Clamped Transmission Eigenvalue Problem

• Clamped Transmission Eigenvalue Problem is a spectral analysis of PDE systems characterized by higher-order operators and clamped boundary conditions in elastic plates and wave scattering scenarios.
• The methodology involves reducing the fourth-order problem into coupled second-order equations, employing variational formulations and operator pencils to analyze non-selfadjoint eigenvalues.
• The framework underpins inverse scattering and computational PDE methods, offering numerical techniques that reveal geometric properties and ensure discreteness of the eigenvalue spectrum.

The clamped transmission eigenvalue problem addresses the spectral analysis of PDE systems where functions are subject to clamped boundary conditions, typically in the context of elastic plates or wave scattering in thin elastic materials. Unlike classical transmission eigenvalue problems, which often couple two second-order operators in bounded domains, the clamped variant involves higher-order operators (notably biharmonic or more generally polyharmonic), non-selfadjoint operator pencils, and may be set in unbounded domains such as all of ℝ². The “clamped” feature enforces that both the function and its normal derivative vanish on the boundary of the scattering obstacle, reflecting physical scenarios like the Kirchhoff–Love plate with an impenetrable, clamped cavity. These problems exhibit unique analytical structure, require careful operator-theoretic treatment, and have strong connections to inverse scattering, spectral theory, and computational PDEs.

1. Problem Definition and Variational Formulation

The archetype for the clamped transmission eigenvalue problem arises in the scattering of time-harmonic biharmonic waves by a clamped obstacle within an infinite Kirchhoff–Love plate (Harris et al., 3 Feb 2025, Harris et al., 28 Aug 2025). The governing equation for the scattered field $u^{\text{scat}}$ is

$$\Delta^2 u^{\text{scat}} - k^4 u^{\text{scat}} = 0 \quad \text{in } \mathbb{R}^2 \setminus \overline{D},$$

subject to clamped boundary conditions on $\partial D$: $$u^{\text{scat}}|_{\partial D} = -u^{\text{inc}}, \quad \partial_{\nu} u^{\text{scat}}|_{\partial D} = -\partial_{\nu} u^{\text{inc}},$$ where $u^{\text{inc}}$ is the incident field.

By operator splitting, this fourth-order equation decouples into two second-order equations: $$\begin{cases} \Delta v + k^2 v = 0 & \text{in } D, \ \Delta w - k^2 w = 0 & \text{in } \mathbb{R}^2 \setminus \overline{D}, \end{cases}$$ with transmission conditions on $\partial D$: $$v + w = 0, \qquad \partial_\nu v + \partial_\nu w = 0,$$ and $w$ satisfying the Sommerfeld radiation condition at infinity. The pair $(k, (v, w))$ is sought such that these conditions are satisfied nontrivially.

The variational setting, adapted to computational and theoretical analysis, uses the energy space

$$X(D, \mathbb{R}^2) = \left\{ (v, w) \in H^1(D) \times H^1(\mathbb{R}^2 \setminus \overline{D}) \colon (v + w)|_{\partial D} = 0 \right\}.$$

For bounded computational domains $B_R \supset D$, an exterior Dirichlet-to-Neumann (DtN) map $T_{i k}$ encodes the modified Helmholtz solution $w$ at the artificial boundary $\partial B_R$, leading to a coupled variational problem.

2. Operator Reduction and Spectral Characterization

The abstract framework for transmission eigenvalues, including the clamped case, is established by reduction to an eigenvalue problem for a non-selfadjoint compact operator (Hitrik et al., 2010). For an elliptic operator $P_0(D)$ of order $m$ with constant coefficients and a sign-definite potential $V$, the problem is formulated as

$$\begin{cases} (P_0 - \lambda ) v = 0 & \text{in } \Omega, \ (P_0(1+V) - \lambda ) w = 0 & \text{in } \Omega, \ v - w \in H^m_0(\Omega). \end{cases}$$

This is reduced to a quadratic eigenvalue problem: $T_\lambda u = (A - \lambda B + \lambda^2 C) u = 0,$ where $A$, $B$, and $C$ are constructed from $P_0$ and $V$. A conjugation using $C^{1/2}$ leads to an operator pencil on a product space, with the transmission eigenvalues corresponding to inverses of the eigenvalues of a compact operator $D$. Sufficient conditions for the existence of transmission eigenvalues use Lidskii’s theorem; completeness of eigenstates is proven under sectorial conditions for the operator.

