Let X and Y be planar Jordan domains of the same finite connectivity, Y being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h:X->Y in the Sobolev space W1,2 extends to a continuous map between closed domains. We prove that there exist homeomorphisms between closed domains which converge to h uniformly and in W1,2. 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 nonlinear 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.
Iwaniec, Tadeusz; Kovalev, Leonid V.; and Onninen, Jani, "Strong Approximation of Homeomorphisms of Finite Dirichlet Energy" (2011). Mathematics Faculty Scholarship. Paper 56.
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