On the classification of Serrin planar domains

TL;DR

The paper develops a global, algebraic classification of Serrin-type planar domains by encoding domain geometry through capillary foliations and holomorphic developing maps, revealing that every Serrin ring or periodic Serrin band is an algebro-geometric mKdV potential. By organizing domains into a spectral-genus hierarchy, the authors obtain explicit moduli spaces: a triangular moduli space ${f T}_n$ for dihedral-symmetric rings and a 1-parameter family of bands transitioning from flat to a disk chain, all described by elliptic data. They establish a deep link between overdetermined elliptic problems and integrable systems, culminating in a finite-type description that parallels Pinkall-Sterling theory for CMC surfaces. The results unify several prior constructions (bifurcation from radial annuli, desingularization of disk chains) and provide explicit analytic descriptions via Weierstrass elliptic functions, with precise geometric limits (necklaces, radial annuli) and symmetry conclusions. This framework offers a roadmap for understanding the global moduli of Serrin domains and suggests avenues for applying capillary foliation techniques to related problems in fluid mechanics and surface theory.

Abstract

We show that all smooth ring domains $Ω\subset \mathbb{R}^2$ that admit a solution to Serrin's classical problem $Δu+2=0$ with locally constant overdetermined boundary conditions along $\partial Ω$ can be described as algebro-geometric potentials of the mKdV hierarchy. The same result holds for periodic unbounded domains with two boundary components. In particular, any such domain is determined by suitable holomorphic data in some algebraic curve. As a consequence, the space of all Serrin ring domains, or periodic Serrin bands, can be ordered into a sequence of finite-dimensional complexity levels. By studying the first non-trivial level, given by elliptic functions, we construct: $(i)$ a global $1$-parameter family of periodic solutions to Serrin's problem that interpolates between a flat band and a chain of disks along an axis, following an unduloid pattern, and $(ii)$ for any $n>1$, a two-dimensional moduli space ${\bf T}_n$ of non-radial Serrin ring domains with a dihedral symmetry group of order $2n$. This moduli space ${\bf T}_n$ is geometrically a triangle, and has radial bands on one side of ${\bf T}_n$, and a necklace of $n$ pairwise tangent disks distributed along the unit circle at its opposite vertex in ${\bf T}_n$.

Figures (8)

• Figure 1.1: Serrin ring domains for $n=2$, $n=3$ and $n=4$ at an intermediate point of their respective moduli spaces. In all figures parametrizations are conformal, and explicit in terms of elliptic functions. The unit circle is always a parameter curve contained in the domain.
• Figure 1.2: Three snapshots of Serrin ring domains for $n=3$. Left: $\tau$ close to zero. Middle: $\tau$ at an intermediate value. Right: $\tau$ close to $1$.
• Figure 1.3: Loss of embeddedness for $\partial \Omega$ at the boundary of the moduli space. Left: interior boundary curve. Right: exterior one.
• Figure 1.4: Top: A Serrin strip domain $\Omega_\tau$ for $\tau$ close to $1$. Middle: $\Omega_\tau$ for an intermediate $\tau\in (0,1)$. Bottom: $\Omega_\tau$ for $\tau>0$ close to zero. In all figures the parameter lines describe a conformal parametrization of the strip. These parametrizations are explicit in terms of a Weierstrass elliptic function $\wp(z)$.
• Figure 6.1: The sectors $S_k$ and the arcs $\gamma_k$ of the closed curve $\gamma$.

Theorems & Definitions (83)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3: Ros-Sicbaldi, RS
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.3
• Definition 2.4