  This paper revives a four-decade-old problem concerning regularity theory for
(continuous) constraint maps with free boundaries. Dividing the map into two
parts, the distance part and the projected image to the constraint, one can
prove various properties for each component. As already pointed out in the
literature, the distance part falls under the classical obstacle problem, which
is well-studied by classical methods. A perplexing issue, untouched in the
literature, is the properties of the projected image and its higher regularity,
which we show to be at most of class $C^{2,1}$. In arbitrary dimensions, we
prove that the image map is globally of class $W^{3,BMO}$, and locally of class
$C^{2,1}$ around the regular part of the free boundary. The issue becomes more
delicate around singular points, and we resolve it in two dimensions. In the
appendix, we extend some of our results to what we call leaky maps.
