Spacelike graphs with prescribed mean curvature
The authors study rotationally symmetric spacelike hypersurfaces with prescribed mean curvature \(f(u)\) for the Minkowski metric \(dx^2_1 + \ldots + dx^2_N - dt^2\), \(N>1\). The problem is reduced to the equations \[ \frac {u''}{(1-u^{'2})^{3/2}} + \frac {N-1}{r} \frac {u'}{(1-u^{'2})^{1/2}} = f(u), \qquad u({| }x{| }) \rightarrow u_0 \quad\text{as } {| }x{| }\rightarrow 0, \] which are thoroughly analyzed by using quite elementary but rather notrivial methods. Any solution \(u=u({| }x{| })\), \(0<x<r_0\), admits a unique extension to \(0<x<\infty \) and either \(u'({| }x{| })\rightarrow 0\) (the regular case) or \(u'({| }x{| })\rightarrow \pm 1\) (the singular case) as \({| }x{| }\rightarrow 0\). For any \(u_0\), there exists a unique regular solution and an infinity of singular ones. The main part of the paper is devoted to the investigation of the global behaviour under various assumptions on the function \(f\).
scientific article; zbMATH DE number 1153444