectrum read

$\mathbf{u}(z;\Lambda)=\frac{1}{\sqrt{4\pi}\sqrt{\Lambda}}\begin{pmatrix}\left(\frac{\Lambda}{\mu}-1\right)^{1/4}\left[c_{1}^{-}e^{-i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}-c_{1}^{+}e^{i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}\right]\\\
\left(\frac{\Lambda}{\mu}-1\right)^{-1/4}\left[c_{1}^{-}e^{-i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}+c_{1}^{+}e^{i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}\right]\end{pmatrix}\\\
+C\,\sqrt{\frac{\lambda+2\mu}{\Lambda}}\begin{pmatrix}1\\\
i\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\end{pmatrix}e^{-\sqrt{1-\frac{\Lambda}{\lambda+2\mu}}\,z}$
(2.33)

for $\Lambda\in I^{(1)}$ and

$\mathbf{u}(z;\Lambda)=\frac{1}{\sqrt{4\pi}\sqrt{\Lambda}}\begin{pmatrix}\left(\frac{\Lambda}{\mu}-1\right)^{1/4}\left[c_{1}^{-}e^{-i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}-c_{1}^{+}e^{i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}\right]\\\
\left(\frac{\Lambda}{\mu}-1\right)^{-1/4}\left[c_{1}^{-}e^{-i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}+c_{1}^{+}e^{i\left(\frac{\Lambda}{\mu}-1\right)^{1/2}z}\right]\end{pmatrix}\\\
+\frac{1}{\sqrt{4\pi}\sqrt{\Lambda}}\begin{pmatrix}\left(\frac{\Lambda}{\lambda+2\mu}-1\right)^{-1/4}\left[c_{2}^{-}e^{-i\left(\frac{\Lambda}{\lambda+2\mu}-1\right)^{1/2}z}+c_{2}^{+}e^{i\left(\frac{\Lambda}{\lambda+2\mu}-1\right)^{1/2}z}\right]\\\
\left(\frac{\Lambda}{\lambda+2\mu}-1\right)^{1/4}\left[-c_{2}^{-}e^{-i\left(\frac{\Lambda}{\lambda+2\mu}-1\right)^{1/2}z}+c_{2}^{+}e^{i\left(\frac{\Lambda}{\lambda+2\mu}-1\right)^{1/2}z}\right]\end{pmatrix}$
(2.34)

for $\Lambda\in I^{(2)}$.

By imposing that (2.33) and (2.34) satisfy mixed boundary conditions (2.1),
(2.3) we obtain the scattering matrices

$S_{\mathrm{DF}}(\Lambda)=\begin{cases}1&\text{for}\quad\Lambda\in(\mu,\lambda+2\mu)\,,\\\
\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}&\text{for}\quad\Lambda\in(\lambda+2\mu,+\infty)\end{cases}$
(2.35)

and

$S_{\mathrm{FD}}(\Lambda)=\begin{cases}-1&\text{for}\quad\Lambda\in(\mu,\lambda+2\mu)\,,\\\
\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}&\text{for}\quad\Lambda\in(\lambda+2\mu,+\infty)\,.\end{cases}$
(2.36)

######  Lemma 2.6.

We have the following:

$\text{the threshold $\Lambda_{*}^{(1)}$
is}\quad\begin{cases}\text{soft}&\text{for}\quad\aleph=\mathrm{DF}\\\
\text{rigid}&\text{for}\quad\aleph=\mathrm{FD}\end{cases}\ ,$ (2.37)
$\text{the threshold $\Lambda_{*}^{(2)}$
is}\quad\begin{cases}\text{rigid}&\text{for}\quad\aleph=\mathrm{DF}\\\
\text{soft}&\text{for}\quad\aleph=\mathrm{FD}\end{cases}\ .$ (2.38)

###### Proof.

In accordance with (2.11), for $\Lambda=\Lambda_{*}^{(1)}$ we seek solutions
in the form

$c_{1}\begin{pmatrix}0\\\ 1\end{pmatrix}+c_{2}\begin{pmatrix}1\\\
i\sqrt{\frac{\lambda+\mu}{\lambda+2\mu}}\end{pmatrix}e^{-\sqrt{\frac{\lambda+\mu}{\lambda+2\mu}}z}\,$
(2.39)

for some constants $c_{1}$ and $c_{2}$. Substituting (2.39) into our boundary
conditions one finds that when $\aleph=\mathrm{DF}$ the function (2.39)
satisfies boundary conditions for any $c_{1}\in\mathbb{R}$ and $c_{2}=0$,
whereas when $\aleph=\mathrm{FD}$ the function (2.39) only satisfies boundary
conditions for $c_{1}=c_{2}=0$. This gives us (2.37).

For $\Lambda=\Lambda_{*}^{(2)}$ we seek solutions in the form

$c\begin{pmatrix}1\\\ 0\end{pmatrix}$ (2.40)

for some constant $c$. Now, it is easy to see that (2.40) satisfies
$\mathrm{FD}$ boundary conditions for any $c\in\mathbb{R}$, whereas it
satisfies $\mathrm{DF}$ boundary conditions only for $c=0$. This gives us
(2.38) and completes the proof. ∎

######  Lemma 2.7.

The operator $\mathcal{L}^{\prime}_{2,\aleph}$ does not have eigenvalues below
or embedded into the continuous spectrum for either set of mixed boundary
conditions $\aleph=\mathrm{DF},\mathrm{FD}$.

###### Proof.

For $\Lambda\in(0,\mu)$ we seek an eigenfunction in the form

$c_{1}\begin{pmatrix}-i\left(1-\frac{\Lambda}{\mu}\right)^{1/2}\\\
1\end{pmatrix}e^{-\sqrt{1-\frac{\Lambda}{\mu}}z}\\\
+c_{2}\,\begin{pmatrix}1\\\
i\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\end{pmatrix}e^{-\sqrt{1-\frac{\Lambda}{\lambda+2\mu}}\,z}\,.$
(2.41)

Substituting (2.41) into the $\mathrm{FD}$ boundary conditions we obtain

$\begin{pmatrix}i\left(1-\frac{\Lambda}{\mu}\right)&-\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\\\
1&i\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\end{pmatrix}\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}\,.$

The latter has a nontrivial solution if and only if

$\chi(\Lambda)=\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\left[-\left(1-\frac{\Lambda}{\mu}\right)+1\right]=\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\frac{\Lambda}{\mu}=0\,.$
(2.42)

But the characteristic equation (2.42) does not admit solutions in $(0,\mu)$.
The case of $\mathrm{DF}$ is analogous, with no eigenfunctions in $(0,\mu)$,
and we omit the details. All in all, there are no solutions below bottom of
the essential spectrum for either set of mixed boundary conditions.

