On a free boundary problem for the \(p\)-Laplacian
The author considers the problem \[ \text{div} \bigl(| Du |^{p-2}Du \bigr) =0\quad\text{in}\quad\Omega_0- \Omega_1, \] \(u\to 1\), \(Du\to c_1\) uniformly a.e. as \(x\to \partial \Omega_1\), \(u\to 0\), \(Du\to c_0\) uniformly a.e. as \(x\to \partial\Omega_0\), where \(\Omega_1 \) and \(\Omega_0\) are bounded connected domains, \(\Omega_1 \subset \Omega_0\), that are starshaped with respect to the origin which is taken inside \(\Omega_1\). The author proves that the problem has a weak solution iff \(\Omega_0\) and \(\Omega_1\) are balls.
scientific article; zbMATH DE number 1123514
scientific article