1.1. The operator of linear elasticity

Let $(M,\,g)$ be a compact connected smooth Riemannian manifold of dimension $d\ge 2$ 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

(1.1)\begin{equation} (\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\,. \end{equation}

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

(1.2)\begin{equation} \mu>0, \quad d\lambda+2\mu>0, \end{equation}

which guarantee strong convexity, see, e.g., [Reference Agranovich, Amosov and Levitin1, Reference Miyanishi and Rozenblum24]. 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 $\mathcal {L}$ readsFootnote ¹

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

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

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

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

(1.5)\begin{equation} \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 \end{equation}

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 [Reference Capoferri and Vassiliev10, formula (8.28)], as well as [Reference Capoferri and Vassiliev9, 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

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

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:

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

• • Free boundary conditions:

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

• • Dirichlet-free (DF) boundary conditions:

  (1.9)\begin{equation} \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\,. \end{equation}

• • Free-Dirichlet (FD) boundary conditions:

  (1.10)\begin{equation} \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\,. \end{equation}

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 [Reference Krupchyk and Tuomela21] 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 $\mathbf {u}\in H^1(\Omega )$, $\mathbf {u} \ne 0$, and $\Lambda \in \mathbb {R}$ such that

   (1.11)\begin{equation} \mathcal{L}\mathbf{u}=\Lambda \mathbf{u} \end{equation}
   subject to the boundary conditions (1.7). The problem (1.11), (1.7) has discrete spectrum, consisting of discrete eigenvalues
   \[ (0<)\Lambda_1^\mathrm{Dir}\le \Lambda_2^\mathrm{Dir}\le \dots \]
   enumerated with account of multiplicity and accumulating to $+\infty$.

2. 2. The free boundary problem (free). The free boundary eigenvalue problem consists in seeking $\mathbf {u}\in H^1(\Omega )$, $\mathbf {u} \ne 0$, 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

   \[ (0\le)\Lambda_1^\mathrm{free}\le \Lambda_2^\mathrm{free}\le \dots \]
   enumerated with account of multiplicity and accumulating to $+\infty$.

3. 3. The Dirichlet-free problem (DF). The Dirichlet-free eigenvalue problem consists in seeking $\mathbf {u}\in H^1(\Omega )$, $\mathbf {u} \ne 0$, 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

   \[ (0\le)\Lambda_1^{\mathrm{DF}}\le \Lambda_2^{\mathrm{DF}}\le \dots \]
   enumerated with account of multiplicity and accumulating to $+\infty$.

4. 4. The free-Dirichlet problem (FD). The free-Dirichlet eigenvalue problem consists in seeking $\mathbf {u}\in H^1(\Omega )$, $\mathbf {u} \ne 0$, 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

   \[ (0\le)\Lambda_1^{\mathrm{FD}}\le \Lambda_2^{\mathrm{FD}}\le \dots \]
   enumerated with account of multiplicity and accumulating to $+\infty$.

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 [Reference Levitin, Monk and Selgas23] 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

(1.12)\begin{equation} N_\aleph(\Lambda):=\# \left\{k\ | \ \Lambda_k^\aleph <\Lambda\right\}. \end{equation}

Clearly, the function (1.12) is monotonically non-decreasing in $\Lambda$ and vanishes identically for $\Lambda \le \Lambda _1^\aleph$.

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 [Reference Strutt and Rayleigh26] in 1877. What started as an investigation prompted by practical questions from physics soon attracted the interest of pure mathematicians, as people realized that the coefficients in these expansions contain geometric invariants (see, e.g., [Reference Levitin, Mangoubi and Polterovich22, Chapter 6]). We refer the reader to [Reference Arendt, Nittka, Peter and Steiner3, Reference Ivrii20, Reference Safarov and Vassiliev27] for historical overviews of the development of the subject.

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

Proposition 1.2 We have

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

where

(1.14)\begin{equation} 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) \end{equation}

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 [Reference Debye14]. A rigorous mathematical proof was provided shortly afterwards by H. Weyl [Reference Weyl30]. It is worth emphasizing that the coefficient $a$ is independent of the choice of boundary conditions.