The threshold $\Lambda=\mu$ is not an eigenvalue. Indeed, an eigenfunction of
the form

$c\begin{pmatrix}1\\\
i\sqrt{\frac{\lambda+\mu}{\lambda+2\mu}}\end{pmatrix}e^{-\sqrt{\frac{\lambda+\mu}{\lambda+2\mu}}z}$

satisfies boundary conditions (2.3) only if $c=0$.

For $\Lambda\in(\mu,\lambda+2\mu)$ we seek an eigenfunction in the form

$c\,\begin{pmatrix}1\\\
i\left(1-\frac{\Lambda}{\lambda+2\mu}\right)^{1/2}\end{pmatrix}e^{-\sqrt{1-\frac{\Lambda}{\lambda+2\mu}}\,z}\,.$

Once again, the latter satisfies either set of mixed boundary conditions (2.3)
only if $c=0$. Therefore, there are no eigenvalues in $(\mu,\lambda+2\mu)$.

Finally, it is easy to see that $\Lambda=\lambda+2\mu$ is not an eigenvalue,
and that there are no square integrable solutions of our one-dimensional
spectral problem for values of the spectral parameter $\Lambda>\lambda+2\mu$.
∎

Now, Lemma 2.7 implies that the one-dimensional counting function vanishes
identically. Therefore, on account of (2.13) and (2.9)–(2.10), combining
(2.35), (2.36) with Lemma 2.6 one arrives at (2.32).

####  2.3.2 Computing $\mathrm{shift}_{\perp,\aleph}$: normally polarised
waves

Let us now examine our one-dimensional spectral problem restricted to the
subspace $\mathbf{P}^{\perp}$ (2.16). The goal of this subsection is to prove
the following.

######  Proposition 2.8.

We have

$\mathrm{shift}_{\perp,\aleph}(\Lambda;\overline{\xi^{\prime}})=@\frac{d-2}{4}\mathbb{1}_{(\mu,+\infty)}(\Lambda)\quad\text{where}\quad@=\begin{cases}-&\text{for
}\aleph=\mathrm{DF}\\\ +&\text{for }\aleph=\mathrm{FD}\end{cases}.$ (2.43)

In order to prove Proposition 2.8 let us implement the algorithm from
subsection 2.1.

One can easliy check that when restricted to normally polarised waves the
operator $\mathcal{L}^{\prime}$ acts as

$\mathcal{L}^{\prime}_{\perp,\aleph}\mathbf{u}=\mu\left(1-\frac{\mathrm{d}^{2}}{\mathrm{d}z^{2}}\right)\mathbf{u}.$

This implies that the one-dimensional spectral problem (2.1) for
$\mathcal{L}^{\prime}=\mathcal{L}^{\prime}_{\perp,\aleph}$ has only one
threshold

$\Lambda_{*}=\mu,$ (2.44)

and the essential spectrum $[\mu,+\infty)$ has multiplicity $d-2$.

For $\Lambda>\mu$ the eigenfunctions of the continuous spectrum read

$\mathbf{u}(z;\Lambda)=\sum_{j=1}^{d-2}\mathbf{e}_{j}\left(c_{j}^{+}e^{i\sqrt{\frac{\Lambda}{\mu}-1}}+c_{j}^{-}e^{-i\sqrt{\frac{\Lambda}{\mu}-1}}\right)\,,$
(2.45)

where $(\mathbf{e}_{j})_{\alpha}=\delta_{j\alpha}$.

By imposing that (2.45) satisfy mixed boundary conditions (2.3) we obtain the
scattering matrices

$S_{\aleph}(\Lambda)=@I_{d-2}\,,\qquad\quad@=\begin{cases}-&\text{for}\
\,\aleph=\mathrm{DF}\\\ +&\text{for}\
\,\aleph=\mathrm{FD}\end{cases},\quad\Lambda\in(\mu,+\infty).$ (2.46)

where $I_{d-2}$ is the $(d-2)$-dimensional identity matrix.

######  Lemma 2.9.

The threshold (2.44) is

$\begin{cases}\text{rigid}&\text{for}\quad\aleph=\mathrm{DF}\\\
\text{soft}&\text{for}\quad\aleph=\mathrm{FD}\end{cases}\ .$ (2.47)

###### Proof.

In accordance with (2.11), for $\Lambda=\Lambda_{*}$ we seek solutions in the
form

$\begin{pmatrix}c_{1}\\\ c_{2}\\\ ...\\\ c_{d-2}\\\ 0\\\ 0\end{pmatrix}$
(2.48)

for some constants $c_{j}$, $j=1,\ldots,d-2$. Substituting the latter into
(2.3) one immediately sees that (2.48) satisfies $\mathrm{DF}$ boundary
conditions only if all the $c_{j}$’s vanish, whereas it satisfies
$\mathrm{FD}$ boundary conditions for any choice of constants $c_{j}$’s.
Hence, one has (2.47). ∎

######  Lemma 2.10.

The operator $\mathcal{L}^{\prime}_{\perp,\aleph}$,
$\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}$, does not have eigenvalues, either
below or embedded into the continuous spectrum.

###### Proof.

For $\Lambda\in(0,\mu)$ we seek an eigenfunction in the form

$\begin{pmatrix}c_{1}\\\ c_{2}\\\ ...\\\ c_{d-2}\\\ 0\\\
0\end{pmatrix}e^{-\sqrt{1-\frac{\Lambda}{\mu}}}\,.$

But the latter does not satisfy either set of mixed boundary conditions unless
all the constants $c_{j}$ vanish; therefore, there are no eigenvalues in
$(0,\mu)$.

Finally, it is easy to see that there are no square integrable solutions of
the one-dimensional spectral problem for values of the spectral parameter
$\Lambda\geq\mu$. Hence, there are no eigenvalues in $[\mu,+\infty)$ either. ∎

As in the previous subsection, Lemma 2.10 implies that the one-dimensional
counting function vanishes identically. Therefore, on account of (2.13) and
(2.9)–(2.10), combining (2.46) with Lemma 2.9 one arrives at (2.43).

