WETTING PHENOMENA AND CONSTANT MEAN CURVATURE SURFACES WITH BOUNDARY

2005 ◽  
Vol 17 (07) ◽  
pp. 769-792 ◽  
Author(s):  
RAFAEL LÓPEZ
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dirichlet Condition ◽  
Microgravity Environment ◽  
Special Focus ◽  
General Shape ◽  
Mean Curvature Equation ◽  
Wetting Phenomena ◽  
Cmc Surfaces ◽  
Flat Portion

In a microscopic scale or microgravity environment, interfaces in wetting phenomena are usually modeled by surfaces with constant mean curvature (CMC surfaces). Usually, the condition regarding the constancy of the contact angle along the line of separation between different phases is assumed. Although the classical capillary boundary condition is the angle made at the contact line, configurations also occur in which a Dirichlet condition is appropriate. In this article, we discuss those with vanishing boundary conditions, such as those that occur on a thin flat portion of a plate of general shape covered with water. In this paper, we review recent works on the existence of CMC surfaces with non-empty boundary, with a special focus on the Dirichlet problem for the constant mean curvature equation.

scholarly journals The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1211 ◽  
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Elliptic Equations ◽  
Constant Mean Curvature ◽  
Euclidean Case ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

scholarly journals Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
Alexander I. Bobenko ◽  
Sebastian Heller ◽  
Nick Schmitt
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Weierstrass Representation ◽  
Geometric Flow ◽  
Constant Mean Curvature Surfaces ◽  
Cmc Surfaces

AbstractWe describe the construction of CMC surfaces with symmetries in $\mathbb {S}^{3}$ S 3 and $\mathbb {R}^{3}$ ℝ 3 using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.

scholarly journals Gradient estimates for the constant mean curvature equation in hyperbolic space

2019 ◽  
Vol 150 (6) ◽  
pp. 3216-3230
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gradient Estimates ◽  
Maximum Principles ◽  
Convex Domains ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

AbstractWe establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.

Boundary tracing and boundary value problems: II. Applications

2007 ◽  
Vol 463 (2084) ◽  
pp. 1925-1938 ◽  
Author(s):  
Michael L Anderson ◽  
Andrew P Bassom ◽  
Neville Fowkes
Keyword(s):  
Partial Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Boundary Value ◽  
Practical Significance ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Boundary Tracing

This is the second of a pair of papers describing the use of boundary tracing for boundary value problems. In the preceding article, the theory of the technique was explained and it was shown how it enables one to use known exact solutions of partial differential equations to generate new solutions. Here, we illustrate the use of the technique by applying it to three equations of practical significance: Helmholtz's equation, Poisson's equation and the nonlinear constant mean curvature equation. A variety of new solutions are obtained and the potential of the technique for further application outlined.

scholarly journals Constant mean curvature surfaces with circular boundary in R³

2006 ◽  
Vol 78 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Pedro A. Hinojosa
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Spherical Cap ◽  
Constant Mean Curvature Surfaces ◽  
Circular Boundary ◽  
Cmc Surfaces ◽  
Global Estimates ◽  
Area And Volume

In this work we deal with surfaces immersed in R³ with constant mean curvature and circular boundary. We improve some global estimates for area and volume of such immersions obtained by other authors. We still establish the uniqueness of the spherical cap in some classes of cmc surfaces.

scholarly journals New Bernstein Type Results in Weighted Warped Products

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Ning Zhang
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Warped Products ◽  
Bernstein Type ◽  
Weighted Mean ◽  
Curvature Equation ◽  
Uniqueness Results ◽  
Mean Curvature Equation ◽  
Weighted Mean Curvature ◽  
Entire Graphs

In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.

scholarly journals Delaunay surfaces in S3(ρ)

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1191-1200 ◽  
Author(s):  
J. Arroyo ◽  
O.J. Garay ◽  
A. Pámpano
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Energy Functional ◽  
Real Space ◽  
Space Forms ◽  
New Approach ◽  
Cmc Surfaces ◽  
Real Space Forms ◽  
Rotational Surfaces

Recently, invariant constant mean curvature (CMC) surfaces in real space forms have been characterized locally by using extremal curves of a Blaschke type energy functional [5]. Here, we use this characterization to offer a new approach to some global results for CMC rotational surfaces in the 3-sphere.

Sign in / Sign up

Export Citation Format

Share Document