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The Dirichlet problem for discontinuous perturbations of the mean curvature operator in Minkowski space  [PDF]
Cristian Bereanu,Petru Jebelean,Calin Serban
Mathematics , 2015,
Abstract: Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.
On multiple positive ground state solutions for a mean curvature equation in Minkowski space  [PDF]
Ruyun Ma,Yanqiong Lu,Tianlan Chen
Mathematics , 2014,
Abstract: In this paper, we show how changes in the sign of nonlinearity leads to multiple radial ground state solutions of the mean curvature equation $ \nabla\cdot \Big[\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big] +\lambda f(u)=0\ \ \text{in} \ \mathbb{R}^N $ for sufficiently large $\lambda$ with $N\geq 2$.
On a prescribed mean curvature equation in Lorentz-Minkowski space  [PDF]
Antonio Azzollini
Mathematics , 2015,
Abstract: We are interested in providing new results on a prescribed mean curvature equation in Lorentz-Minkowski space set in the whole R^N, with N >2. We study both existence and multiplicity of radial ground state solutions for p>1, emphasizing the fundamental difference between the subcritical and the supercritical case. We also study speed decay at infinity of ground states, and give some decay estimates. Finally we provide a multiplicity result on the existence of sign-changing bound state solutions for any p>1.
Entire downward translating solitons to the mean curvature flow in Minkowski space  [PDF]
Joel Spruck,Ling Xiao
Mathematics , 2015,
Abstract: In this paper, we study entire translating solutions $u(x)$ to a mean curvature flow equation in Minkowski space. We show that if $\Sigma=\{(x, u(x))| x\in\mathbb{R}^n\}$ is a strictly spacelike hypersurface, then $\Sigma$ reduces to a strictly convex rank k soliton in $\mathbb{R}^{k, 1}$ (after splitting off trivial factors) whose "blowdown" converges to a multiple $\lambda\in(0, 1)$ of a positively homogeneous degree one convex function in $\mathbb{R}^k$. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation.
Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature  [PDF]
Rafael López,Esma Demir
Mathematics , 2010,
Abstract: In this work we find all helicoidal surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is $0$ or $1$ and that the surface is ruled. If the generating curve is a Lorentzian circle, we show that the only possibility is that the axis is spacelike and the center of the circle lies in the axis.
Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space  [PDF]
David Brander
Mathematics , 2009, DOI: 10.1017/S030500411100007
Abstract: We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Bj\"orling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the partial derivatives $f_x$ and $f_y$ are given by $\frac{\dd f_0}{\dd x}$ and $v$ along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in $L^3$. We use this to find the Bj\"orling data -- and holomorphic potentials -- which characterize cuspidal edge, swallowtail and cross cap singularities.
Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space  [PDF]
Huaiyu Jian
Mathematics , 2006,
Abstract: In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.
Timelike surfaces with zero mean curvature in Minkowski 4-space  [PDF]
Georgi Ganchev,Velichka Milousheva
Mathematics , 2011, DOI: 10.1007/s11856-012-0169-y
Abstract: On any timelike surface with zero mean curvature in the four-dimensional Minkowski space we introduce special geometric (canonical) parameters and prove that the Gauss curvature and the normal curvature of the surface satisfy a system of two natural partial differential equations. Conversely, any two solutions to this system determine a unique (up to a motion) timelike surface with zero mean curvature so that the given parameters are canonical. We find all timelike surfaces with zero mean curvature in the class of rotational surfaces of Moore type. These examples give rise to a one-parameter family of solutions to the system of natural partial differential equations describing timelike surfaces with zero mean curvature.
Surfaces of annulus type with constant mean curvature in Lorentz-Minkowski space  [PDF]
Rafael Lopez
Mathematics , 2005,
Abstract: In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space $\l^3$ and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in $\l^3$ of constant mean curvature bounded by two concentric circles in parallel planes. As conclusion, we characterize spacelike surfaces of revolution with constant mean curvature as the only that either i) are the solutions of the exterior Dirichlet problem for constant boundary data or ii) have an isolated conical-type singularity.
Inverse anisotropic mean curvature flow and a Minkowski type inequality  [PDF]
Chao Xia
Mathematics , 2015,
Abstract: In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we prove a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.
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