Global rigidity of two-dimensional bubbles

Technical Summary: Global Rigidity of Two-Dimensional Bubbles

Problem Statement
This paper investigates stationary hollow vortices with surface tension in two dimensions. The physical model describes a vacuum phase (a bubble) $B_{in}$ submerged in an incompressible, irrotational fluid phase $B_{out} = \mathbb{R}^2 \setminus B_{in}$. The interface $S = \partial B_{in}$ is a free boundary governed by the Euler equations with surface tension. The problem is formulated as an overdetermined elliptic free boundary value problem for the stream function $\psi$ in the exterior domain.

The governing equations require $\psi$ to be harmonic in $B_{out}$, constant on the boundary $S$, and to exhibit logarithmic asymptotics at infinity corresponding to a fixed circulation. The boundary condition on $S$ is the Young-Laplace law, relating the pressure jump (via Bernoulli's law) to the mean curvature $H$ and the surface tension coefficient. This leads to the dimensionless formulation (1.2), parameterized by the Weber number $We = \alpha^2 / (\sigma R)$, where $\alpha$ is the circulation and $R$ is a characteristic length scale.

The central question addressed is the global rigidity of the unit circle as a solution to this problem. Specifically, the paper seeks to determine the range of the Weber number $We$ for which the unit disk is the unique solution (up to translation). This supports a conjecture by Crowdy and Wegmann [31] that circular solutions are unique for most values of circulation, except for a discrete sequence of bifurcation points.

Methodology
The author employs a combination of geometric analysis, elliptic PDE identities, complex analysis, and variational calculus.

1. Geometric Identities and Pohozaev Identity: For the direct rigidity proof (Theorem 1.1), the author combines the Pohozaev identity for the harmonic stream function with geometric identities involving the curvature and the normal vector. By integrating the jump condition against specific test functions (such as $x \cdot n$) and utilizing the divergence theorem, the author derives integral relations linking the perimeter, the Weber number, and the circulation.
2. Linear Stability Analysis: To understand the local behavior near the circular solution, a linearization of the free boundary condition is performed using Fourier modes. This identifies the eigenvalues of the linearized operator, revealing bifurcation points where non-circular solutions may emerge.
3. Variational Approach: The paper establishes a connection between the free boundary problem and a variational problem: minimizing the functional $F_{We}(E) = We \pi I(E) + P(E)$, where $I(E)$ is the logarithmic potential energy and $P(E)$ is the perimeter. The critical points of this functional correspond to solutions of the free boundary problem.
4. Complex Analysis and Capacity: For the sharp rigidity result (Theorem 1.3), the author utilizes conformal mapping techniques. The exterior domain is mapped to the exterior of the unit disk via a conformal map $\Phi$. The problem is reformulated in terms of the derivative of this map, $\Phi'$. The proof relies on Grönwall's area theorem, Hardy space norms, and Bergman space inequalities to bound the perimeter in terms of the logarithmic capacity.

Key Contributions and Results

• Sharp Global Rigidity for $We \le 2$ (Theorem 1.1): The paper proves that if a solution to the free boundary problem exists with $0 \le We \le 2$, the interface $S$ must be a circle. The proof relies on the isoperimetric inequality and the Cauchy-Schwarz inequality applied to derived integral identities.
• Variational Classification (Theorem 1.2): In the class of convex sets with fixed area $\pi$, the unit disk is the unique minimizer of the functional $F_{We}$ if and only if $0 \le We \le 3$. This result improves upon previous work [5] which established a similar result only for an implicit, small constant $We_0$. The proof introduces a new sharp inequality relating the perimeter and the logarithmic capacity of sets bounded by a Jordan curve (Lemma 4.4).
• Sharp Global Rigidity for $We \le 3$ (Theorem 1.3): The main result of the paper is the extension of the rigidity threshold to $We \le 3$. The author proves that if $S$ is not a circle, then $We > 3$. This bound is sharp because non-circular solutions (bifurcation branches) are known to exist for $We > 3$ [31]. The proof involves a detailed analysis of the conformal map derivative, showing that for non-circular domains, the derived inequality forces $We > 3$.
• Linear Analysis and Bifurcation: The paper confirms that the bifurcation points for non-circular solutions occur exactly at integer values $We = m+1$ for $m \ge 2$. The first bifurcation occurs at $We = 3$. The analysis shows that the non-circular branches emanating from $We=3$ immediately lie in the regime $We > 3$, consistent with the global rigidity result at $We=3$.

Significance and Claims
The paper claims to resolve the global rigidity question for two-dimensional hollow vortices with surface tension by establishing the sharp threshold $We = 3$.

• It confirms the conjecture of Crowdy and Wegmann [31] that the bubble is circular for most values of circulation, specifically for all $We \le 3$.
• It provides a new, sharp isoperimetric-isocapacitary inequality (Lemma 4.4) that relates the perimeter deficit to the logarithmic capacity deficit, which is of independent interest in geometric function theory.
• The work distinguishes between the variational problem (minimizing energy) and the free boundary problem. While the variational problem is well-posed under convexity constraints, the paper notes that it is not immediately clear if all minimizers of the variational problem satisfy the pointwise free boundary equations, though every strictly convex minimizer does yield a solution.
• The author emphasizes that the proof of Theorem 1.1 (for $We \le 2$) uses only real-variable methods (geometric identities and PDE estimates) and does not rely on complex analysis, suggesting potential applicability to higher-dimensional problems, whereas the sharp result for $We \le 3$ relies heavily on complex analysis techniques specific to two dimensions.

The paper concludes by noting that for $We > 3$, non-circular solutions exist, and numerical experiments suggest these solutions are elongated shapes resembling a stadium (the Minkowski sum of a line segment and a disk). The exact characterization of these non-critical solutions remains an open problem.