On two free boundary problems in potential theory
The authors consider two free boundary value problems for the equation \[ \sum^ N_{k=1}{\partial\over\partial x_ k}\left(g\left(|\nabla u|^ 2\right){\partial u\over\partial x_ k}\right)=0 \tag{1} \] in the cylinder \(\Omega\subset R^ N\) with the free bases \(\Gamma_ 0\), \(\Gamma_ 1\) and the lateral surface \(\Gamma_ c\). It is assumed that \(g\in C^ \infty\) and \(g(\xi)+2\xi g'(\xi)>0\) \(\forall\xi\geq 0\). On the free boundary simultaneously the Dirichlet and Neumann conditions with constant data are prescribed. It is proved that the boundary value problem is solvable and that the free boundaries are contained in two different horizontal hyperplanes and that the solution depends only on \(x_ N\). The second problem is considered for the equation (1) with the right-hand side equal to \(-2\) in \(\Omega=\Omega_ 0\backslash\bigcup^ m_{k=1}\overline\Omega_ k\), where \(\Omega_ k\) are disconnected contractible regions in \(R^ N\), contained in a bounded domain \(\Omega_ 0\subset R^ N\). The boundary conditions are \[ u=0,\quad{\partial u\over\partial n}=a_ 0=\text{const}\quad \text{ on }\partial\Omega_ 0, \] \[ u=c_ k,\quad{\partial u\over\partial n}=ak=\text{const}\quad\text{ on } \partial\Omega_ k,\quad k=1,\dots,m. \] It is shown that the problem is solvable iff \(\Omega\) is spherically symmetric.
scientific article; zbMATH DE number 57715
scientific article