Fundamental Tone of the Bilaplacian

• The fundamental tone is defined as the first nonzero eigenvalue of a fourth-order operator, determining plate vibration frequencies and stability in elasticity theory.
• It is computed via variational methods over function spaces with complex boundary conditions, including Dirichlet, Navier, and free plate models.
• The analysis leverages isoperimetric inequalities, symmetry considerations, and numerical methods, with significant applications in spectral geometry and physical plate design.

The fundamental tone of the Bilaplacian refers to the first nonzero eigenvalue (and its associated eigenfunction) in the spectrum of fourth-order elliptic operators such as $\Delta^{2}$ (or more generally, $\Delta\Delta u - \tau\Delta u$ for the free plate with tension). This value characterizes the lowest frequency of vibration for plates, the spectral gap for fourth-order PDEs, and is central in spectral geometry, elasticity theory, and PDE analysis. Unlike the second-order case, the bilaplacian introduces distinct analytic, geometric, and physical complexity: its eigenfunctions need not be of a fixed sign and its boundary conditions are higher-order and more intricate, leading to rich phenomena in spectral optimization and stability theory.

1. Mathematical Formulation and Boundary Conditions

The canonical bilaplacian eigenvalue problems are formulated as:

• Free Plate (Physical Plate Model):

$$\Delta\Delta u - \tau\Delta u = \omega u \quad \text{in } \Omega$$

with natural boundary conditions, arising from variational principles involving

$$Q[u] = \frac{\int_\Omega (|D^2u|^2 + \tau|\nabla u|^2)\,dx}{\int_\Omega u^2\,dx}$$

which produce: - $u_{nn} = \partial_n^2 u = 0$ on $\partial\Omega$ - $$Vu \equiv \tau u_n - P[(D^2u)n] - \partial_n(\Delta u) = 0$$ on $\partial\Omega$ (Chasman, 2010)

• Dirichlet Bilaplacian: $u = 0$ and $\partial_n u = 0$ on $\partial\Omega$; i.e., clamped plates.
• Navier Bilaplacian: $u = 0$ and $(1+\sigma) \partial_{n}^2 u + \sigma \Delta u = 0$ on $\partial\Omega$ for Poisson ratio $\sigma$.

The fundamental tone $\lambda_1$ (or $\omega_1$) is defined variationally as the minimum Rayleigh quotient over appropriate function spaces, such as $H^2_0(\Omega)$.

2. Isoperimetric Inequalities and Domain Optimization

Spectral geometry for the Bilaplacian builds on isoperimetric principles:

• Maximality of the Ball: For the free plate problem, among all domains of a given volume, the ball maximizes the fundamental tone (Chasman, 2010). The proof adapts Weinberger’s method, using trial functions formed from spherical harmonics and radial ultraspherical Bessel functions:

$u_1(r,\theta) = R(r) Y_1(\theta)$

where $R(r) = j_1(ar) + \gamma i_1(br)$ (with $j_1$, $i_1$ the ultraspherical Bessel and modified Bessel functions, $a^2 b^2 = \omega$, $b^2 - a^2 = \tau$, $\gamma$ chosen to satisfy the boundary conditions).

• Quantitative Stability: Deviations from the ball decrease the fundamental tone at least quadratically in the Fraenkel asymmetry $\mathcal{A}(\Omega)$

$$\lambda_2(\Omega) \leq \lambda_2(\Omega^*) [1-\eta_{N,\tau,|\Omega|}\mathcal{A}(\Omega)^2]$$

with explicit, sharp constants $\eta_{N,\tau,|\Omega|}$, and $\Omega^*$ a ball of matching volume (Buoso et al., 2016).

• Shape Derivative Analysis: For problems with Robin parameters and general boundary conditions, the analytic dependence of the fundamental tone on domain shape is given by Hadamard formulas for shape derivatives, involving surface integrals of quantities such as

$$G(v) = (1-\sigma)|D^2v|^2 + \sigma(\Delta v)^2 + \alpha|\nabla v|^2 + \text{boundary terms}$$

where $v$ runs over orthonormal eigenfunctions (Buoso et al., 2021).

3. Spectral Properties: Simplicity, Multiplicity, and Symmetry Breaking

Recent work has addressed the simplicity of the fundamental tone and symmetry properties, especially on radially symmetric domains:

• Dimension $N\ge3$: The fundamental tone is always simple (one-dimensional eigenspace) and realized by a radial eigenfunction (angular mode $\ell=0$).
• Dimension $N=2$: Simplicity holds for large inner radii; for thin annuli, the fundamental tone may be double (mode $\ell=1$). The critical threshold for simplicity is precisely characterized (Buoso et al., 6 Oct 2025).

The separation of variables yields eigenfunctions of the form $u(x) = f(r) S_\ell(\theta)$, with angular dependence determined by spherical harmonics and the radial part $f(r)$ solving an ODE involving Bessel functions. The Rayleigh quotient depends monotonically on $\ell$ for $N\ge3$, ensuring minimality at $\ell=0$.

