Bounded and unbounded capillary surfaces derived from the catenoid

Abstract

We construct two kinds of capillary surfaces by using a perturbation method. Surfaces of first kind are embedded in a solid ball B of $\mathbb{R}^3$ with assigned mean curvature function and whose boundary curves lie on $\partial B.$ The contact angle along such curves is a non-constant function. Surfaces of second kind are unbounded and embedded in $\mathbb{R}^3 \setminus \tilde B,$ $\tilde B$ being a deformation of a solid ball in $\mathbb{R}^3.$ These surfaces have assigned mean curvature function and one boundary curve on $\partial \tilde B.$ Also in this case the contact angle along the boundary is a non-constant function.