Spectral asymptotics for linear elasticity: the case of mixed boundary conditions

This paper establishes two-term spectral asymptotics for the linear elasticity operator with mixed boundary conditions on smooth compact Riemannian manifolds of arbitrary dimension, validating the general formulae through explicit analytical and numerical examples in two and three dimensions.

The Gist

The Big Picture: Listening to the Shape of a Drum

Imagine you have a drum. If you hit it, it vibrates and produces a sound. That sound isn't just one note; it's a complex chord made up of many specific frequencies (pitches). In physics and math, these frequencies are called eigenvalues.

If you could count how many distinct notes the drum can play below a certain pitch (say, below the highest note a human ear can hear), you would get a number called the eigenvalue counting function.

For a long time, mathematicians have known a "rule of thumb" (called Weyl's Law) for predicting this count. It basically says: "The number of notes depends mostly on how big the drum is." If you double the size of the drum, you get roughly eight times as many notes (in 3D).

But this paper asks a deeper question: What if we want to be more precise? What if we want to know the second most important factor?

The authors, Matteo Capoferri and Isabel Mann, discovered that the shape of the edge (the boundary) of the drum matters just as much as the size of the drum itself, but only when the edge is treated in a very specific, "mixed" way.

The Drum with a "Mixed" Edge

Usually, a drum edge is either:

1. Clamped (Dirichlet): The edge is glued down tight. It can't move at all.
2. Free (Neumann): The edge is loose. It can wiggle freely in the air.

This paper looks at a Mixed Boundary Condition. Imagine a drum where:

• The edge is glued down so it can't slide sideways (tangential movement is blocked).
• But, the edge is allowed to bounce up and down (normal movement is free).

Or the reverse:

• The edge can slide sideways freely.
• But it is glued tight so it can't bounce up and down.

The authors wanted to find a formula to predict the "second term" of the note count for these specific, mixed-edge drums.

The "Elastic" Drum

This isn't just a simple drum skin; it's a solid block of rubber or steel (an elastic body). When you shake a solid block, it doesn't just vibrate like a drum skin. It sends out two types of waves:

1. Compression waves (P-waves): Like a sound wave in air, squishing and stretching the material. These travel fast.
2. Shear waves (S-waves): Like shaking a rope side-to-side. These travel slower.

The math gets messy because these two types of waves usually get tangled up at the edges. If you clamp the edge, both waves get stuck. If you leave it free, both waves bounce off.

The Big Discovery:
The authors found that for these mixed conditions, the math surprisingly simplifies. Even though the material is complex, the "mixed" edge acts like a filter that separates the waves cleanly.

• One type of wave sees the edge as "clamped."
• The other type sees the edge as "free."

Because they separate so cleanly, the authors could derive a very simple, elegant formula for the second term of the prediction. It's like finding out that a complicated machine actually runs on a single, simple gear.

The Analogy: The Traffic Jam at the Border

Imagine the vibrations (waves) are cars trying to leave a city (the drum).

• The City Size: Determines how many cars are inside to begin with (the first term of the formula).
• The Border: Determines how fast they can leave.

In a normal "clamped" city, all cars are stuck at the border. In a "free" city, they all zoom out.
In this mixed city, the border has two lanes:

• Lane A: Cars must stop completely (clamped).
• Lane B: Cars can drive through freely (free).

The authors calculated exactly how much the "traffic jam" at the border slows down the total count of cars leaving the city. They found that the slowdown depends on the length of the border and the speed of the cars (which depends on the material's stiffness).

Why This Matters

1. It's a "Second Term" Breakthrough: The first term (size) was known for a century. The second term (edge shape) is much harder to calculate, especially for complex materials like rubber or steel.
2. It's Surprisingly Simple: Usually, mixing boundary conditions creates a mathematical nightmare. The authors found that for these specific mixed conditions, the answer is clean and doesn't require messy integrals or complicated trigonometry.
3. Real-World Verification: They didn't just do the math on paper. They tested it on two shapes:
   • A Disk (2D).
   • A Cylinder (3D).
     They calculated the exact notes these shapes produce and compared them to their new formula. The formula matched perfectly, both on paper and in computer simulations.