Domain Type	Dimension	Fundamental Tone	Eigenfunction Branch
Ball/Annulus	$N\ge 3$	Simple	Radial ($\ell=0$)
Annulus (thin)	$N=2$	Double	Nonradial ($\ell=1$)
Annulus (thick)	$N=2$	Simple	Radial ($\ell=0$)

4. Geometric and Probabilistic Perspectives

• Bilaplacian on Manifolds: On compact Riemannian manifolds, the spectrum of the Bilaplacian coincides with the squares of the Laplace-Beltrami eigenvalues ($\Delta^2\varphi = \lambda^2\varphi$ for $\Delta\varphi = \lambda\varphi$) (Leandre, 2022). This relationship establishes the fundamental tone of the Bilaplacian as the square of the lowest nonzero Laplacian eigenvalue (modulo the kernel).
• Eells-Elworthy–Malliavin Construction: The probabilistic representation of the Bilaplacian semigroup uses horizontal lifts to the frame bundle $\mathrm{SO}(M)$, relating $(\Delta^2)$-generated diffusion to projected horizontal Laplacians. Thus, spectral and stochastic analysis of the Bilaplacian can utilize tools developed for second-order operators (heat kernel, exit times, spectral gap estimates) (Leandre, 2022).
• Stochastic PDEs: Monotonicity inequalities for bilaplacian operators govern existence, uniqueness, and energy dissipation in stochastic PDEs. Coercivity estimates like

$$-\langle\phi, \partial^4\phi\rangle_p + \|\partial^2\phi\|_p^2 \le C\|\phi\|_p^2$$

yield spectral gap bounds and underpin decay estimates for solutions, connecting the fundamental tone to probabilistic representations and statistical physics (Bhar et al., 2023).

5. Nonlinear and Nonsmooth Variants

• $p$-Bilaplacian and the 1-Bilaplacian: Asymptotics of nonlinear systems (e.g., Lane-Emden systems with large exponents) show that least-energy solutions converge to those of nonsmooth ("1-bilaplacian") eigenvalue problems:

$$\Delta \left( \frac{\Delta u}{|\Delta u|} \right) = \Lambda_{1,q}|u|^{q-2}u \;\;\; \text{in }\Omega,\quad u = \frac{\Delta u}{|\Delta u|} = 0 \text{ on } \partial \Omega$$

where the eigenvalue is minimized on the ball (Faber-Krahn type result):

$\Lambda_{1,q}(B) = \frac{1}{\|G_B(\cdot,0)\|_q}$

(with $G_B$ the Dirichlet Green function) (Abatangelo et al., 2023). The limit profile of least-energy solutions is explicitly determined in terms of the Green function, emphasizing the fundamental tone’s role even in nonsmooth regimes.

• Fractional Bilaplacian ($(-\Delta)^s$ for $s\in(1,2)$): Fractional powers interpolate the rigidity of the Laplacian ($s=1$) and flexiblity of the Bilaplacian ($s=2$). Loss of maximum principles occurs for $s>1$, so that ground states or fundamental tones may exhibit sign changes—unlike the "positivity preserving" property at $s=1$ (Saldaña, 2018).

6. Physical Interpretation and Applications

• Plate Theory: The fundamental tone models the lowest frequency of vibration for elastic plates under various boundary constraints (clamped, free, elastically supported—Robin, Navier, Steklov-type). Knowing its value and simplicity or multiplicity is critical for predicting mode shapes, uniqueness of vibration, and structural stability.
• Shape Optimization: The ball, by virtue of maximizing (or minimizing, depending on boundary condition) the fundamental tone, is optimal in many contexts. This underpins engineering design principles—choosing optimal shapes for plates (membranes, shells) to avoid buckling or undesirable resonance modes (Chasman, 2010, Buoso et al., 2016).
• Numerical Analysis: Precise decay and boundary behavior of discrete bilaplacian Green’s functions, proven via compactness and discrete Sobolev-Poincaré methods, inform the convergence analysis of finite difference/finite element schemes for fourth-order PDEs (Müller et al., 2017).

7. Open Problems and Contemporary Directions

• Boundary Condition Complexity: The interaction of multiple boundary parameters (Robin, third-order coupling, bulk-boundary terms) leads to rich bifurcation phenomena and spectral transitions. Explicit characterization of critical domains remains open for certain parameter regimes (Buoso et al., 2021, Buoso et al., 2021).
• Multiplicity and Symmetry Breaking: For thin planar annuli, multiplicity of the fundamental tone suggests underlying symmetry breaking and degeneracy in vibrational modes. Extension to higher dimensions and more general geometries continues to be explored (Buoso et al., 6 Oct 2025).
• Extensions to Nonlinear and Fractional Problems: Analysis of the fundamental tone for nonlinear ($p$-Bilaplacian) and fractional ($s$-Bilaplacian) operators points toward deeper links between spectral theory, regularity, and geometric measure theory.

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The fundamental tone of the Bilaplacian thus encapsulates the intersection of spectral optimization, geometric analysis, plate and elasticity theory, probabilistic representation, and nonlinear PDE. Its paper yields both sharp analytic results (isoperimetric inequalities, shape sensitivity, spectral gap estimates) and practical physical insights (uniqueness and symmetry of vibrational modes, optimal design principles). The domain geometry, boundary condition regime, and analytic structure of the operator collectively determine the spectrum’s rigidity and flexibility, with ongoing advances addressing open problems and extending the theory to broader settings.