# American Institute of Mathematical Sciences

February  2018, 38(2): 589-614. doi: 10.3934/dcds.2018026

## Bounded and unbounded capillary surfaces derived from the catenoid

 1 KAIST, Department of Mathematical Sciences, 291 Daehak-ro, Yuseong-gu, Daejeon, South Korea 2 KIAS, School of Mathematics, 87 Hoegi-ro, Dongdaemun-gu, Seoul, South Korea

Received  April 2017 Revised  August 2017 Published  February 2018

Fund Project: This research was supported by Basic Science Research Program through the National Research Foundation of South Korea (NRF) funded by the Ministry of Education, Grant NRF- 2016R1A1A005299.

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.

Citation: Filippo Morabito. Bounded and unbounded capillary surfaces derived from the catenoid. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 589-614. doi: 10.3934/dcds.2018026
##### References:
 [1] C. Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 157-175. [2] G. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commun. Partial Diff. Equations, 13 (1988), 33-59.  doi: 10.1080/03605308808820537. [3] F. Morabito, Higher genus capillary surfaces in the unit ball of $\mathbb{R}^3$ Boundary Value Problems 2014 (2014), 23pp. doi: 10.1186/1687-2770-2014-130. [4] F. Morabito, Singly periodic minimal surfaces in a solid cylinder of $\mathbb{R}^3$, Discrete Continuous Dynamical Systems, 35 (2015), 4987-5001.  doi: 10.3934/dcds.2015.35.4987. [5] F. Morabito, Free boundaries surfaces and Saddle Tower minimal surfaces in ${\mathbb S}^2 × \mathbb{R}$, Journal of Mathematical Analysis and Applications, 443 (2016), 478-525.  doi: 10.1016/j.jmaa.2016.05.006. [6] L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 61 (1976), 19-34.  doi: 10.1007/BF00251860. [7] J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 28 (1975), 189-200.  doi: 10.1002/cpa.3160280202. [8] N. Uraltseva, Solvability of the capillary problem, Vestnik Leningrad. Univ., 19 (1973), 54-64,152.

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##### References:
 [1] C. Gerhardt, Global regularity of the solutions to the capillarity problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 157-175. [2] G. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commun. Partial Diff. Equations, 13 (1988), 33-59.  doi: 10.1080/03605308808820537. [3] F. Morabito, Higher genus capillary surfaces in the unit ball of $\mathbb{R}^3$ Boundary Value Problems 2014 (2014), 23pp. doi: 10.1186/1687-2770-2014-130. [4] F. Morabito, Singly periodic minimal surfaces in a solid cylinder of $\mathbb{R}^3$, Discrete Continuous Dynamical Systems, 35 (2015), 4987-5001.  doi: 10.3934/dcds.2015.35.4987. [5] F. Morabito, Free boundaries surfaces and Saddle Tower minimal surfaces in ${\mathbb S}^2 × \mathbb{R}$, Journal of Mathematical Analysis and Applications, 443 (2016), 478-525.  doi: 10.1016/j.jmaa.2016.05.006. [6] L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 61 (1976), 19-34.  doi: 10.1007/BF00251860. [7] J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 28 (1975), 189-200.  doi: 10.1002/cpa.3160280202. [8] N. Uraltseva, Solvability of the capillary problem, Vestnik Leningrad. Univ., 19 (1973), 54-64,152.
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