Initial value problem for the constant mean curvature equation in the Reissner-NordstrÖm spacetime

2021 ◽  
Vol 87 (3) ◽  
pp. 369-395
Author(s):  
Kuo-Wei Lee
Keyword(s):  
Initial Value Problem ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Initial Value ◽  
Curvature Equation ◽  
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 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 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.

Spacelike Graphs of Prescribed Mean Curvature in the Steady State Space

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Rafael López
Keyword(s):  
Steady State ◽  
Dirichlet Problem ◽  
State Space ◽  
Bounded Domain ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation

AbstractWe study the Dirichlet problem of the constant mean curvature equation in the steady state space over a bounded domain of a slice. Under suitable conditions on the convexity of the domain, we prove the existence of a spacelike graph with constant mean curvature

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