####  2.3.3 Putting things together

Combining Proposition 2.5, Proposition 2.8, and formulae (2.29), (2.27),
(2.31) we obtain

$\mathrm{shift}_{\aleph}(\Lambda;\xi^{\prime})=@\begin{cases}0&\text{for }\
\Lambda<\mu\,|\xi^{\prime}|^{2}\\\ \frac{d-3}{4}&\text{for }\
\mu\,|\xi^{\prime}|^{2}<\Lambda<(\lambda+2\mu)\,|\xi^{\prime}|^{2}\\\
\frac{d-2}{4}&\text{for }\
\Lambda>(\lambda+2\mu)\,|\xi^{\prime}|^{2}\end{cases}\quad\text{with}\quad@=\begin{cases}-&\text{for}\
\,\aleph=\mathrm{DF},\\\ +&\text{for}\ \,\aleph=\mathrm{FD}.\end{cases}$
(2.49)

Substituting (2.49) into (2.14) and integrating we arrive at (1.21). This
completes the proof of Theorem 1.8.

##  Appendix 3.§3. Explicit examples

In this section we verify our formulae for the second Weyl coefficients by
examining the asymptotics of the eigenvalue counting function for explicit
examples: the disk, and flat cylinders in dimensions $d=2$ and $d=3$.

The choice of examples is motivated by the fact that they possess the
following properties.

  1. (i)

They allow for separation of variables for the operator of linear elasticity
with mixed boundary conditions.

  2. (ii)

They satisfy the conditions on branching Hamiltonian billiards from Theorem
1.7, so that the two-term asymptotics for the counting function is valid.

  3. (iii)

For flat cylinders, variables separate completely and one can write down the
full spectrum explicitly. Therefore, unlike in [6], we can verify our formulae
_analytically_ , using asymptotic expansions for certain number-theoretic
series determined by our eigenvalues.

### §3.1. Two-dimensional examples

####  3.1.1 The disk

Let $M\subset\mathbb{R}^{2}$ be the unit disk and let us work in standard
polar coordinates $(r,\theta)$. Following [24, Chapter XIII] (see also [22]),
we introduce a fictitious third coordinate $z$ orthogonal to the disk and seek
solutions in the form

$\mathbf{u}(r,\theta)=\operatorname{grad}\psi_{1}(r,\theta)+\operatorname{curl}\left(\psi_{2}(r,\theta)\,\hat{\mathbf{z}}\right)\,,$
(3.1)

where $\hat{\mathbf{z}}$ is the unit vector in the direction of $z$ and
$\psi_{j}$, $j=1,2$, are auxiliary scalar potentials. Substituting (3.1) into
(1.11) one obtains that the scalar potentials must satisfy the Helmholtz
equations

$-\Delta\psi_{j}=\omega_{j,\Lambda}\psi_{j},\qquad j=1,2,$ (3.2)
$\omega_{1,\Lambda}:=\frac{\Lambda}{\lambda+2\mu},\qquad\omega_{2,\Lambda}:=\frac{\Lambda}{\mu}\,.$
(3.3)

But now the general solution to (3.2) regular at $r=0$ reads

$\psi_{j}(r,\phi)=c_{j,0}J_{0}\left(\sqrt{\omega_{j,\Lambda}}r\right)+\sum_{k=1}^{\infty}J_{k}\left(\sqrt{\omega_{j,\Lambda}}r\right)\left(c_{j,k,+}\mathrm{e}^{\mathrm{i}k\theta}+c_{j,k,-}\mathrm{e}^{-\mathrm{i}k\theta}\right),$
(3.4)

where the $J_{k}$’s are Bessel functions of the first kind. By substituting
(3.4) and imposing that (3.1) satisfies $\mathrm{DF}$ boundary conditions

$\left.\begin{pmatrix}(\lambda+2\mu)\,\partial_{r}u_{1}+\lambda(u_{1}+\partial_{\theta}u_{2})\\\
u_{2}\end{pmatrix}\right|_{r=1}=\begin{pmatrix}0\\\ 0\end{pmatrix}\,,$

one obtains the secular equation

$\mu
kJ_{k}\left(\sqrt{\omega_{2,\Lambda}}\right)\left[\omega_{2,\Lambda}J_{k}\left(\sqrt{\omega_{1,\Lambda}}\right)-2\sqrt{\omega_{1,\Lambda}}J_{k+1}\left(\sqrt{\omega_{1,\Lambda}}\right)\right]\\\
+\mu\sqrt{\omega_{2,\Lambda}}J_{k+1}\left(\sqrt{\omega_{2,\Lambda}}\right)\left[2\sqrt{\omega_{1,\Lambda}}J_{k+1}\left(\sqrt{\omega_{1,\Lambda}}\right)-(2k+\omega_{2,\Lambda})J_{k}\left(\sqrt{\omega_{1,\Lambda}}\right)\right]=0.$
(3.5)

For $\mathrm{FD}$ boundary conditions one obtains an analogous formula, which
we omit.

One can use Mathematica to find the zeroes of (3.5) (and the corresponding
equation for $\mathrm{FD}$ boundary conditions) numerically and compute the
eigenvalue counting function $N_{\aleph}(\Lambda)$,
$\aleph\in\\{\mathrm{DF},\mathrm{FD}\\}$, for reasonably large values of the
parameter $\Lambda$.

The numerical results are shown in Figures 1 and 2.

Figure 1: The DF eigenvalue problem for the disk. In all images $\mu=1$.
Figure 2: The FD eigenvalue problem for the disk. In all images $\mu=1$.