Let $\mathcal {A}$ be an elliptic semibounded differential operator of even order $2\,m$ 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 [Reference Agranovich and Vishik2]. Then it is known [Reference Greiner16, 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

(1.15)\begin{align} \mathcal{Z}_\mathcal{B}(t)& :=\int_M \operatorname{tr} G(x,x,t) \,\operatorname{dVol}_M\sim \widetilde{c}_{d-1} \, t^{-{d}/{2m}}\nonumber\\ & \quad + \widetilde{c}_{d-2} \, t^{-{d}/{2m}+ {1}/{2m}}+\dots+\widetilde{c}_{d-k}t^{-{d}/{2m}+ ({n-1})/{2m}}+\dots \nonumber\\ & \quad\text{as}\quad t\to 0^+, \end{align}

where $\mathrm {tr}$ stands for matrix trace. Moreover, all coefficients in (1.15) are locally determined [Reference Greiner16, §2]. We refer the reader to [Reference Grubb18], [Reference Grubb17, §4.2] and references therein for further details and generalizations.

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

Definition 1.3 For $1\le n\le d$, we define the $n$-th Weyl coefficient for the elliptic boundary value problem $(\mathcal {A},\,\mathcal {B})$ to be the number

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

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

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

(1.17)\begin{align} N(\Lambda)& =C_{d}\Lambda^{{d}/{2m}}+C_{d-1}\Lambda^{{d-1}/{2m}}+ \dots +C_{d-j+1}\Lambda^{{d-j+1}/{2m}}\nonumber\\ & \quad + o(\Lambda^{{d-j+1}/{2m}}) \quad \text{as}\quad \Lambda \to+\infty \end{align}

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

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

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

\begin{align*} \mathcal{Z}_\aleph(t)& =\sum_{k=1}^\infty {\rm e}^{-\Lambda_k^\aleph t}=\Gamma\left(\frac{d}2+1\right)\,a \operatorname{Vol}_d(M)t^{-{d}/{2}}\\ & \quad + c_{d-2}^\aleph \,t^{-{d-1}/{2}}+ o(t^{-{d-1}/{2}})\,\quad\text{as}\quad t\to 0^+, \end{align*}

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 [Reference Vasil'ev29] 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

(1.19)\begin{equation} 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, \end{equation}

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

Theorem 1.7 is a special case of [Reference Vasil'ev28, 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 $\partial M$, $d\ge 2$. The second Weyl coefficient for the elliptic boundary value problem $(\mathcal {L},\, \mathcal {B}_\aleph )$, $\aleph \in \{\mathrm {DF},\, \mathrm {FD}\}$, is given by

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

where

(1.21)\begin{align} b_{\aleph} & := @ \frac{1}{2^{d+1}\pi^{{d-1}/2} \Gamma\left(\frac{d+1}{2}\right)} \left( \frac{d-3}{\mu^{{d-1}/2}} + \frac1{(\lambda+2\mu)^{{d-1}/{2}}} \right) \quad \text{with}\nonumber\\ & \quad @= \begin{cases} - & \text{for}\ \,\aleph=\mathrm{DF},\\ + & \text{for}\ \,\aleph=\mathrm{FD}. \end{cases} \end{align}

The above theorem, whose proof will be given in § 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 [Reference Capoferri, Friedlander, Levitin and Vassiliev7, Theorem 1.8]Footnote ² .

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 $\alpha :={\mu }/{\lambda +2\mu }$ in a nontrivial fashion, see [Reference Capoferri, Friedlander, Levitin and Vassiliev7, 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 § 2.1, and formula (2.3) in particular.

3. (iii) Note that

   (1.22)\begin{equation} b_{FD}<0< b_{DF} \end{equation}
   for $d=2$, whereas
   (1.23)\begin{equation} b_{DF}<0< b_{FD} \end{equation}
   for $d\ge 3$.

4. (iv) We should like to emphasize 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 [Reference Chervova, Downes and Vassiliev13, Section 11] for a historical overview.

5. (v) In this paper we prove theorem 1.8 by studying the eigenvalue counting function. There are, of course, alternative approaches to the problem available in the literature, for instance by means of heat kernel techniques — see, e.g., [Reference Branson, Gilkey, Ørsted and Pierzchalski5].

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.