  We are concerned with the stability of multidimensional (M-D) transonic
shocks in steady supersonic flow past multidimensional wedges. One of our
motivations is that the global stability issue for the M-D case is much more
sensitive than that for the 2-D case, which requires more careful rigorous
mathematical analysis. In this paper, we develop a nonlinear approach and
employ it to establish the stability of weak shock solutions containing a
transonic shock-front for potential flow with respect to the M-D perturbation
of the wedge boundary in appropriate function spaces. To achieve this, we first
formulate the stability problem as a free boundary problem for nonlinear
elliptic equations. Then we introduce the partial hodograph transformation to
reduce the free boundary problem into a fixed boundary value problem near a
background solution with fully nonlinear boundary conditions for second-order
nonlinear elliptic equations in an unbounded domain. To solve this reduced
problem, we linearize the nonlinear problem on the background shock solution
and then, after solving this linearized elliptic problem, develop a nonlinear
iteration scheme that is proved to be contractive.