####  3.1.2 Flat cylinders

Consider the two-dimensional cylinder $M:=\mathbb{T}\times[0,h]$, where
$\mathbb{T}$ is the one-dimensional torus and $h>0$ is the height of the
cylinder, equipped with coordinates $(x^{1},x^{2})\in[0,2\pi)\times[0,h]$. Of
course,

$\operatorname{Vol}_{2}(M)=2\pi h,\qquad\operatorname{Vol}_{1}(\partial
M)=4\pi.$ (3.6)

We separate variables by seeking a solution in the form

$\mathbf{u}(x^{1},x^{2})=\operatorname{grad}\psi_{1}(x^{1},x^{2})+\operatorname{curl}\left(\psi_{2}(x^{1},x^{2})\,\hat{\mathbf{z}}\right)\,,$
(3.7)

where $\hat{\mathbf{z}}$ is the unit vector in the auxiliary coordinate
$x^{3}$ (orthogonal to the $(x^{1},x^{2})$-plane) and $\psi_{j}$, $j=1,2$, are
scalar potentials. As in subsection 3.1.1, the scalar potentials satisfy
Helmholtz equation (3.2), (3.3). The general solution for $\psi_{j}$, $j=1,2$,
reads

$\psi_{j}(x^{1},x^{2})=\sum_{\xi\in\mathbb{Z}}\left(c_{j,\xi,+}e^{i\left(x^{1}\xi+\sqrt{\omega_{j,\Lambda}-\xi^{2}}x^{2}\right)}+c_{j,\xi,-}e^{i\left(x^{1}\xi-\sqrt{\omega_{j,\Lambda}-\xi^{2}}x^{2}\right)}\right)\,.$
(3.8)

Substituting (3.8) into (3.7) and, in turn, imposing boundary conditions
$\mathcal{B}_{\mathrm{DF}}$ yields the secular equation

$\Lambda^{2}\left(\frac{\Lambda}{\mu}-\xi^{2}\right)\sin\left(h\sqrt{\frac{\Lambda}{\mu}-\xi^{2}}\right)\sin\left(h\sqrt{\frac{\Lambda}{\lambda+2\mu}-\xi^{2}}\right)=0\,.$
(3.9)

Similarly, imposing boundary conditions $\mathcal{B}_{\mathrm{FD}}$ yields the
secular equation

$\Lambda^{2}\left(\frac{\Lambda}{\lambda+2\mu}-\xi^{2}\right)\sin\left(h\sqrt{\frac{\Lambda}{\mu}-\xi^{2}}\right)\sin\left(h\sqrt{\frac{\Lambda}{\lambda+2\mu}-\xi^{2}}\right)=0\,.$
(3.10)

A careful examination of (3.7)–(3.10) yields the following.

######  Theorem 3.1.

The eigenvalues of the Dirichlet-free (DF) eigenvalue problem for the operator
of linear elasticity on the two-dimensional cylinder are:

  1. (i)

Eigenvalues

$\frac{k^{2}\pi^{2}}{h^{2}}(\lambda+2\mu),\qquad k=1,2,\dots,$ (3.11)

with multiplicity $1$.

  2. (ii)

Eigenvalues

$\frac{k^{2}\pi^{2}}{h^{2}}\mu,\qquad k=1,2,\dots,$ (3.12)

with multiplicity $1$.

  3. (iii)

Eigenvalues

$n^{2}\,\mu,\qquad n=1,2,\dots,$ (3.13)

with multiplicity $2$.

  4. (iv)

Eigenvalues

$\left(n^{2}+\frac{k^{2}\pi^{2}}{h^{2}}\right)\mu,\qquad n,k=1,2,\dots,$
(3.14)

with multiplicity $2$.

  5. (v)

Eigenvalues

$\left(n^{2}+\frac{k^{2}\pi^{2}}{h^{2}}\right)(\lambda+2\mu),\qquad
n,k=1,2,\dots,$ (3.15)

with multiplicity $2$.

######  Theorem 3.2.

The eigenvalues of the free-Dirichlet (FD) eigenvalue problem for the operator
of linear elasticity on the two-dimensional cylinder are:

  1. (i)

Eigenvalues

$\frac{k^{2}\pi^{2}}{h^{2}}(\lambda+2\mu),\qquad k=1,2,\dots,$ (3.16)

with multiplicity $1$.

  2. (ii)

Eigenvalues

$\frac{k^{2}\pi^{2}}{h^{2}}\mu,\qquad k=1,2,\dots,$ (3.17)

with multiplicity $1$.

  3. (iii)

Eigenvalues

$\left(n^{2}+\frac{k^{2}\pi^{2}}{h^{2}}\right)\mu,\qquad n,k=1,2,\dots,$
(3.18)

with multiplicity $2$.

  4. (iv)

Eigenvalues

$n^{2}(\lambda+2\mu),\qquad n=1,2,\dots,$ (3.19)

with multiplicity $2$.

  5. (v)

Eigenvalues

$\left(n^{2}+\frac{k^{2}\pi^{2}}{h^{2}}\right)(\lambda+2\mu),\qquad
n,k=1,2,\dots,$ (3.20)

with multiplicity $2$.

Figure 3: The DF eigenvalue problem for 2D flat cylinders. In all images
$\mu=1$. Figure 4: The FD eigenvalue problem for 2D flat cylinders. In all
images $\mu=1$.

Observe that the DF (Theorem 3.1) and the FD (Theorem 3.2) spectra coincide,
except for fact that the series of eigenvalues (3.13) in the DF spectrum is
replaced by the series of eigenvalues (3.19) in the FD spectrum.

Theorems 3.4 and 3.5 allow us to write down the eigenvalue counting functions
$N_{\mathrm{DF}}$ and $N_{\mathrm{FD}}$ explicitly. They read

$N_{\mathrm{DF}}(\Lambda)=\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\lambda+2\mu}}\right\rfloor+\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\mu}}\right\rfloor+2\left\lfloor\sqrt{\frac{\Lambda}{\mu}}\right\rfloor\\\
+2\sum_{n=1}^{\left\lfloor\sqrt{\frac{\Lambda}{\mu}}\right\rfloor}\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\mu}-n^{2}}\right\rfloor+2\sum_{n=1}^{\left\lfloor\sqrt{\frac{\Lambda}{\lambda+2\mu}}\right\rfloor}\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\lambda+2\mu}-n^{2}}\right\rfloor,$
(3.21)

$N_{\mathrm{FD}}(\Lambda)=N_{\mathrm{DF}}(\Lambda)+2\left(\left\lfloor\sqrt{\frac{\Lambda}{\lambda+2\mu}}\right\rfloor-\left\lfloor\sqrt{\frac{\Lambda}{\mu}}\right\rfloor\right)\,.$
(3.22)

Here $\lfloor\,\cdot\,\rfloor$ denotes the integer part (floor function).

Let us verify formula (1.21) by computing the asymptotic expansions of (3.21)
and (3.22) as $\Lambda\to+\infty$.

