We consider a sectorial Riemann problem for the isentropic gas dynamics equations leading to strong regular reflection. We write the problem in self-similar coordinates and derive a mixed type system and a free boundary problem for the unknown reflected shock and the subsonic state behind the reflected shock. We present preliminary ideas for proving local existence of solutions to this free boundary problem using the theory of second order elliptic equations with mixed boundary conditions and fixed point theorems. This is a joint work with Barbara Lee Keyfitz and Suncica Canic.