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arXiv · 1108.0199

Approximation up to the boundary of homeomorphisms of finite Dirichlet energy

Tadeusz Iwaniec·Leonid V. Kovalev·Jani Onninen
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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.

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Tadeusz Iwaniec, Leonid V. Kovalev, Jani Onninen. 2013-02-09. Approximation up to the boundary of homeomorphisms of finite Dirichlet energy. https://doi.org/10.1112/blms%2Fbds016

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