Approximation up to the boundary of homeomorphisms of finite Dirichlet energy

Let ⊂ and ⊂ be Jordan domains of the same finite connectivity, being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h: → in the Sobolev space ^{1, 2} extends to a continuous map h: →. We prove that there exist homeomorphisms hk: → that converge to h uniformly and in ^{1, 2}. The problem of approximation of Sobolev homeomorphisms, raised by J. M. Ball and L. C. Evans, is deeply rooted in a study of energy-minimal deformations in non-linear elasticity. The new feature of our main result is that approximation takes place also on the boundary, where the original map need not be a homeomorphism.