  An approximation theorem of Youngs (1948) asserts that a continuous map
between compact oriented topological 2-manifolds (surfaces) is monotone if and
only if it is a uniform limit of homeomorphisms. Analogous approximation of
Sobolev mappings is at the very heart of Geometric Function Theory (GFT) and
Nonlinear Elasticity (NE). In both theories the mappings in question arise
naturally as weak limits of energy-minimizing sequences of homeomorphisms. As a
result of this, the energy-minimal mappings turn out to be monotone. In the
present paper we show that, conversely, monotone mappings in the Sobolev space
$W^{1,p}, 1<p < \infty$, are none other than $W^{1,p}$-weak (also strong)
limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way
of illustration, we establish the existence of energy-minimal deformations
within the class of Sobolev monotone mappings for $\,p\,$-harmonic type energy
integrals.
