  We study a class of elastic energy functionals for maps between planar
domains (among them the so-called squared distance functional) whose critical
points (elastic maps) allow a far more complete theory than one would expect
from general elasticity theory. For some of these functionals elastic maps even
admit a "Weierstrass representation" in terms of holomorphic functions,
reminiscent of the one for minimal surfaces. We also prove a global uniqueness
theorem that does not seem to be known in other situations.
