Bounded and unbounded capillary surfaces derived from the catenoid

We construct two kinds of capillary surfaces by using a perturbation method. Surfaces of first kind are embedded in a solid ball B of ℝ3 with assigned mean curvature function and whose boundary curves lie on the boundary of B: The contact angle along such curves is a non-constant function. Surfaces of second kind are unbounded and embedded in ℝ3 \ B'; B' being a deformation of a solid ball in ℝ3: These surfaces have assigned mean curvature function and one boundary curve on the boundary of B' : Also in this case the contact angle along the boundary is a non-constant function.