Approximation up to the boundary of homeomorphisms of finite Dirichlet energy

Abstract

Let 핏⊂ℂ and 핐⊂ℂ be Jordan domains of the same finite connectivity, 핐 being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h: 핏→핐 in the Sobolev space 풲1, 2 extends to a continuous map h: 핏→핐. We prove that there exist homeomorphisms hk: 핏→핐 that converge to h uniformly and in 풲1, 2(핏, 핐). 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.