A free boundary value problem arisen in unsteady compressible flow

We study a free boundary value problem of multidimensional unsteady potential Row equation. The problem is defined in a domain bounded by two conical surfaces, one is given, and other is to be determined. In self-similar coordinates the problem can be reduced to a free boundary value problem of an elliptic equation. Then the existence of the solution is proved by using partial hodograph transformation and nonlinear alternating iteration. This result indicates that the structure of the weak solution with discontinuity is stable under small perturbation of data.