Technical Summary: Simplicity of Eigenvalues for Elliptic Problems with Mixed Steklov-Robin Boundary Conditions

1. Problem Statement

The paper investigates the spectral properties of two specific classes of elliptic eigenvalue problems defined on a bounded open domain $\Omega \subset \mathbb{R}^n$ with a $C^{1,1}$ boundary $\partial\Omega$. The boundary is partitioned into two disjoint connected components, $S$ and $W$ (with non-empty interiors). The study focuses on proving that for a generic domain (i.e., for almost all domain shapes), all eigenvalues of these problems are simple (multiplicity one).

The two primary problems analyzed are:

1. Mixed Steklov-Robin Problem for the Laplacian:
   $$\begin{cases} -\Delta u = 0 & \text{in } \Omega \\ \partial_\nu u = \lambda u & \text{on } S \\ \partial_\nu u + u = 0 & \text{on } W \end{cases}$$
   This is a modification of the classical sloshing problem, incorporating a Robin condition on $W$ to overcome technical obstacles related to the zero-mass case.

2. Steklov-type Problem for the Operator $-\Delta + I$:
   $$\begin{cases} -\Delta u + u = 0 & \text{in } \Omega \\ \partial_\nu u = \lambda u & \text{on } S \\ \partial_\nu u = 0 & \text{on } W \end{cases}$$
   This problem features a mass term and Neumann conditions on $W$.

The authors also extend their analysis to problems with non-constant coefficients (perturbations of the potential $a(x)$) and anisotropic elliptic operators in divergence form ($Lu = \partial_i(a_{ij}(x)\partial_j u)$).

2. Methodology

The core methodology relies on domain perturbation techniques and the analysis of the transversality of associated operators. The approach follows the framework established by Micheletti (referenced as [11, 12, 9] in the text) for proving the genericity of simple eigenvalues.

2.1 Variational Formulation and Operator Definition

The problems are reformulated in a variational setting within the Sobolev space $H^1(\Omega)$.

• For Problem (1), a scalar product is defined as $a_\Omega(u, v) = \int_\Omega \nabla u \nabla v \, dx + \int_{\partial\Omega} uv \, d\sigma$.
• For Problem (2), the scalar product includes a volume term: $a_\Omega(u, v) = \int_\Omega \nabla u \nabla v \, dx + \int_\Omega uv \, dx$.

An operator $E_\Omega$ is constructed such that the eigenvalue problem is equivalent to finding eigenpairs $(\mu, e)$ for $E_\Omega$ where $\mu = 1/\lambda$. This transforms the problem into finding the spectrum of a compact, self-adjoint operator on a Hilbert space.

2.2 Domain Perturbation and Pullback

To analyze the dependence of eigenvalues on the domain shape, the authors consider perturbations $\psi \in C^2(\mathbb{R}^n, \mathbb{R}^n)$ with small norm, defining a perturbed domain $\Omega_\psi = (I + \psi)\Omega$.

• Functions on $\Omega_\psi$ are pulled back to the fixed domain $\Omega$ via the diffeomorphism $\gamma_\psi$.
• This allows the definition of a family of operators $T_\psi$ acting on the fixed space $H^1(\Omega)$.
• The differentiability of these operators with respect to the parameter $\psi$ at $\psi=0$ is established.

2.3 The "No-Splitting" Condition

The proof hinges on Theorem 6, an abstract result stating that if an eigenvalue $\bar{\lambda}$ of multiplicity $m > 1$ maintains its multiplicity under all small perturbations, then a specific "no-splitting" condition must hold. Specifically, for an orthonormal basis $\{e_1, \dots, e_m\}$ of the eigenspace, there must exist a scalar $\rho$ such that:
$$\langle T'(0)[\psi]e_j, e_i \rangle = \rho \delta_{ij}$$
for all perturbations $\psi$.

The authors demonstrate that this condition cannot hold for the specific problems studied. They derive explicit integral formulas (involving terms $L_1, I_1, I_2, G$) for the derivative of the operator and show that assuming the condition holds leads to a contradiction via the Unique Continuation Principle.

2.4 Iterative Splitting Strategy

The proof proceeds in two stages:

1. Single Eigenvalue Splitting: It is shown that for any multiple eigenvalue, there exists a small perturbation (supported either in $S$ or $W$) that reduces its multiplicity.
2. Global Simplicity: By iterating this process countably many times with a converging sequence of perturbations (ensuring the total perturbation remains small), it is proven that one can construct a domain where all eigenvalues are simple.

3. Key Contributions and Results

3.1 Main Theorems on Domain Perturbations

• Theorem 1: For the mixed Steklov-Robin problem (1), for any $\epsilon > 0$, there exists a perturbation $\psi$ with $\|\psi\|_{C^2} < \epsilon$ such that all eigenvalues of the problem on $\Omega_\psi$ are simple. Crucially, $\psi$ can be chosen to leave either the component $S$ or the component $W$ unchanged.
• Theorem 2: For the Steklov-type problem (2), a similar result holds: all eigenvalues can be made simple via a small perturbation that leaves the component $W$ unchanged.
  • Note: The authors note a technical limitation for Problem (2): while they can prove simplicity by perturbing $S$, they encounter difficulties deriving a contradiction when attempting to perturb only $W$ while keeping $S$ fixed.

3.2 Results on Coefficient Perturbations

The paper extends the genericity of simple eigenvalues to problems with variable coefficients:

• Theorem 3: For problems (6) and (7) involving a potential $a(x)$, there exists a small perturbation $b(x)$ such that the modified coefficient $\tilde{a} = a + b$ yields a problem with simple eigenvalues.
• Theorem 4: For the anisotropic operator $Lu = \partial_i(a_{ij}\partial_j u)$, there exists a small perturbation $B(x)$ of the coefficient matrix $A(x)$ such that the resulting eigenvalues are simple.
• Corollary 5: A similar result is stated for the anisotropic operator with a mass term (proof omitted).

4. Significance and Claims

The paper claims to establish that the simplicity of eigenvalues is a generic property for these classes of mixed boundary value problems. This is significant because:

• Stability and Interpretability: Simple eigenvalues ensure that the qualitative and perturbative analysis of the PDEs is stable and physically consistent, as the system possesses a unique shape (eigenfunction) for each energy level.
• Generalization: The results generalize the classical Steklov problem and address the "sloshing" problem with mixed conditions, which are relevant in fluid dynamics and engineering.
• Methodological Robustness: The work demonstrates the applicability of Micheletti's abstract transversality approach to complex mixed boundary conditions and anisotropic operators, overcoming technical hurdles related to the specific boundary interactions (e.g., the interface between $S$ and $W$).

The authors explicitly state that the inclusion of the linear term in the Neumann condition for Problem (1) was a necessary modification to overcome technical obstacles present in the pure Neumann (zero-mass) case, noting that a dedicated study of the classical sloshing problem (3) is currently in preparation.