######  Proposition 3.3.

The functions (3.21) and (3.22) admit the following two-term asymptotic
expansion:

$N_{\aleph}(\Lambda)=\frac{h}{2}\left(\frac{1}{\mu}+\frac{1}{\lambda+2\mu}\right)\Lambda\pm\left(\frac{1}{\mu^{1/2}}-\frac{1}{(\lambda+2\mu)^{1/2}}\right)\Lambda^{1/2}+o(\Lambda^{1/2})\quad\text{as}\quad\Lambda\to+\infty$
(3.23)

with sign

$\begin{cases}+&\text{for}\ \,\aleph=\mathrm{DF},\\\ -&\text{for}\
\,\aleph=\mathrm{FD}.\end{cases}$ (3.24)

###### Proof.

Formula (3.23) follows from (3.22), (3.21), and the estimate

$\sum_{n=1}^{\lfloor\sqrt{x}\rfloor}\left\lfloor
a\sqrt{x-n^{2}}\right\rfloor=\frac{\pi
a}{4}x-\frac{1}{2}\left(a+1\right)x^{1/2}+o(x^{1/2})\quad\text{as}\quad
x\to+\infty\,,\quad a>0\,.$

∎

On account of (3.6), Proposition 3.3 agrees with Theorem 1.8 as well as
formula (1.22).

Figures 3 and 4 show a comparison between the actual counting functions
(3.21), (3.22) and the two-term asymptotic expansions (3.23), (3.24).

### §3.2. Three-dimensional examples

####  3.2.1 Flat cylinders

Consider the three-dimensional cylinder $M:=\mathbb{T}^{2}\times[0,h]$, where
$\mathbb{T}^{2}$ is the flat two-dimensional torus with side $2\pi$ and $h>0$
is the height of the cylinder, equipped with coordinates
$(x^{1},x^{2},x^{3})\in[0,2\pi)^{2}\times[0,h]$. Of course,

$\operatorname{Vol}_{3}(M)=4\pi^{2}h,\qquad\operatorname{Vol}_{2}(\partial
M)=8\pi^{2}.$ (3.25)

We separate variables by seeking a solution in the form

$\mathbf{u}(x^{1},x^{2},x^{3})=\operatorname{grad}\psi_{1}(x^{1},x^{2},x^{3})+\operatorname{curl}\left(\psi_{2}(x^{1},x^{2},x^{3})\,\hat{\mathbf{z}}\right)+\operatorname{curl}\operatorname{curl}\left(\psi_{3}(x^{1},x^{2},x^{3})\,\hat{\mathbf{z}}\right)\,,$
(3.26)

where $\hat{\mathbf{z}}$ is the unit vector in the (positive) direction
$x^{3}$ and $\psi_{j}$, $j=1,2,3$, are scalar potentials. Once again, the
scalar potentials satisfy Helmholtz equation (3.2), with $\omega_{1,\Lambda}$
and $\omega_{2,\Lambda}$ defined in accordance with (3.3), and
$\omega_{3,\Lambda}:=\omega_{2,\Lambda}$. The general solution for $\psi_{j}$,
$j=1,2,3$, reads

$\psi_{j}(x^{1},x^{2},x^{3})=\sum_{(\xi_{1},\xi_{2})\in\mathbb{Z}^{2}}\left(c_{j,\xi_{1},\xi_{2},+}e^{i\left(x^{1}\xi_{1}+x^{2}\xi_{2}+\sqrt{\omega_{j,\Lambda}-n}x^{3}\right)}+c_{j,\xi_{1},\xi_{2},-}e^{i\left(x^{1}\xi_{1}+x^{2}\xi_{2}+\sqrt{\omega_{j,\Lambda}-n}x^{2}\right)}\right)\,,$
(3.27)

where $n:=\xi_{1}^{2}+\xi_{2}^{2}$. Substituting (3.27) into (3.26) and, in
turn, imposing boundary conditions $\mathcal{B}_{\mathrm{DF}}$ at $x^{3}=0$
and $x^{3}=h$ yields the secular equation

$\Lambda^{2}n^{2}\left(\frac{\Lambda}{\mu}-n\right)\sin^{2}\left(h\sqrt{\frac{\Lambda}{\mu}-n}\right)\sin\left(h\sqrt{\frac{\Lambda}{\lambda+2\mu}-n}\right)=0\,.$
(3.28)

Similarly, imposing boundary conditions $\mathcal{B}_{\mathrm{FD}}$ yields the
secular equation

$\Lambda^{2}n^{2}\left(\frac{\Lambda}{\mu}-n\right)\left(\frac{\Lambda}{\lambda+2\mu}-n\right)\sin^{2}\left(h\sqrt{\frac{\Lambda}{\mu}-n}\right)\sin\left(h\sqrt{\frac{\Lambda}{\lambda+2\mu}-n}\right)=0\,.$
(3.29)

The $\mathrm{DF}$ (resp. $\mathrm{FD}$) spectrum is a subset of the zeroes of
(3.28) (resp. (3.29)). A direct examination of solutions of (3.28) and (3.29)
yields the following.

Let $r_{2}:\mathbb{N}\to\mathbb{N}$ be the sum of squares function:

$r_{2}(n):=\\#\left\\{(a,b)\in\mathbb{Z}^{2}\ |\ n=a^{2}+b^{2}\right\\}\,.$

######  Theorem 3.4.

The eigenvalues of the Dirichlet-free eigenvalue problem for the operator of
linear elasticity on the three-dimensional flat cylinder are:

  1. (i)

Eigenvalues

$\frac{k^{2}\pi^{2}}{h^{2}}(\lambda+2\mu),\qquad k=1,2,\dots,$ (3.30)

with multiplicity $1$.

  2. (ii)

Eigenvalues

$n\,\mu,\qquad n=1,2,\dots,$ (3.31)

with multiplicity $r_{2}(n)$.

  3. (iii)

Eigenvalues

$\left(n+\frac{k^{2}\pi^{2}}{h^{2}}\right)\mu,\qquad n,k=1,2,\dots,$ (3.32)

with multiplicity $2r_{2}(n)$.

  4. (iv)

Eigenvalues

$\left(n+\frac{k^{2}\pi^{2}}{h^{2}}\right)(\lambda+2\mu),\qquad
n,k=1,2,\dots,$ (3.33)

with multiplicity $r_{2}(n)$.

######  Theorem 3.5.

The eigenvalues of the free-Dirichlet eigenvalue problem for the operator of
linear elasticity on the three-dimensional flat cylinder are:

  1. (i)

Eigenvalues

$\frac{k^{2}\pi^{2}}{h^{2}}(\lambda+2\mu),\qquad k=1,2,\dots,$ (3.34)

with multiplicity $1$.

