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 The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

2017 ◽  
Vol 24 (1) ◽  
pp. 113-134 ◽  
Author(s):  
Chiara Corsato ◽  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Differential Operators ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Mean Curvature Equations

AbstractWe discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

The Mean Curvature Equation with Oscillating Nonlinearity

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Anderson L. A. de Araujo ◽  
Marcelo Montenegro
Keyword(s):  
Dirichlet Problem ◽  
Mean Curvature ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
The Mean

AbstractWe find a solution of the Dirichlet problem for the prescribed mean curvature equation

scholarly journals Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions

2004 ◽  
Vol 2004 (18) ◽  
pp. 913-948
Author(s):  
Fei-Tsen Liang
Keyword(s):  
Dirichlet Problem ◽  
Mean Curvature ◽  
Boundary Regularity ◽  
Gradient Estimates ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Global Estimates ◽  
The Mean ◽  
Boundary Contact ◽  
Global Boundedness

We obtain global estimates for the modulus, interior gradient estimates, and boundary Hölder continuity estimates for solutionsuto the capillarity problem and to the Dirichlet problem for the mean curvature equation merely in terms of the mean curvature, together with the boundary contact angle in the capillarity problem and the boundary values in the Dirichlet problem.

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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