Bifurcations in Isoperimetric Problems with Nonlocal Interactions

1. Problem Statement and Context

The paper investigates the existence of non-spherical critical points for an energy functional modeling the liquid drop model with nonlocal interactions, subject to a fixed volume constraint.

• The Energy Functional: For a smooth domain $\Omega \subset \mathbb{R}^N$ ($N \ge 2$), the energy is defined as:
  $$E(\Omega) = \text{Per}(\Omega) + D(\Omega)$$
  where $\text{Per}(\Omega)$ is the perimeter (surface area) and $D(\Omega)$ represents a nonlocal interaction term.
• Interaction Terms:
  • Coulomb-type: $D(\Omega) = \frac{1}{2} \iint_{\Omega \times \Omega} \frac{1}{|x-y|^\lambda} dx dy$ for $0 < \lambda < N-1$.
  • Logarithmic (2D): $D(\Omega) = \frac{1}{2\pi} \iint_{\Omega \times \Omega} \log\left(\frac{1}{|x-y|}\right) dx dy$ (relevant for $N=2$).
• Physical Motivation:
  • In $N=3$ with $\lambda=1$, this models atomic nuclei where the perimeter term represents surface tension (holding the nucleus together) and the nonlocal term represents Coulomb repulsion between protons.
  • In $N=2$, the logarithmic interaction arises in biological pattern formation.
• The Challenge: While balls (spheres) are always critical points (minimizing perimeter while maximizing the interaction term), the competition between these two effects leads to a delicate variational problem. It is known that for large volumes, balls cease to be global minimizers. The paper focuses on local bifurcations: identifying specific radii where non-spherical solutions branch off from the family of balls.

2. Methodology

The authors employ a rigorous bifurcation theory approach, specifically utilizing the Crandall–Rabinowitz Theorem. The methodology proceeds as follows:

1. Geometric Parametrization: The domain $\Omega$ is assumed to be star-shaped with respect to the origin, represented as:
   $$\Omega_\phi = \{ x \in \mathbb{R}^N : |x| < \phi(\frac{x}{|x|}) \}$$
   where $\phi: S^{N-1} \to \mathbb{R}$ is a positive function. The ball of radius $R$ corresponds to $\phi \equiv R$.
2. Reduction to PDE on the Sphere: The condition for $\Omega_\phi$ to be a critical point of $E$ under a volume constraint is reduced to a semilinear elliptic equation on the sphere $S^{N-1}$:
   $$F(\phi) = \mu$$
   where $F$ involves the mean curvature of the boundary and a nonlocal integral term.
3. Operator Formulation: The problem is reformulated as finding zeros of an operator $\Phi(R, u) = 0$, where $\phi = R + u$ and $u$ is a small perturbation.
   • $\Phi(R, 0) = 0$ for all $R > 0$ (balls are always solutions).
   • The linearized operator $L_R = D_u \Phi(R, 0)$ is analyzed.
4. Spectral Analysis: The authors analyze the eigenvalues of $L_R$. Bifurcation occurs when the kernel of $L_R$ becomes non-trivial (i.e., an eigenvalue crosses zero). This happens at a discrete sequence of radii $\{R_n\}$.
5. Symmetry Reduction: To apply the Crandall–Rabinowitz theorem, the authors restrict the operator to specific subspaces of functions with particular symmetries (invariant under subgroups of the orthogonal group $O(N)$). This ensures the kernel is one-dimensional (simple eigenvalue crossing), a necessary condition for the theorem.
6. Lyapunov–Schmidt Reduction: To prove the non-existence of solutions near radii that are not in the bifurcation sequence, they use a reduction argument to show that any nearby solution must be trivial (a rigid motion of the ball).

3. Key Contributions and Results

The paper establishes the existence of an unbounded sequence of radii $\{R_n\}_{n \ge 2}$ at which non-spherical solutions bifurcate from the family of balls. The results are presented for three distinct cases:

A. The Liquid Drop Model ($N=3, \lambda=1$)

• Bifurcation Radii: $R_n = \left( \frac{3(2n+1)(n+2)}{8\pi} \right)^{1/3}$ for $n \ge 2$.
  • For $n=2$, $R_2$ corresponds to the volume $m=10$, recovering a known result by Frank [12].
• Symmetry of Solutions:
  • Even $n$: Solutions bifurcate with $O(2) \times O(1)$ symmetry (invariant under rotation in the $x_1x_2$-plane and reflection in $x_3$). The shape transitions from prolate to oblate. The bifurcation is transcritical (volume changes at the linear level).
  • Odd $n$: Solutions bifurcate with distinct symmetries depending on $m \in \{(n+1)/2, \dots, n\}$. The bifurcation is likely subcritical or supercritical (volume change occurs at second order).
• Multiplicity: For odd $n$, multiple branches of solutions emerge, revealing a richer structure than previously known.

B. The Planar Case ($N=2$, Logarithmic Interaction)

• Bifurcation Radii: $R_n = (2n(n+1))^{1/3}$ for $n \ge 2$.
• Symmetry: Solutions exhibit $D_n$ symmetry (invariant under rotation by $2\pi/n$ and reflection).
• Volume Behavior: Unlike the 3D even case, the volume of the bifurcating solutions remains equal to the ball's volume at the linear level for all $n$. The authors conjecture volume preservation holds at the second order as well.

C. General Dimension ($N \ge 2$, General $\lambda$)

• Bifurcation Radii: A general formula is derived for $R_n(\lambda, N)$ involving Gamma functions and the interaction exponent $\lambda$.
• Higher Dimensions ($N \ge 4$): The paper highlights that for $N \ge 4$, the bifurcation structure is significantly more complex.
  • Even $n$: Solutions exist with $O(k) \times O(N-k)$ symmetry for any $k=1, \dots, N-1$.
  • Odd $n$: Solutions exist with specific symmetries involving Gegenbauer polynomials.
• Non-Existence: For any radius $R$ not in the sequence $\{R_n\}$, there are no non-trivial solutions in a neighborhood of the ball (up to rigid motions).

4. Significance and Impact

1. Extension of Known Results: The work significantly extends the seminal result of Frank [12] (which identified the first bifurcation at $m=10$ in 3D) by proving that bifurcations occur at an infinite, unbounded sequence of radii.
2. Richness of Equilibria: It demonstrates that the liquid drop model possesses a vast landscape of non-spherical equilibria, not just the "pearl necklace" or toroidal shapes previously constructed numerically or in specific cases.
3. Symmetry Breaking: The paper provides a detailed classification of the symmetry breaking patterns (transcritical vs. sub/supercritical, different symmetry groups) depending on the dimension and the parity of the mode number $n$.
4. Mathematical Rigor: By combining spectral analysis of nonlocal operators with the Crandall–Rabinowitz theorem and careful symmetry arguments, the authors provide a rigorous existence proof for these complex shapes, addressing a long-standing gap in the analytical understanding of the liquid drop model.
5. Universality: The results apply to a broad class of nonlocal interactions (Coulomb and logarithmic) and dimensions, suggesting that bifurcation from spherical symmetry is a generic feature of such isoperimetric problems with competing local and nonlocal terms.

In summary, this paper resolves the question of whether non-spherical solutions exist beyond the first instability point, proving they do so infinitely often, and characterizing the geometry and symmetry of these new solutions across various dimensions and interaction types.