On the weak differentiability of $u\circ f^{-1}$

Abstract

Let $p\geq n-1$ and suppose that $f:\Omega\to{\mathsf R}^n$ is a homeomorphism in the Sobolev space $W^{1,p}_{(\mathrm{loc}}(\Omega,{\mathsf R}^n)$. Further let $u\in W^{1,q}_{(\mathrm{loc}}(\Omega)$ where $q=\frac{p}{p-(n-1)}$ and for $q>n$ we also assume that $u$ is continuous. Then $u\circ f^{-1}\in (\mathrm{BV}_{(\mathrm{loc}}(f(\Omega))$ and if we moreover assume that $f$ is a mapping of finite distortion, then $u\circ f^{-1}\in W^{1,1}_{(\mathrm{loc}}(f(\Omega))$.