Bounded and unbounded capillary surfaces derived from the catenoid
Filippo Morabito
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 589-614 doi: 10.3934/dcds.2018026

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.

keywords: Capillary surfaces Jacobi operator perturbation method contact angle fixed point theorem
Singly periodic free boundary minimal surfaces in a solid cylinder of $\mathbb{R}^3$
Filippo Morabito
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 4987-5001 doi: 10.3934/dcds.2015.35.4987
The aim of this work is to show the existence of free boundary minimal surfaces of Saddle Tower type which are embedded in a vertical solid cylinder in $\mathbb{R}^3$ and invariant with respect to a vertical translation. The number of boundary curves equals $2l$, $l \ge 2$. These surfaces come in families depending on one parameter and they converge to $2l$ vertical stripes having a common vertical intersection line. Such surfaces are obtained by perturbing the symmetrically modified Saddle Tower minimal surfaces.
keywords: fixed point theorem. perturbation method Singly periodic minimal surfaces contact angle free boundary

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