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Energy-critical NLS with potentials of quadratic growth
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 |
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.
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.
|
show all references
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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