The Takeaway

If you have a solid object (like a bridge, a building, or a musical instrument) and you want to know how it vibrates, you usually need to know its size. This paper tells us that if the edges of that object are "mixed" (partially glued, partially free), you must also know the length of the edge to get an accurate prediction.

The authors provided the "recipe" (the formula) to calculate this extra bit of information, proving that even in the complex world of elastic vibrations, there is a hidden simplicity waiting to be found.

Technical Summary

1. Problem Statement and Context

The paper addresses the spectral asymptotics of the operator of linear elasticity on a smooth, compact, connected Riemannian manifold $(M, g)$ of dimension $d \ge 2$ with a boundary $\partial M$.

• The Operator: The operator $L$ acts on vector fields $u$ and is defined by the Lamé parameters $\lambda$ and $\mu$ (satisfying strong convexity conditions $\mu > 0, d\lambda + 2\mu > 0$). In local coordinates, it is given by:
  $$(L u)_\alpha := -\mu (\nabla_\beta \nabla^\beta u_\alpha + \text{Ric}_{\alpha\beta} u^\beta) - (\lambda+\mu) \nabla_\alpha \nabla_\beta u^\beta$$
• The Goal: The authors aim to derive an explicit formula for the second term (the second Weyl coefficient) in the asymptotic expansion of the eigenvalue counting function $N_\aleph(\Lambda)$ as $\Lambda \to +\infty$.
• Boundary Conditions: While previous work (specifically reference [6]) covered "pure" Dirichlet ($u|_{\partial M}=0$) and free ($T u|_{\partial M}=0$) conditions, this paper focuses on mixed boundary conditions:
  1. DF (Dirichlet-Free): Tangential displacement is zero ($u^\top = 0$), and normal traction is zero ($T u \cdot n = 0$).
  2. FD (Free-Dirichlet): Tangential traction is zero ($T u \cdot \tau = 0$), and normal displacement is zero ($u \cdot n = 0$).

The eigenvalue counting function is defined as $N_\aleph(\Lambda) = \#\{k \mid \Lambda_k^\aleph < \Lambda\}$. The asymptotic expansion is expected to take the form:
$$N_\aleph(\Lambda) = a \text{Vol}_d(M) \Lambda^{d/2} + C_\aleph^{d-1} \Lambda^{(d-1)/2} + o(\Lambda^{(d-1)/2})$$
The first term (Weyl's law) is known and independent of boundary conditions. The paper solves for the second coefficient $C_\aleph^{d-1}$ (or equivalently the Weyl coefficient $c_{d-2}$).

2. Methodology

The proof relies on a constructive algorithm developed by Vassiliev for calculating the second Weyl coefficient for systems of PDEs. The methodology proceeds through the following steps:

A. Reduction to Local Boundary Analysis

Using the fact that the first two Weyl coefficients depend only on local geometry, the authors assume $M$ is a domain in $\mathbb{R}^d$ with the Euclidean metric. They introduce boundary coordinates $(x', z)$ where $z$ is the distance to the boundary.

B. Decomposition into Invariant Subspaces

A key technical innovation is the decomposition of the $d$-dimensional problem into simpler components based on wave polarization relative to the boundary normal.

• Invariant Subspaces: The authors define a subspace $P$ (spanned by the tangential momentum vector and the normal vector) and its orthogonal complement $P^\perp$.
• Decomposition: They prove that the operator and the mixed boundary conditions preserve these subspaces. This allows the spectral shift function to be split:
  $$\text{shift}_\aleph = \text{shift}_{P, \aleph} + \text{shift}_{\perp, \aleph}$$
  • $\text{shift}_{P, \aleph}$ corresponds to a 2-dimensional problem (involving longitudinal and transverse waves coupled in the plane of propagation).
  • $\text{shift}_{\perp, \aleph}$ corresponds to normally polarized waves (shear waves perpendicular to the propagation plane), which decouple into a simpler $(d-2)$-dimensional problem.

