Approximation up to the boundary of homeomorphisms of finite Dirichlet energy

TL;DR

This paper proves that Sobolev homeomorphisms between planar Jordan domains can be uniformly and energetically approximated by homeomorphisms that also match boundary behavior, even when the original map isn't a boundary homeomorphism.

Contribution

It establishes boundary approximation of Sobolev homeomorphisms, extending previous results to include boundary behavior where the original map may not be a homeomorphism.

Findings

Existence of boundary-approximating homeomorphisms in $W^{1,2}$

Uniform convergence of approximations to the original map

Approximation preserves energy in the Sobolev space

Abstract

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 $W^{1,2}$ extends to a continuous map between closed domains. We prove that there exist homeomorphisms between closed domains which converge to h uniformly and in $W^{1,2}$. 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 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.