The uniqueness of transonic shocks in supersonic flow past a 2-D wedge

We have proved the uniqueness of transonic shocks in steady supersonic flows past a slightly perturbed two-dimensional infinite wedge, under appropriate conditions on the downstream subsonic flows. We formulate it to a mathematical problem of the uniqueness of solutions of nonlinear partial differential equations of hyperbolic-elliptic mixed type with a free boundary. By working on several elliptic equations of physical quantities separately, we obtain a priori estimates of them, and then prove the uniqueness without assumptions on high regularity. Moreover, uniform estimates on the ellipticity and the positive lower bound of the speed are achieved under a geometrical condition on the wedge. The mathematical ideas and techniques developed here will also be useful for other related problems involving similar analytical difficulties.