  For a certain class of genuinely nonlinear two-by-two planar hyperbolic
systems we show that any classical solution on a smoothly bounded domain has
nontangential boundary limits except on a set whose Hausdorff dimension is
bounded by some system-dependent constant which is strictly less than 1 and, on
the other hand, that for any system of the kind considered there is in fact a
solution on a half-plane which fails to have nontangential limits at a set of
boundary points of positive Hausdorff dimension.