C. One-Dimensional Spectral Problem

For each subspace, the problem reduces to a 1D spectral problem on the half-line $z \in [0, \infty)$ with a parameter $\xi'$ (tangential momentum).

1. Continuous Spectrum: The authors identify thresholds (branch points) of the continuous spectrum determined by the Lamé parameters.
2. Scattering Matrix: They construct the scattering matrix $S(\Lambda)$ by imposing the mixed boundary conditions on the general solutions of the 1D ODEs.
3. Phase Shift: The spectral shift function is computed via the argument of the determinant of the scattering matrix and the number of discrete eigenvalues below the continuous spectrum.
4. Integration: The second Weyl coefficient is obtained by integrating the spectral shift function over the cotangent bundle of the boundary.

3. Key Contributions and Results

Main Theorem (Theorem 1.8)

The paper establishes the explicit formula for the second Weyl coefficient $c_{d-2}^\aleph$ for mixed boundary conditions:
$$c_{d-2}^\aleph = \frac{d-1}{2} b_\aleph \text{Vol}_{d-1}(\partial M)$$
where the coefficient $b_\aleph$ is given by:
$$b_\aleph = \frac{1}{2^{d+1} \pi^{\frac{d-1}{2}} \Gamma(\frac{d+1}{2})} \left( \frac{d-3}{\mu^{\frac{d-1}{2}}} + \frac{1}{(\lambda+2\mu)^{\frac{d-1}{2}}} \right) \times \begin{cases} -1 & \text{for } \aleph = \text{DF} \\ +1 & \text{for } \aleph = \text{FD} \end{cases}$$

Significant Observations:

• Simplicity: Unlike the "pure" Dirichlet and free cases (where the coefficients involve complex integrals of inverse trigonometric functions depending on the ratio $\mu/(\lambda+2\mu)$), the mixed boundary condition coefficients are algebraically simple.
• Reason for Simplicity: The mixed conditions (DF and FD) do not mix the components of the vector field when reduced to the 1D spectral problem, unlike the pure conditions which couple longitudinal and transverse modes non-trivially.
• Sign Reversal: The sign of the second term depends on the dimension $d$ and the type of mixed condition:
  • For $d=2$: $b_{\text{FD}} < 0 < b_{\text{DF}}$.
  • For $d \ge 3$: $b_{\text{DF}} < 0 < b_{\text{FD}}$.

Explicit Examples (Section 3)

To verify the general formula, the authors explicitly solve the spectral problem for:

1. The Disk ($d=2$): Using separation of variables and Bessel functions, they derive the secular equation and compute the counting function numerically, confirming the asymptotic behavior.
2. Flat Cylinders ($d=2$ and $d=3$):
   • For cylinders, the variables separate completely, allowing the authors to write down the full spectrum explicitly.
   • They derive exact formulas for the eigenvalue counting functions $N_{\text{DF}}(\Lambda)$ and $N_{\text{FD}}(\Lambda)$ involving floor functions and sums over lattice points.
   • By performing asymptotic analysis on these exact counting functions (using number-theoretic estimates for sums of squares), they analytically recover the second Weyl coefficient, perfectly matching the general formula derived in Section 2.

4. Significance

• Completeness of Theory: This work completes the picture of two-term spectral asymptotics for linear elasticity by covering the mixed boundary conditions, which were previously an open problem.
• Physical Relevance: Mixed boundary conditions (DF and FD) model realistic physical scenarios, such as bodies clamped tangentially but free to move normally, or vice versa (sliding boundaries).
• Methodological Advancement: The paper demonstrates the power of decomposing elastic systems into invariant subspaces to simplify the calculation of spectral invariants for systems of PDEs.
• Verification: The rigorous analytical verification using flat cylinders provides a strong validation of the abstract algorithm, bridging the gap between general spectral theory and explicit solvable models.

In summary, Capoferri and Mann provide a definitive formula for the second term in the Weyl expansion for linear elasticity with mixed boundary conditions, revealing a surprising algebraic simplicity compared to pure boundary conditions, and validating their results through explicit spectral analysis of cylindrical domains.