  The shock reflection problem is one of the most important problems in
mathematical fluid dynamics, since this problem not only arises in many
important physical situations but also is fundamental for the theory of
multidimensional conservation laws. However, most of the fundamental issues for
shock reflection have not been understood. Therefore, it is important to
establish the regularity of solutions to shock reflection in order to
understand fully the phenomena of shock reflection. On the other hand, for a
regular reflection configuration, the potential flow governs the exact behavior
of the solution in $C^{1,1}$ across the pseudo-sonic circle even starting from
the full Euler flow, that is, both of the nonlinear systems are actually the
same in an physically significant region near the pseudo-sonic circle; thus, it
becomes essential to understand the optimal regularity of solutions for the
potential flow across the pseudo-sonic circle and at the point where the
pseudo-sonic circle meets the reflected shock. In this paper, we study the
regularity of solutions to regular shock reflection for potential flow. In
particular, we prove that the $C^{1,1}$-regularity is optimal for the solution
across the pseudo-sonic circle and at the point where the pseudo-sonic circle
meets the reflected shock. We also obtain the $C^{2,\alpha}$ regularity of the
solution up to the pseudo-sonic circle in the pseudo-subsonic region. The
problem involves two types of transonic flow: one is a continuous transition
through the pseudo-sonic circle from the pseudo-supersonic region to the
pseudo-subsonic region; the other a jump transition through the transonic shock
as a free boundary from another pseudo-supersonic region to the pseudo-subsonic
region.
