On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions

1. Problem Statement

The paper addresses a shape optimization problem concerning the first Laplacian eigenvalue on doubly connected domains in $\mathbb{R}^n$ ($n \geq 2$). Specifically, the authors consider a domain $\Omega = \Omega_{out} \setminus \Omega_{in}$, where $\Omega_{in}$ and $\Omega_{out}$ are open, bounded, and convex sets.

The study focuses on the eigenvalue problem with mixed Robin boundary conditions:
$$\begin{cases} -\Delta u = \lambda_{RR}(\Omega) u & \text{in } \Omega, \\ \frac{\partial u}{\partial \nu} + \beta_{in} u = 0 & \text{on } \partial \Omega_{in}, \\ \frac{\partial u}{\partial \nu} + \beta_{out} u = 0 & \text{on } \partial \Omega_{out}, \end{cases}$$
where $\beta_{in}, \beta_{out} \in (0, +\infty]$ (including limiting cases for Neumann and Dirichlet conditions).

The Core Question: Under suitable geometric constraints, does the spherical shell (an annulus formed by two concentric balls) maximize the first eigenvalue $\lambda_{RR}(\Omega)$ among all doubly connected domains with fixed volume and specific boundary geometric parameters?

2. Methodology

The authors develop a higher-dimensional extension of the "Effectless Cut" technique, originally introduced by Hersch for planar membranes. The methodology proceeds in three main stages:

A. The Effectless Cut Construction (Higher Dimensions)

The central technical innovation is proving the existence of an intermediate surface (a "cut") that decomposes the domain without decreasing the eigenvalue.

• Gradient Flow: They utilize the gradient flow lines of the positive eigenfunction $u$ associated with $\lambda_{RR}(\Omega)$.
• Axisymmetry: A crucial assumption is that the domain $\Omega$ is axisymmetric. This allows the classification of critical points of $u$ (isolated points on the symmetry axis or analytic varieties of codimension 1 or 2).
• Decomposition: By analyzing the flow lines, they construct an open set $G$ such that $\Omega_{in} \subset \subset G \subset \subset \Omega_{out}$.
• The "Effectless" Property: The boundary $\partial^* G$ (essential boundary) is constructed such that the normal derivative of the eigenfunction vanishes almost everywhere on it:
  $$\frac{\partial u}{\partial \nu} = 0 \quad \text{on } \partial^* G.$$
  This implies that the eigenvalue of the original domain $\Omega$ is equal to the eigenvalues of the two subdomains created by the cut, specifically:
  $$\lambda_{RR}(\Omega) = \lambda_{RN}(G \cap \Omega) = \lambda_{NR}(\Omega \setminus G).$$

B. Isoperimetric Inequalities for Mixed Eigenvalues

Once the domain is decomposed into two subdomains with Neumann conditions on the cut, the problem reduces to maximizing mixed eigenvalues ($\lambda_{RN}$ and $\lambda_{NR}$).

• The authors rely on known reverse Faber-Krahn type inequalities for mixed boundary conditions established in previous literature (e.g., Della Pietra and Piscitelli).
• These inequalities state that for a fixed volume and fixed "inner" geometric parameter (related to the inner boundary), the spherical shell maximizes the mixed eigenvalue.

C. Geometric Constraints (Class $\mathcal{S}$)

To apply the isoperimetric inequalities effectively, the authors define a specific class of domains $\mathcal{S}$. A domain belongs to $\mathcal{S}$ if its volume and boundary geometric parameters satisfy a specific relation involving Quermassintegrals (intrinsic volumes):
$$\frac{|\Omega|}{\omega_n} = \left( \frac{P(\Omega_{out})}{n\omega_n} \right)^{\frac{n}{n-1}} - \left( \frac{W_{n-1}(\Omega_{in})}{\omega_n} \right)^n$$
where $P$ is the perimeter and $W_{n-1}$ is the $(n-1)$-th Quermassintegral. This class ensures that the "optimal" spherical shell can be uniquely defined to match the volume and boundary constraints of the arbitrary domain.

3. Key Contributions

1. Higher-Dimensional Effectless Cut: The paper successfully generalizes Hersch's planar effectless cut argument to dimensions $n \geq 2$. This was previously an open problem due to the lack of direct generalization of planar topological tools.
2. Optimization of Robin Eigenvalues: It establishes that spherical shells are the maximizers for the first Robin-Robin eigenvalue in doubly connected domains, provided the domains are axisymmetric and satisfy the geometric constraints of class $\mathcal{S}$.
3. Rigidity Results: The paper proves that equality in the eigenvalue bound holds if and only if the domain is a spherical shell (up to translation).
4. Non-emptiness of Admissible Class: The authors demonstrate that the intersection of axisymmetric domains ($\mathcal{A}$) and the geometrically constrained class ($\mathcal{S}$) is non-empty and contains domains that are not simply concentric shells (e.g., non-concentric configurations).

4. Main Results

Theorem 1.4 (Main Result):
Let $\Omega \in \mathcal{S} \cap \mathcal{A}$ and $(\beta_{in}, \beta_{out}) \in (0, +\infty]^2$. Let $A_{R_1, R_2}$ be the spherical shell such that:

• $|A_{R_1, R_2}| = |\Omega|$ (Same volume),
• $P(B_{R_2}) = P(\Omega_{out})$ (Same outer perimeter),
• $W_{n-1}(B_{R_1}) = W_{n-1}(\Omega_{in})$ (Same inner Quermassintegral).

Then:
$$\lambda_{RR}(\Omega) \leq \lambda_{RR}(A_{R_1, R_2})$$
Equality holds if and only if $\Omega$ is a spherical shell (up to translation).

Theorem 1.2 (Existence of the Cut):
For any axisymmetric doubly connected domain and positive eigenfunction $u$, there exists an intermediate set $G$ such that the normal derivative of $u$ vanishes on the essential boundary of $G$, effectively splitting the domain into two parts with Neumann conditions on the interface.

5. Significance and Applications

• Mathematical Physics: The results are directly applicable to the study of vibrating membranes and heat conduction in multiply connected media with Robin boundary conditions (modeling heat transfer with convection).
• Shape Optimization: The paper provides a rigorous framework for solving isoperimetric problems in higher dimensions for mixed boundary conditions, a notoriously difficult area where classical symmetrization techniques often fail.
• Generalization of Classical Inequalities: It extends the classical Faber-Krahn (minimization for Dirichlet) and Szegő-Weinberger (maximization for Neumann) inequalities to the context of doubly connected domains with Robin conditions.
• Methodological Advancement: The "Effectless Cut" method offers a new tool for analyzing eigenvalues in complex geometries, potentially applicable to other spectral optimization problems where symmetry plays a role.

6. Limitations and Future Directions

• Axisymmetry: The proof relies heavily on the assumption that the domain is axisymmetric to classify the critical points of the eigenfunction. The authors note that removing this assumption remains an open challenge.
• Convexity: The convexity of the inner and outer boundaries is required to ensure the validity of the isoperimetric inequalities used in the final step.
• Simply Connected Counterexamples: The paper notes that if the convexity assumption is dropped (e.g., in 2D), the result may fail due to homogenization effects where eigenvalues can be made arbitrarily large by adding small holes.

In summary, this paper represents a significant advancement in spectral geometry by bridging the gap between planar techniques and higher-dimensional spectral optimization, proving that spherical shells are the optimal configurations for maximizing the first Robin eigenvalue in a specific, well-defined class of doubly connected domains.