Abstract. In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the (SQG)α equations with α ∈ (0, 1). From the numerical experiments implemented for Euler equations in [12,36,39] it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle [27,41], however they do not give enough information about the pairs such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the (SQG)α equation when α ∈ (0, 1). The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.

In this paper we present a new characterization of Sobolev spaces on R n . Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of R n and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.

Abstract. We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is C ∞ provided the patch is close to the bifurcation circle in the Lipschitz norm. The rotating patch is also convex if it is close to the bifurcation circle in the C 2 norm. Our proof is based on Burbea's approach to V -states.

In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α ∈ R. The case α = 0 corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α ∈ (0, 1) the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an ellipse. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses. Therefore we show a surprising connection between vortices and dislocations.

In this paper we discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. A particular example are those of the form f (z)χ Ω (z), where Ω is a bounded domain with boundary of class C 1+ε and f a function in Lip(ε, Ω) satisfying f ∞ < 1. An important point is that there is no restriction whatsoever on the Lip(ε, Ω) norm of f besides the requirement on Beltrami coefficients that the supremum norm be less than 1. The crucial fact in the proof is the extra cancellation enjoyed by even homogeneous Calderón-Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far reaching consequences.An application to a Lipschitz regularity result for solutions of second order elliptic equations in divergence form in the plane is presented.