Positive radial solutions of a mean curvature equation in Lorentz–Minkowski space with strong singularity

2018 ◽  
Vol 99 (9) ◽  
pp. 1631-1637
Author(s):  
Minghe Pei ◽  
Libo Wang
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Radial Solutions ◽  
Curvature Equation ◽  
Strong Singularity ◽  
Mean Curvature Equation ◽  
Positive Radial Solutions

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 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.

Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space

2019 ◽  
Vol 21 (03) ◽  
pp. 1850003 ◽  
Author(s):  
Xuemei Zhang ◽  
Meiqiang Feng
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
The One ◽  
Exact Multiplicity

In this paper, bifurcation diagrams and exact multiplicity of positive solution are obtained for the one-dimensional prescribed mean curvature equation in Minkowski space in the form of [Formula: see text] where [Formula: see text] is a bifurcation parameter, [Formula: see text], the radius of the one-dimensional ball [Formula: see text], is an evolution parameter. Moreover, we make a comparison between the bifurcation diagram of one-dimensional prescribed mean curvature equation in Euclid space and Minkowski space. Our methods are based on a detailed analysis of time maps.

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{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} 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.

Sign in / Sign up

Export Citation Format

Share Document