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MiS Preprint
45/2003
Mappings of finite distortion:The degree of regularity.
Daniel Faraco, Pekka Koskela and Xiao Zhong
Abstract
Recently, a rich theory of mappings of finite distortion has been established. It has been proved that under natural assumptions on the distortion these mappings share with the familiar mappings of bounded distortion interesting topological and and analytical properties. In this paper we concentrate in the self-improving integrability of these mappings since the existing methods have only yield partial results (see the Monograph of Iwaniec and Martin for a detailed account about what was known about these mappings). We prove the following theorem: Let $K(x)$ be such that $\textrm{exp}(\beta K(x)) \in L^1_{loc}$, $\beta >0$. Then there exists two universal constants $c_1(n), c_2(n)$ with the following property. Let $f$ be in $W^{1,1}_{loc}$ with $|Df(x)|^n\le K(x)J(x,f)$ and the Jacobian determinant $J(x,f)$ in $L^1\log^{-c_1(n)\beta}L$. Then automatically $J(x,f)$ is in $L^1\log^{c_2(n)\beta}L$.
As a consequence we obtained novel results on the size of removable sets for bounded functions of finite distortion and on the area distortion under these type of mappings.
