Hollow vortices and minimal surfaces

TL;DR

This paper establishes a correspondence between solutions to the hollow vortex problem and minimal graphs with horizontal symmetry, providing new examples of domains with hollow vortices.

Contribution

It introduces a novel link between the hollow vortex problem and minimal surface theory, enabling the construction of new vortex domain examples.

Findings

Solutions correspond to minimal graphs with symmetry

New examples of hollow vortex domains are constructed

Theoretical link between vortex problems and minimal surfaces

Abstract

We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.