Geometric properties of solutions of the Levi curvature equation in \(\mathbb C^ 2\)
The authors study for \(L(u)=-\text{det}\left(\begin{smallmatrix} 0 & u_{z_1} & u_{z_2}\\ u_{z_1} & u_{z_1z_1} & u_{z_1z_2}\\ u_{z_2} & u_{z_2z_1} & u_{z_2z_2}\end{smallmatrix}\right)\) the Dirichlet problem: \[ L(u)= k|\partial u|^3\text{ in }\Omega\subset\mathbb C^2,\;u=g\text{ on }b\Omega,\tag{\(*\)} \] where \(\Omega\) is bounded and strictly pseudoconvex, \(k=k(z,t)\) is a real function on \(\Omega\times\mathbb R\) and \(g:b\Omega\to\mathbb R\). Introducing the notion of weak solution of \(L(u)=k|\partial u|^3\), they prove an existence theorem for the Dirichlet problem \((*)\). Furthermore, they study weak solutions of the homogeneous problem \(L(u)=0\) in terms of geometric properties of their level and sublevel sets.
scientific article; zbMATH DE number 889773