  2. (ii)

Eigenvalues

$n\,\mu,\qquad n=1,2,\dots,$ (3.35)

with multiplicity $r_{2}(n)$777Here and further on by multiplicity zero we
mean that the corresponding number is not an eigenvalue..

  3. (iii)

Eigenvalues

$\left(n+\frac{k^{2}\pi^{2}}{h^{2}}\right)\mu,\qquad n,k=1,2,\dots,$ (3.36)

with multiplicity $2r_{2}(n)$.

  4. (iv)

Eigenvalues

$n\,(\lambda+2\mu),\qquad n=1,2,\dots,$ (3.37)

with multiplicity $r_{2}(n)$.

  5. (v)

Eigenvalues

$\left(n+\frac{k^{2}\pi^{2}}{h^{2}}\right)(\lambda+2\mu),\qquad
n,k=1,2,\dots,$ (3.38)

with multiplicity $r_{2}(n)$.

Observe that the DF (Theorem 3.4) and the FD (Theorem 3.5) spectra coincide,
except for the additional series of eigenvalues (3.37) in the FD spectrum.

Theorems 3.4 and 3.5 allow us to write down the eigenvalue counting functions
$N_{\mathrm{DF}}$ and $N_{\mathrm{FD}}$ explicitly. They read

$N_{\mathrm{DF}}(\Lambda)=\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\lambda+2\mu}}\right\rfloor+\sum_{n=1}^{\left\lfloor\frac{\Lambda}{\mu}\right\rfloor}r_{2}(n)\left(2\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\mu}-n}\right\rfloor+1\right)\\\
+\sum_{n=1}^{\left\lfloor\frac{\Lambda}{\lambda+2\mu}\right\rfloor}r_{2}(n)\left\lfloor\frac{h}{\pi}\sqrt{\frac{\Lambda}{\lambda+2\mu}-n}\right\rfloor\,,$
(3.39)
$N_{\mathrm{FD}}(\Lambda)=N_{\mathrm{DF}}(\Lambda)+\sum_{n=1}^{\left\lfloor\frac{\Lambda}{\lambda+2\mu}\right\rfloor}r_{2}(n).$
(3.40)

Let us verify formula (1.21) by computing the asymptotic expansions of (3.40)
and (3.39) as $\Lambda\to+\infty$.

######  Proposition 3.6.

The functions (3.40) and (3.39) admit the following two-term asymptotic
expansion:

$N_{\aleph}(\Lambda)=\frac{2h}{3}\left(\frac{2}{\mu^{3/2}}+\frac{1}{(\lambda+2\mu)^{3/2}}\right)\Lambda^{3/2}\mp\frac{\pi}{2(\lambda+2\mu)}\Lambda+o(\Lambda)\quad\text{as}\quad\Lambda\to+\infty$
(3.41)

with sign

$\begin{cases}-&\text{for}\ \,\aleph=\mathrm{DF},\\\ +&\text{for}\
\,\aleph=\mathrm{FD}.\end{cases}$ (3.42)

###### Proof.

Formula (3.42) follows from (3.40), (3.40), and the estimates

$\sum_{n=1}^{\lfloor x\rfloor}r_{2}(n)=\pi x+O(x^{1/3})\quad\text{as}\quad
x\to+\infty,$ $\sum_{n=1}^{\lfloor x\rfloor}\left\lfloor
a\sqrt{x-n}\right\rfloor r_{2}(n)=\frac{2\pi
a}{3}x^{3/2}-\frac{\pi}{2}x+o(x)\quad\text{as}\quad x\to+\infty,\quad a>0.$

∎

On account of (3.25), Proposition 3.6 agrees with Theorem 1.8 as well as
formula (1.23).

Figures 5 and 6 show a comparison between the actual counting functions
(3.39), (3.40) and the two-term asymptotic expansions (3.41), (3.42).

Figure 5: The DF eigenvalue problem for 3D flat cylinders. In all images
$\mu=1$. Figure 6: The FD eigenvalue problem for 3D flat cylinders. In all
images $\mu=1$.

## Acknowledgements

We are indebted to Michael Levitin, Yiannis Petridis and Dmitri Vassiliev for
insightful conversations on aspects of this paper, and to Gerd Grubb and
Grigori Rozenblum for useful bibliographic suggestions.

MC was partially supported by a grant of the Heilbronn Institute for
Mathematical Research (HIMR) via the UKRI/EPSRC and by EPSRC Fellowship
EP/X01021X/1. IM was supported by a MAC-MIGS Summer Internship (MAC-MIGS CDT,
Maxwell Institute Graduate School, Edinburgh).

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* •

Charles R. Ferenbaugh, Experiments in Sustainable Software Practices for
Future Architectures [83]

  * •

Eric G Stephan, Todd O Elsethagen, Kerstin Kleese van Dam, Laura Riihimaki.
What Comes First, the OWL or the Bean? [88]

  * •

Derek R. Gaston, John Peterson, Cody J. Permann, David Andrs, Andrew E.
Slaughter, Jason M. Miller, Continuous Integration for Concurrent
Computational Framework and Application Development [89]

  * •

Anshu Dubey, B. Van Straalen. Experiences from Software Engineering of Large
Scale AMR Multiphysics Code Frameworks [39]

  * •

Markus Blatt. DUNE as an Example of Sustainable Open Source Scientific
Software Development [30]

  * •

David Koop, Juliana Freiere, Cláudio T. Silva, Enabling Reproducible Science
with VisTrails [27]

  * •

Sean Ahern, Eric Brugger, Brad Whitlock, Jeremy S. Meredith, Kathleen Biagas,
Mark C. Miller, Hank Childs, VisIt: Experiences with Sustainable Software [26]

  * •

Sou-Cheng (Terrya) Choi. MINRES-QLP Pack and Reliable Reproducible Research
via Staunch Scientific Software [90]

  * •

Michael Crusoe, C. Titus Brown. Walking the talk: adopting and adapting
sustainable scientific software development processes in a small biology lab
[31]

  * •

Dhabaleswar K. Panda, Karen Tomko, Karl Schulz, Amitava Majumdar. The MVAPICH
Project: Evolution and Sustainability of an Open Source Production Quality MPI
Library for HPC [28]

  * •

Eric M. Heien, Todd L. Miller, Becky Gietzel, Louise H. Kellogg. Experiences
with Automated Build and Test for Geodynamics Simulation Codes [91]

