Shape Derivatives of Eigenvalues

Updated 30 June 2026

Shape derivatives of eigenvalues are first- and higher-order variations that describe how spectral values change in response to smooth domain perturbations.
They are derived using both volume and boundary integral formulas, including classical Hadamard-type representations, to ensure rigorous spectral analysis.
Numerical approximations, such as Galerkin FEM methods, validate convergence rates and support optimization in applications from Laplacian to Maxwell operators.

Shape derivatives of eigenvalues describe the first-order (and higher-order) variations of spectral quantities associated to partial differential operators with respect to smooth deformations of the underlying domain or geometric data. The shape derivative encodes how the spectral data—most notably the eigenvalues—of operators such as the Laplacian, biharmonic operator, or Maxwell’s system depend on infinitesimal domain perturbations, and it underpins the rigorous analysis and optimization of eigenvalue functionals in shape optimization. Both volume-type and boundary-type shape derivative formulas are available, depending on the level of regularity of the domain, the operator, and the considered perturbation.

1. Mathematical Framework for Shape Derivatives of Eigenvalues

Consider a smoothly bounded domain $\Omega \subset \mathbb{R}^d$ and a linear elliptic operator (e.g., $-\Delta$ ) with boundary conditions (e.g., Dirichlet, Neumann). For a family of diffeomorphisms $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ , $\Phi_0 = \mathrm{Id}$ , define the perturbed domain $\Omega_t = \Phi_t(\Omega)$ . The eigenvalue problem on $\Omega_t$ is typically of the form: $\begin{cases} -\Delta u_t = \lambda_t u_t & \text{in } \Omega_t, \ \mathcal{B}(u_t) = 0 & \text{on } \partial\Omega_t. \end{cases}$ The shape derivative of a simple eigenvalue $\lambda$ in the direction of a velocity field $V$ (i.e., $\partial_t \Phi_t|_{t=0} = V$ ) is defined as

$-\Delta$ 0

The shape derivative can be represented as either a volume or a boundary integral, whose explicit form depends on the operator, boundary conditions, and regularity.

The elementary symmetric functions of possibly multiple eigenvalues ("eigenvalue clusters") depend analytically on the domain under suitable separation of the spectrum, and real-analytic dependence extends naturally to these symmetric functions (Buoso, 2014, Buoso, 2016, Lamberti et al., 2020, Riva et al., 1 Apr 2025).

2. Hadamard-Type Formulas: Volume and Boundary Representations

Scalar Laplacian and Second-Order Systems

For the Dirichlet Laplacian on a $-\Delta$ 1 domain, with a normalized eigenfunction $-\Delta$ 2, the classical Hadamard formula for a simple eigenvalue reads: $-\Delta$ 3 For Neumann or Robin-type conditions, the integrand contains additional terms involving $-\Delta$ 4, $-\Delta$ 5, and, for Robin, boundary curvature and Robin parameter effects (Bandle et al., 2015).

Volume formulations, which are more general (requiring less regularity), express the derivative as: $-\Delta$ 6 valid in minimal regularity settings (Zhu et al., 2018).

Higher-Order and Systems

For elliptic systems and the biharmonic operator, Hadamard-type boundary integrals involve the relevant energy densities: $-\Delta$ 7 with $-\Delta$ 8 explicitly given in terms of second derivatives of $-\Delta$ 9 and, for Neumann, additional terms depending on operator parameters (Buoso, 2014, Buoso, 2016).

For Maxwell and Helmholtz operators, Hadamard formulas (under sufficient regularity) take the form (Lamberti et al., 2024, Lamberti et al., 18 Feb 2025, Lamberti et al., 2020): $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 0

3. Analytic Dependence, Symmetric Functions, and Multiple Eigenvalues

For a tuple of eigenvalues (possibly coming from a multiple or clustered eigenvalue), the elementary symmetric functions

$\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 1

depend real-analytically on domain perturbations, under eigenvalue separation. The Hadamard-type formula for the shape derivative is (Buoso, 2014, Buoso, 2016, Lamberti et al., 2020, Lamberti et al., 18 Feb 2025): $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 2 with $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 3 the relevant quadratic form in the eigenfunctions. For systems, similar formulas appear for the Neumann–Poincaré operator and Grushin-type Laplacians (Riva et al., 1 Apr 2025, Lamberti et al., 2020).

For clusters of eigenvalues (multiplicity $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 4), Rellich–Nagy theory shows these branches split into $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 5 analytic functions under generic perturbation, with first derivatives given by the eigenvalues of an $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 6 matrix of boundary integrals (derivatives with respect to $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 7) (Lamberti et al., 18 Feb 2025, Lamberti et al., 2020, Lamberti et al., 2020).

