Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is then natural to ask which homeomorphisms admit a homeomorphic extension that has some Sobolev regularity. A simplified question is: given a homeomorphic embedding $\varphi$ of the unit circle into the plane, when can we find a homeomorphism $f$ from the unit disk with the same boundary values so that the first order distributional derivatives are $p$-integrable?

 

The motivation for this problem partially comes from trying to understand which boundary values can correspond to deformations of finite energy.

 

I will describe the recent results by myself and my collaborators on this problem.