  We establish the existence and stability of cylindrical transonic shock
solutions under three dimensional perturbations of the incoming flows and the
exit pressure without any further restrictions on the background transonic
shock solutions. The strength and position of the perturbed transonic shock are
completely free and uniquely determined by the incoming flows and the exit
pressure. The optimal regularity is obtained for all physical quantities, and
the velocity, the Bernoulli's quantity, the entropy and the pressure share the
same regularity. The approach is based on the deformation-curl decomposition to
the steady Euler system introduced by the authors to decouple the hyperbolic
and elliptic modes effectively. However, one of the key elements in application
of the deformation-curl decomposition is to find a decomposition of the
Rankine-Hugoniot conditions, which shows the mechanism of determining the shock
front uniquely by an algebraic equation and also gives an unusual second order
differential boundary conditions on the shock front for the first order
deformation-curl system. After homogenizing the curl system and introducing a
potential function, this unusual condition on the shock front becomes the
Poisson equation with homogeneous Neumann boundary condition on the
intersection of the shock front and the cylinder walls from which an oblique
boundary condition for the potential function can be uniquely derived.
