Degenerate elliptic problems in a class of free domains (Q1568989)

The authors introduce a quasilinear degenerate elliptic equation of the form \[ -\nabla [\alpha(|\nabla u|\nabla u)]=1. \tag{1} \] An example for the operator on the left hand side of (1) is the \(p\)-Laplacian, which corresponds to the choice \(\alpha (|\nabla u|)= |\nabla u|^{p-2}\) with \(1<p< \infty\). The authors establish the existence and the uniqueness of a solution to a free boundary value problem for equation (1) with mixed boundary conditions. Furthermore, the authors consider two examples for the obtained results in fluid mechanics, namely to the study of steady non-Newtonian flows in open channels or valleys.