  We consider a surveillance-evasion game in an environment with obstacles. In
such an environment, a mobile pursuer seeks to maintain the visibility with a
mobile evader, who tries to get occluded from the pursuer in the shortest time
possible. In this two-player zero-sum game setting, we study the
discontinuities of the value of the game near the boundary of the target set
(the non-visibility region). In particular, we describe the transition between
the usable part of the boundary of the target (where the value vanishes) and
the non-usable part (where the value is positive). We show that the value
enjoys a different behaviour depending on the regularity of the obstacles
involved in the game. Namely, we prove that the boundary profile is continuous
for the case of smooth obstacles, and that it exhibits a jump discontinuity
when the obstacle contains corners. Moreover, we prove that, in the latter
case, there is a semi-permeable barrier emanating from the interface between
the usable and the non-usable part of the boundary of the target set.