In settings with clamped boundary conditions, these reduction arguments persist, with the spectral theory of compact operators (Schatten classes, trace class) providing tools for existence, counting, and completeness of eigenstates.

3. Existence, Discreteness, and Spectral Estimates

Existence and discreteness of clamped transmission eigenvalues in infinite plates are established via variational reformulation and operator-theoretic arguments (Harris et al., 28 Aug 2025). The key steps involve:

• Reformulating the PDEs as variational problems with coupled domains,
• Defining bounded, self-adjoint, coercive operators (e.g., $A_k$) and compact perturbations ($B$),
• Employing continuity in $k$ and intermediate value arguments on subspaces spanned by Dirichlet eigenfunctions to prove at least $m$ eigenvalues exist in given intervals,
• Ensuring the set of transmission eigenvalues is infinite and discrete (with accumulation only at infinity).

Analytically, the operator $A_k$ is shown to be coercive at $k = 0$ and nonpositive at specific $k = \tau_2$ corresponding to the Dirichlet spectrum on small balls contained in $D$, thus connecting transmission eigenvalues to classical Laplace spectral data.

Estimated bounds and gap control for the classical clamped plate eigenvalues are obtained via variational and rearrangement techniques (Cheng et al., 2012, Ji et al., 2020). Lower bounds for transmission eigenvalues can, in principle, be adapted from these polynomial inequalities, provided the necessary adjustments for interface or transmission conditions are handled rigorously.

4. Relationships with Classical Spectra

A significant structural property of the clamped transmission eigenvalue problem is its relationship to Dirichlet and Neumann Laplacian eigenvalues for the scatterer (Harris et al., 28 Aug 2025). Specifically, for $k_1$ the smallest clamped transmission eigenvalue and $\lambda_1$ the first Dirichlet eigenvalue for $-\Delta$ in $D$, it is proved that

$k_1^2 \leq \lambda_1.$

Moreover, numerical experiments suggest an interlacing property: $\mu_j \leq k_j^2 \leq \lambda_j,$ where $\mu_j$, $\lambda_j$ are Neumann and Dirichlet eigenvalues, respectively.

This spectral interlacing has implications for inverse problems since transmission eigenvalues reflect both internal structure and boundary characteristics of the scatterer, and their distribution carries geometric information.

5. Numerical Methods and Computational Challenges

Rigorous computational approaches to clamped transmission eigenvalue problems rely on boundary integral formulations and contour-integral methods (Harris et al., 3 Feb 2025). By representing solutions in terms of single-layer potentials and discretizing the resulting boundary integral equations, the nonlinear matrix eigenvalue problem can be attacked via methods like the Beyn algorithm or recursive integral methods (Huang et al., 2015).

For problems set in the whole $\mathbb{R}^2$, compactness must be recovered by introducing artificial boundaries and nonlocal operators (exterior DtN maps). Challenges unique to clamped transmission eigenvalues include the interplay of radiative and exponentially decaying solutions, non-selfadjoint operator pencils, and the need for robust numerical localization in the face of closely clustered eigenvalues. Numerical experiments confirm theoretical predictions regarding existence, discreteness, and monotonicity properties.

6. Monotonicity and Inverse Problem Implications

Numerical evidence indicates a monotonic relationship between the first transmission eigenvalue and the measure of the scatterer (Harris et al., 3 Feb 2025). Specifically, shrinking the scatterer increases the first eigenvalue, analogous to the behavior of classical Dirichlet spectra. This monotonicity, if proven generally, would be significant in inverse spectral theory, as it suggests transmission eigenvalues can serve as quantitative indicators of geometric properties of the obstacle.

7. Generalizations and Further Directions

The framework and analysis for clamped transmission eigenvalues extend to higher-order and more complex systems, such as sixth-order problems arising in elastic thin films (Papanicolaou et al., 3 Aug 2025). Here, biorthogonal spectral methods are constructed by exploiting Petrov–Galerkin formulations with bases of eigenfunctions satisfying clamped boundary conditions and their adjoint counterparts. These spectral approaches are proven to converge rapidly and are adaptable to a wide class of non-selfadjoint transmission eigenvalue problems, including those with nonstandard boundary conditions.

Research continues to develop sharper bounds for eigenvalue gaps (Chen et al., 2016), deeper connections to inverse problems via far-field operators (Harris et al., 3 Feb 2025), and extension to problems with multiple interface or conductivity parameters (Ayala et al., 2022). The theoretical and computational frameworks established for the clamped transmission eigenvalue problem have broad relevance across spectral theory, mathematical physics, and computational PDE methods.