Degenerate elliptic problems in a class of free domains

We study a mixed boundary value problem for an operator of p-laplacian type . The main feature of the problem is the fact that the exact domain where i t is considered is not known a priori and is to be determined so that a cer tain integral condition is satisfied. We establish the existence of a uniqu e solution to the problem, by means of the analysis of the range of an appr opriate real function, and we show the continuous dependence with respect t o a family of operators. These results can be applied to the study of unidi rectional non-Newtonian flows of power-law type, in particular to solve a s implified problem arising in theoretical glaciology and to show the existen ce of a Bingham flow in an open channel; the uniqueness in this case is an open problem. (C) Elsevier, Paris.