#### Deployment, Support, and Maintenance of Existing Software

  * •

Henri Casanova, Arnaud Giersch, Arnaud Legrand, Martin Quinson, Frédéric
Suter. SimGrid: a Sustained Effort for the Versatile Simulation of Large Scale
Distributed Systems [92]

  * •

Erik Trainer, Chalalai Chaihirunkarn, James Herbsleb. The Big Effects of
Short-term Efforts: A Catalyst for Community Engagement in Scientific Software
[38]

  * •

Jeremy Cohen, Chris Cantwell, Neil Chue Hong, David Moxey, Malcolm
Illingworth, Andrew Turner, John Darlington, Spencer Sherwin. Simplifying the
Development, Use and Sustainability of HPC Software [93]

  * •

Jaroslaw Slawinski, Vaidy Sunderam. Towards Semi-Automatic Deployment of
Scientific and Engineering Applications [94]

#### Best Practices, Challenges, and Recommendations

  * •

Andreas Prlić, James B. Procter. Ten Simple Rules for the Open Development of
Scientific Software [41]

  * •

Anshu Dubey, S. Brandt, R. Brower, M. Giles, P. Hovland, D. Q. Lamb, F.
Löffler, B. Norris, B. O’Shea, C. Rebbi, M. Snir, R. Thakur, Software
Abstractions and Methodologies for HPC Simulation Codes on Future
Architectures [95]

  * •

Jeffrey Carver, George K. Thiruvathukal. Software Engineering Need Not Be
Difficult [84]

  * •

Craig A. Stewart, Julie Wernert, Eric A. Wernert, William K. Barnett, Von
Welch. Initial Findings from a Study of Best Practices and Models for
Cyberinfrastructure Software Sustainability [96]

  * •

Jed Brown, Matthew Knepley, Barry Smith. Run-time extensibility: anything less
is unsustainable [97]

  * •

Shel Swenson, Yogesh Simmhan, Viktor Prasanna, Manish Parashar, Jason Riedy,
David Bader, Richard Vuduc. Sustainable Software Development for Next-Gen
Sequencing (NGS) Bioinformatics on Emerging Platforms [98]

### Policy

#### Modeling Sustainability

  * •

Coral Calero, M. Angeles Moraga, Manuel F. Bertoa. Towards a Software Product
Sustainability Model [43]

  * •

Colin C. Venters, Lydia Lau, Michael K. Griffiths, Violeta Holmes, Rupert R.
Ward, Jie Xu. The Blind Men and the Elephant: Towards a Software
Sustainability Architectural Evaluation Framework [44]

  * •

Marlon Pierce, Suresh Marru, Chris Mattmann. Sustainable Cyberinfrastructure
Software Through Open Governance [45]

  * •

Daniel S. Katz, David Proctor. A Framework for Discussing e-Research
Infrastructure Sustainability [46]

  * •

Christopher Lenhardt, Stanley Ahalt, Brian Blanton, Laura Christopherson, Ray
Idaszak. Data Management Lifecycle and Software Lifecycle Management in the
Context of Conducting Science [47]

  * •

Nicholas Weber, Andrea Thomer, Michael Twidale. Niche Modeling: Ecological
Metaphors for Sustainable Software in Science [48]

#### Credit, Citation, Impact

  * •

Matthew Knepley, Jed Brown, Lois Curfman McInnes, Barry Smith. Accurately
Citing Software and Algorithms Used in Publications [51]

  * •

Jason Priem, Heather Piwowar. Toward a comprehensive impact report for every
software project [50]

  * •

Daniel S. Katz. Citation and Attribution of Digital Products: Social and
Technological Concerns [49]

  * •

Neil Chue Hong, Brian Hole, Samuel Moore. Software Papers: improving the
reusability and sustainability of scientific software [55]

In addition, the following paper from another area were also discussed in this
area.

  * •

Frank Löffler, Steven R. Brandt, Gabrielle Allen and Erik Schnetter. Cactus:
Issues for Sustainable Simulation Software [70]

#### Reproducibility

  * •

Sou-Cheng (Terrya) Choi. MINRES-QLP Pack and Reliable Reproducible Research
via Staunch Scientific Software [90]

  * •

Victoria Stodden, Sheila Miguez. Best Practices for Computational Science:
Software Infrastructure and Environments for Reproducible and Extensible
Research [99]

#### Implementing Policy

  * •

Randy Heiland, Betsy Thomas, Von Welch, Craig Jackson. Toward a Research
Software Security Maturity Model [57]

  * •

Brian Blanton, Chris Lenhardt, A User Perspective on Sustainable Scientific
Software [58]

  * •

Daisie Huang, Hilmar Lapp. Software Engineering as Instrumentation for the
Long Tail of Scientific Software [59]

  * •

Chandra Krintz, Hiranya Jayathilaka, Stratos Dimopoulos, Alexander Pucher,
Rich Wolski. Developing Systems for API Governance [56]

### Communities, Models, and Education

#### Communities

  * •

Reagan Moore. Extensible Generic Data Management Software [100]

  * •

Karen Cranston, Todd Vision, Brian O’Meara, Hilmar Lapp. A grassroots approach
to software sustainability [33]

  * •

J.-L. Vay, C. G. R. Geddes, A. Koniges, A. Friedman, D. P. Grote, D. L.
Bruhwiler. White Paper on DOE-HEP Accelerator Modeling Science Activities [65]

  * •

Ketan Maheshwari, David Kelly, Scott J. Krieder, Justin M. Wozniak, Daniel S.
Katz, Mei Zhi-Gang, Mainak Mookherjee. Reusability in Science: From Initial
User Engagement to Dissemination of Results [66]

  * •

Edmund Hart, Carl Boettiger, Karthik Ram, Scott Chamberlain. rOpenSci – a
collaborative effort to develop R-based tools for facilitating Open Science
[67]

  * •

L. Christopherson, R. Idaszak, S. Ahalt. Developing Scientific Software
through the Open Community Engagement Process [32]

  * •

Marlon Pierce, Suresh Marru, Mark A. Miller, Amit Majumdar, Borries Demeler.
Science Gateway Operational Sustainability: Adopting a Platform-as-a-Service
Approach [68]

  * •

Lynn Zentner, Michael Zentner, Victoria Farnsworth, Michael McLennan, Krishna
Madhavan, and Gerhard Klimeck, nanoHUB.org: Experiences and Challenges in
Software Sustainability for a Large Scientific Community [69]

  * •

Andy Terrel. Sustaining the Python Scientific Software Community [34]

  * •

Frank Löffler, Steven R. Brandt, Gabrielle Allen and Erik Schnetter. Cactus:
Issues for Sustainable Simulation Software [70]

  * •

Nancy Wilkins-Diehr, Katherine Lawrence, Linda Hayden, Marlon Pierce, Suresh
Marru, Michael McLennan, Michael Zentner, Rion Dooley, Dan Stanzione. Science
Gateways and the Importance of Sustainability [71]

  * •

Sou-Cheng (Terrya) Choi. MINRES-QLP Pack and Reliable Reproducible Research
via Staunch Scientific Software [90]

In addition, the following paper from another area was also discussed in this
area.

