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The question of whether the equations governing the motion of an inviscid, incompressible fluid admit solutions representing steady vortex rings has not been studied widely, despite the central place of such rings in the theory of vortex motion initiated by Helmholtz [1] By a steady vortex ring we mean a figure of revolution si that is expected to be homeomorphic to a solid torus in most cases, and is associated with a continuous, axi-symmetric, solenoidal vector field q (the fluid velocity) defined in a cylinder V or in the whole of R 3 , and having the following properties when we take axes fixed in the ring si. (a) Both si and q do not vary with time; (b) the vorticity

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Abstract. We consider a nonlinear, elliptic, free-boundary problem involving an initially unknown set A that represents, for example, the cross-section of a steady vortex ring or of a confined plasma in equilibrium. The solutions are characterized by a variational principle which allows us to describe their behaviour under a limiting process such that the diameter of A tends to zero, while the solutions degenerate to the solution of a related linear problem. This limiting solution is the sum of the Green function of the linear operator and of a smooth function satisfying the boundary conditions. Mathematically speaking, this limiting process, that we call "nonlinear desingularization", is a novel kind of bifurcation phenomenon since the nonlinear effect here involves smoothing the singularity of the associated linear problem. "Nonlinear desingularization" is an interesting, hitherto little studied phenomenon in nonlinear partial differential equations. By nonlinear desingularization we mean that a linear boundary-value problem, whose solution possesses one or more isolated singularities, is a degenerate form of a family of nonlinear problems whose solutions are smooth and, moreover, converge to the solution of the linear system upon degeneration. This phenomenon occurs, for example, in the study of vortex motion in ideal fluids; there a circular vortex filament is used to approximate a steady vortex ring of small cross-section. The Stokes stream function of the filament is the Green function of an elliptic operator, and hence has an isolated singularity, while the Stokes stream function for the vortex ring of small crosssection satisfies a non-linear partial differential equation, and is smooth. In this paper we study this situation from a global point of view that is independent of the elaborate singular-perturbation techniques often used for such problems.Nonlinear desingularization arises in problems of theoretical physics other than the classical problem mentioned above. For example, it occurs in the onset of flux penetration, as determined by the Ginzberg-Landau theory, in supercon- has discussed it in connection with new classes of static, Euclidean, SU(2) Yang-Mills fields (monopoles) that are not spherically synmaetric and have unit Pontrjagin index. We hope to consider these applications in a later paper.