4. Minimal Regularity, Abstract Frameworks, and Extensions

Shape-differentiability extends in modern theory to arbitrary (possibly infinite-dimensional) parameter spaces and Lipschitz domains, formulated via abstract self-adjoint operator theory in Banach spaces (Lamberti et al., 18 Feb 2025, Lamberti et al., 2024). Under Lipschitz regularity:

Volume integral (algebraic) formulas for shape derivatives remain valid and derive from differentiability of parameter-dependent forms pulled back to a fixed reference domain.
Surface (boundary) integral representations become available only when the domain and eigenfunctions possess sufficient regularity for integration by parts and appropriate trace theorems.
The abstract Hellmann–Feynman theorem generalizes first-order spectral perturbation results to both simple and multiple eigenvalues, even for infinite-dimensional perturbations.

These techniques apply directly to de Rham complexes (Maxwell, Helmholtz), the Grushin Laplacian, and other hypoelliptic or degenerate-elliptic settings (Lamberti et al., 18 Feb 2025, Lamberti et al., 2020).

5. Numerical Approximation and Convergence of Shape Gradients

Galerkin FEM discretization of shape gradient formulas, both volume and boundary types, has been rigorously analyzed (Zhu et al., 2018):

Volume formulas are more robust: under mild regularity ( $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 8), the error in discrete shape gradients decreases as $\Phi_t: \mathbb{R}^d \to \mathbb{R}^d$ 9, reaching $\Phi_0 = \mathrm{Id}$ 0 when $\Phi_0 = \mathrm{Id}$ 1 is $\Phi_0 = \mathrm{Id}$ 2-regular.
Boundary formulas converge as $\Phi_0 = \mathrm{Id}$ 3 (up to logarithmic factors), and do not achieve the convergence rates of volume formulas, except in exceptional geometries (such as standard domains for Neumann problems).
For multiple eigenvalues, convergence extends to the eigenvalues of the shape-Hessian matrix, which encode the sensitivity in multiple directions.
Numerical experiments validate theory: for Dirichlet problems, boundary-type shape gradients display strictly slower convergence than volume-type; for Neumann, some domains exhibit unexpectedly fast convergence for boundary-type from cancellation phenomena.

6. Applications, Critical Shapes, and Overdetermined Boundary Conditions

The shape derivative plays a pivotal role in optimization and characterization of critical shapes:

Stationary domains under volume or perimeter constraints satisfy overdetermined boundary conditions: the sum over a basis of eigenfunction fluxes (e.g., $\Phi_0 = \mathrm{Id}$ 4) is constant (or proportional to mean curvature) on $\Phi_0 = \mathrm{Id}$ 5 (Lamberti et al., 2020, Buoso, 2016, Buoso, 2014, Lamberti et al., 2020).
For operators exhibiting rotation-invariance, the ball is always a critical domain for symmetric functions of eigenvalues.
For the Neumann-Poincaré operator, spheres satisfy criticality for all symmetric eigenvalue sums, and the analytic formula for shape derivatives recovers and generalizes earlier normal-perturbation approaches (Riva et al., 1 Apr 2025).

7. Geometric and Operator-Theoretic Generalizations

Extensions of shape derivatives include:

Shape derivatives of intrinsic geometric invariants, e.g., Gauss curvature, principal curvatures (eigenvalues of the Weingarten map): first and second order shape derivatives are accessible via matrix perturbation theory (Chicco-Ruiz et al., 2017).
Operators beyond canonical elliptic cases: Zaremba problems, Grushin operators, and boundary integral operators such as Neumann–Poincaré, where the only the normal component of the velocity affects the derivative (Anoop et al., 2020, Riva et al., 1 Apr 2025).
For cavity electromagnetic eigenvalues (Maxwell), mapping–and–adjoint approaches facilitate efficient evaluation of shape gradients and are compatible with mixed finite element schemes and isogeometric analysis (Herter et al., 2024, Dölz et al., 2024).

Summary Table: Shape Derivative Formulas for Simple Eigenvalues

Operator	Volume Form	Boundary (Hadamard) Form
Laplace, Dirichlet	$\Phi_0 = \mathrm{Id}$ 6	$\Phi_0 = \mathrm{Id}$ 7
Laplace, Neumann	(same as above)	$\Phi_0 = \mathrm{Id}$ 8
Maxwell	$\Phi_0 = \mathrm{Id}$ 9	$\Omega_t = \Phi_t(\Omega)$ 0
Biharmonic, Dirichlet	—	$\Omega_t = \Phi_t(\Omega)$ 1
Robin	—	$\Omega_t = \Phi_t(\Omega)$ 2

All formulas require appropriate regularity assumptions on $\Omega_t = \Phi_t(\Omega)$ 3, $\Omega_t = \Phi_t(\Omega)$ 4, and $\Omega_t = \Phi_t(\Omega)$ 5. The volume form is always valid; boundary forms require higher regularity.

The rigorous theory of shape derivatives for eigenvalues thus synthesizes functional analytic spectral theory, PDE boundary value problems, and differential geometry, providing the analytic and computational foundation for a wide spectrum of applications in spectral optimization, geometric analysis, and engineering (Zhu et al., 2018, Lamberti et al., 2024, Lamberti et al., 18 Feb 2025, 2601.11890, Buoso, 2014, Buoso, 2016, Lamberti et al., 2020, Riva et al., 1 Apr 2025, Chicco-Ruiz et al., 2017).