  * •

Marcus Hanwell, Amitha Perera, Wes Turner, Patrick O’Leary, Katie Osterdahl,
Bill Hoffman, Will Schroeder. Sustainable Software Ecosystems for Open Science
[25]

#### Industry & Economic Models

  * •

Anne C. Elster. Software for Science: Some Personal Reflections on Funding,
Licensing, Publishing and Teaching [81]

  * •

Ian Foster, Vas Vasiliadis, Steven Tuecke. Software as a Service as a path to
software sustainability [80]

  * •

Marcus Hanwell, Amitha Perera, Wes Turner, Patrick O’Leary, Katie Osterdahl,
Bill Hoffman, Will Schroeder. Sustainable Software Ecosystems for Open Science
[25]

In addition, the following papers from other areas were also discussed in this
area.

  * •

Brian Blanton, Chris Lenhardt, A User Perspective on Sustainable Scientific
Software [58]

  * •

Markus Blatt. DUNE as an Example of Sustainable Open Source Scientific
Software Development [30]

  * •

Dhabaleswar K. Panda, Karen Tomko, Karl Schulz, Amitava Majumdar. The MVAPICH
Project: Evolution and Sustainability of an Open Source Production Quality MPI
Library for HPC [28]

  * •

Andy Terrel. Sustaining the Python Scientific Software Community [34]

#### Education & Training

  * •

Ivan Girotto, Axel Kohlmeyer, David Grellscheid, Shawn T. Brown. Advanced
Techniques for Scientific Programming and Collaborative Development of Open
Source Software Packages at the International Centre for Theoretical Physics
(ICTP) [40]

  * •

Thomas Crawford. On the Development of Sustainable Software for Computational
Chemistry [37]

In addition, the following papers from other areas were also discussed in this
area.

  * •

Charles R. Ferenbaugh, Experiments in Sustainable Software Practices for
Future Architectures [83]

  * •

David Koop, Juliana Freiere, Cláudio T. Silva, Enabling Reproducible Science
with VisTrails [27]

  * •

Sean Ahern, Eric Brugger, Brad Whitlock, Jeremy S. Meredith, Kathleen Biagas,
Mark C. Miller, Hank Childs, VisIt: Experiences with Sustainable Software [26]

  * •

Sou-Cheng (Terrya) Choi. MINRES-QLP Pack and Reliable Reproducible Research
via Staunch Scientific Software [90]

  * •

Frank Löffler, Steven R. Brandt, Gabrielle Allen and Erik Schnetter. Cactus:
Issues for Sustainable Simulation Software [70]

  * •

Erik Trainer, Chalalai Chaihirunkarn, James Herbsleb. The Big Effects of
Short-term Efforts: A Catalyst for Community Engagement in Scientific Software
[38]

##  Appendix C Attendees

The following is a partial list of attendees who were recorded on the document
[1] that was being used for live note taking at the workshop, or by the SC13
student volunteers, with some additions also made by the authors of this
report.

Jay Alameda

Gabrielle Allen

David Andrs

Brian Austin

Lorena A. Barba

David Bernholdt

Phil Bourne

Karl Broman

Sharon Broude Geva

Jed Brown

Maxine Brown

David Bruhwiler

Bruno Bzeznik

Alexandru Calotoiu

Jeffrey Carver

Shreyas Cholia

Peng Huat Chua

Neil Chue Hong

John W. Cobb

Timothy Cockerill

Karen Cranston

Rion Dooley

Anshu Dubey

Marat Dukhan

Anne C. Elster

Ian Foster

Juliana Freire

Jeffrey Frey

Derek Gaston

Allison Gehrke

Brian Glendenning

Christian Godenschwager

Derek Groen

Edmund Hart

Magne Haveraaen

Steven Heaton

Oscar Hernandez

James Hetherington

Simon Hettrick

Jonathan Hicks

Kenneth Hoste

James Howison

Daisie Huang

Shao-Ching Huang

Tsutomu Ikegami

Kaxuya Ishimura

Christian Iwainsky

Craig Jackson

Wesley Jones

Randall Judd

Shusuke Kasamatsu

Daniel S. Katz

Kerk Kee

Kellie Kercher

Mads Kristensen

Carolyn Lauzon

Arnaud Legrand

Chris Lenhardt

Michael Levy

Frank Löffler

Monica Lücke

Simon A. F. Lund

Arthur Maccabe

Paul Madden

Louis Maddox

Philip Maechling

Ketan Maheshwari

Brian Marker

Suresh Marru

Cezary Mazurek

James McClure

Matt McKenzie

Chris Mentzel

Paul Messina

Mike Mikailov

J. Yates Monteith

Reagan More

Rafael Morizawa

Pierre Neyron

Lucas Nussbaum

Patrick O’Leary

Manish Parashar

Cody Permann

Jack Perdue

John Peterson

Quan Pham

Marlon Pierce

Heather Piwowar

David Proctor

Sara Rambacher

Nicolas Renon

Jason Riedy

Todd Rimer

Bill Sacks

Andreas Schreiber

William Scullin

Andrew Slaughter

Jaraslaw Slawinski

Arfon Smith

Spencer Smith

James Spencer

Eric Stahlberg

Timothy Stitt

Hyoshin Sung

Frédéric Suter

Shel Swenson

Yoshio Tanaka

Andy Terrel

George Thiruvathukal

Keren Tomko

John Towns

Erik Trainer

Satori Tsuzuki

Matthew Turk

Eric van Wyk

Colin C. Venters

Brice Videau

Tajendra Vir Singh

Von Welch

Nancy Wilkins-Diehr

Theresa Windus

Felix Wolf

Rich Wolski

Lynn Zentner