Two-sided hypersurfaces, entire Killing graphs and the mean curvature equation in warped products with density

2020 ◽  
Vol 70 ◽  
pp. 101623
Author(s):  
Henrique F. de Lima ◽  
André F.A. Ramalho ◽  
Marco A.L. Velásquez
Keyword(s):  
Mean Curvature ◽  
Warped Products ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

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 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.

scholarly journals Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1185-1205
Author(s):  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Mean Curvature ◽  
Solution Method ◽  
The Novel ◽  
Open Ball ◽  
Radial Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Radially Symmetric Solutions ◽  
The Mean ◽  
Curvature Problem

Abstract This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{&#x03A9;},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{&#x03A9;},\end{array}\right. with \text{&#x03A9;} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

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