  This paper considers the formulation of the adjoint problem in two dimensions
when there are shocks in the flow solution. For typical cost functions, the
adjoint variables are continuous at shocks, where they have to obey an internal
boundary condition, but their derivatives may be discontinuous. The derivation
of the adjoint shock equations is reviewed and detailed predictions for the
behavior of the gradients of the adjoint variables at shocks are obtained as
jump conditions for the normal adjoint gradients in terms of the tangent
gradients. Several numerical computations on a very fine mesh are used to
illustrate the behavior of numerical adjoint solutions at shocks.
