Symmetry breaking bifurcations for an overdetermined boundary value problem on an exterior domain issued from electrodynamics

We consider an electrically charged fluid occupying a solid cylindrical region Ω of R^3. Outside the domain Ω there is an electric field with electric potential which solves the Laplace equation and diverges as the distance from the axis tends to infinity. At ∂Ω the potential is constant and there is a balance between the pressure difference inside and outside the fluid, capillary forces proportional to the mean curvature and electrostatic repulsion of charges. We are interested in showing the existence of domains different from the solid cylinder Ω and satisfying the conditions described above. This problem is equivalent to an overdetermined elliptic boundary value problem on an exterior domain. We show the bifurcation phenomenon occurs and produces the deformation of the solid cylinder into rippled cylinders.