We consider relative equilibrium solutions of the two-dimensional Euler
equations in which the vorticity is concentrated on a union of finite-length
vortex sheets. Using methods of complex analysis, more specifically the theory
of the Riemann-Hilbert problem, a general approach is proposed to find such
equilibria which consists of two steps: first, one finds a geometric
configuration of vortex sheets ensuring that the corresponding circulation
density is real-valued and also vanishes at all sheet endpoints such that the
induced velocity field is well-defined; then, the circulation density is
determined by evaluating a certain integral formula. As an illustration of this
approach, we construct a family of rotating equilibria involving different
numbers of straight vortex sheets rotating about a common center of rotation
and with endpoints at the vertices of a regular polygon. This equilibrium
generalizes the well-known solution involving single rotating vortex sheet.
With the geometry of the configuration specified analytically, the
corresponding circulation densities are obtained in terms of a integral
expression which in some cases lends itself to an explicit evaluation. It is
argued that as the number of sheets in the equilibrium configuration increases
to infinity, the equilibrium converges in a certain distributional sense to a
hollow vortex bounded by a constant-intensity vortex sheet, which is also a
known equilibrium solution of the two-dimensional Euler equations.