Bifurcation domains from any high eigenvalue for an overdetermined elliptic problem

1. Problem Statement

The paper investigates an overdetermined eigenvalue problem on unbounded domains in $\mathbb{R}^{N+1}$ ($N \ge 1$). The problem is formulated as:
$$\begin{cases} -\Delta u = \lambda u & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \\ \partial_\nu u = \text{const} & \text{on } \partial\Omega, \end{cases}$$
where $\Omega$ is an unbounded domain, $\nu$ is the unit outward normal, and $\lambda$ is an eigenvalue.

Context and Motivation:

• Serrin's Theorem: For bounded domains, the existence of a solution to this problem implies the domain is a ball.
• Berenstein Conjecture: This conjecture posits that for unbounded domains, if a nontrivial solution exists, the domain must be a ball, half-space, generalized cylinder, or their complements.
• Previous Work: Sicbaldi [25] and Schlenk & Sicbaldi [23] constructed counterexamples to the Berenstein conjecture by perturbing the straight cylinder $B_1^N \times \mathbb{R}$ to create periodic unbounded domains admitting positive solutions corresponding to the first eigenvalue ($\lambda_1$).
• Gap: The authors address whether such counterexamples exist for higher eigenvalues ($\lambda_k$ with $k \ge 3$) and whether the resulting solutions can be sign-changing (nodal).

2. Methodology

The authors employ bifurcation theory (specifically the Crandall-Rabinowitz theorem) to construct nontrivial solutions by perturbing the straight cylinder.

A. Domain Perturbation

They consider a family of modified cylinders $\Omega_s$ defined by:
$$\Omega_s = \left\{ (x, t) \in \mathbb{R}^N \times \mathbb{R} : |x| < 1 + s \cos\left(\frac{2\pi}{T}t\right) + s w\left(\frac{2\pi}{T}t\right) \right\}$$
where $s$ is a small bifurcation parameter, $T$ is the period, and $w$ is a smooth periodic function.

B. Linearization and Dirichlet-to-Neumann Operator

1. Operator Formulation: The problem is transformed into an abstract operator equation $F(v, T) = 0$, where $v$ represents the boundary perturbation.
2. Linearization: They analyze the linearized operator $D_v F(0, T)$ at the trivial solution (the straight cylinder). This involves studying the spectral properties of the operator, which reduces to analyzing the eigenvalues $\sigma_m(T)$ of a specific Sturm-Liouville type problem involving Bessel functions.
3. Bessel Function Analysis: A significant portion of the technical work involves the detailed analysis of Bessel functions $J_\nu$ and modified Bessel functions $I_\nu$. The authors study the zeros and monotonicity of the function $\sigma_1(T)$ (the principal eigenvalue of the linearized operator) to locate bifurcation points.
   • They establish that $\sigma_1(T)$ has exactly $k$ zeros $T_{1,*}, \dots, T_{k,*}$ in specific intervals determined by the eigenvalues of the unit ball.
   • They prove the non-degeneracy of these zeros (i.e., $\sigma'(T_{i,*}) \neq 0$), which is crucial for applying the bifurcation theorem.

C. Handling High Eigenvalues and Kernel Dimensions

• Case $k=1, 2$: The kernel of the linearized operator is one-dimensional (spanned by $\cos(t)$).
• Case $k \ge 3$: The analysis becomes more complex. The kernel dimension can be higher than 1 if a bifurcation point $T_{i,*}$ is an integer multiple of a previous bifurcation point $T_{j,*}$ (resonance).
• Strategy: When the kernel is multidimensional, the authors restrict the function space to a subspace where the kernel becomes one-dimensional, allowing them to still apply the Crandall-Rabinowitz theorem.

D. Dimensional Cases

• $N \ge 2$: The analysis relies heavily on the properties of Bessel functions $J_\nu$ where $\nu = (N-2)/2$.
• $N = 1$: The problem reduces to trigonometric functions (cosine/sine). The authors provide explicit formulas for the bifurcation points and verify the results hold in this case as well.

3. Key Contributions and Results

Main Theorems

• Theorem 1.1 (First Bifurcation Branch): For any $k \ge 3$, there exists a smooth family of unbounded domains bifurcating from the cylinder at a specific period $T_{1,*}$. These domains admit a sign-changing solution $u$ that changes sign $k-1$ times radially.
• Theorem 1.2 (Higher Bifurcation Branches): There exist $k-1$ distinct bifurcation points $T_{2,*}, \dots, T_{k,*}$. At each point, a new family of domains bifurcates.
  • If $T_{i,*}$ is not an integer multiple of previous periods, the solution is a perturbation of the $k$-th eigenfunction.
  • If resonance occurs ($T_{i,*} = l T_{j,*}$), the bifurcation involves a combination of modes.

Specific Findings

1. Existence of Sign-Changing Solutions: The paper proves the existence of nontrivial unbounded domains supporting solutions that change sign $k-1$ times. This contrasts with previous results which focused on positive solutions ($k=1$).
2. Counterexamples to Berenstein Conjecture: These constructed domains serve as counterexamples to the Berenstein conjecture for unbounded domains, specifically for any high eigenvalue $\lambda_k$.
3. Multiplicity of Bifurcation: The authors identify $k-1$ distinct branches of bifurcation for each $k \ge 3$, whereas previous literature only established the first branch.
4. Analytic Ingredients: The proof introduces novel analytic techniques for handling the zeros of Bessel functions in the context of high eigenvalues, specifically proving convexity-type inequalities ($J_\nu^2(s) > J_{\nu-1}(s)J_{\nu+1}(s)$) over extended intervals.

4. Significance

• Generalization: This work significantly generalizes the results of Sicbaldi [25] and the authors' own previous work [9] (which covered $k=2$) to any integer $k \ge 3$.
• Complexity of High Eigenvalues: It demonstrates that the bifurcation structure for high eigenvalues is rich and complex, involving multiple branches and potential resonance phenomena (multidimensional kernels).
• Refutation of Conjectures: It provides a robust class of counterexamples to the Berenstein conjecture, showing that the geometry of domains admitting overdetermined solutions is far more diverse than previously thought, even for sign-changing solutions.
• Methodological Advancement: The paper refines the bifurcation technique for cases where the linearized operator has a high-dimensional kernel, offering a template for similar problems in spectral geometry and nonlinear PDEs.

In summary, the paper establishes that for any high eigenvalue $\lambda_k$ of the Laplacian on a unit ball, one can construct unbounded, periodic, non-cylindrical domains that admit sign-changing solutions to the overdetermined eigenvalue problem, thereby expanding the known landscape of counterexamples to symmetry conjectures in unbounded domains.