A Bernstein property of solutions to a class of prescribed affine mean curvature equations

2006 ◽  
Vol 32 (2) ◽  
pp. 147-165 ◽  
Author(s):  
James Alexander McCoy
Keyword(s):  
Mean Curvature ◽  
Bernstein Property ◽  
Mean Curvature Equations ◽  
Affine Mean Curvature

scholarly journals Asymptotic analysis of singular solutions of the scalar and mean curvature equations

2005 ◽  
Vol 2005 (5) ◽  
pp. 679-698
Author(s):  
Gonzalo García ◽  
Hendel Yaker
Keyword(s):  
Asymptotic Analysis ◽  
Mean Curvature ◽  
Related Problem ◽  
Singular Solutions ◽  
Conformal Deformation ◽  
Conformal Metric ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Mean Curvature Equations ◽  
Sobolev Critical Exponent

We show that positive solutions of a semilinear elliptic problem in the Sobolev critical exponent with Newmann conditions, related to conformal deformation of metrics inℝ+n, are asymptotically symmetric in a neighborhood of the origin. As a consequence, we prove for a related problem of conformal deformation of metrics inℝ+nthat if a solution satisfies a Kazdan-Warner-type identity, then the conformal metric can be realized as a smooth metric onS+n.

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.

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