The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space
The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space
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DOI：
10.1515/gmj-2016-0078
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发表时间：
2017-03
影响因子：
0.7
通讯作者：
Chiara Corsato;F. Obersnel;P. Omari
中科院分区：
文献类型：
--
作者：
Chiara Corsato;F. Obersnel;P. Omari
摘要 我们讨论了洛伦兹-闵可夫斯基空间中与梯度相关的规定平均曲率方程相关的狄利克雷问题解的存在性、多重性、局域性和稳定性性质 { - div ( ∇ u 1 - | ∇ u | 2 ) = f ( x , u , ∇ u ) in Ω , u = 0 on ∂Ω。 $\left\{\begin{对齐} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial%\Omega.\end{aligned}\right.$ 得到的结果显示出各种特殊性，这是由于所涉及的微分算子的特殊性，并且对于其他拟线性微分算子驱动的椭圆问题没有对应物。这项研究还受到某些弗里德曼-勒梅特-罗伯逊-沃克以及史瓦西-赖斯纳-诺德斯特伦时空中规定平均曲率图研究的一些最新成果的推动。
Abstract We 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 { - div ( ∇ u 1 - | ∇ u | 2 ) = f ( x , u , ∇ u ) in Ω , u = 0 on ∂ Ω . $\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$ 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.