arXiv subjects
Publications and source records attributed to Luděk Kleprlík.
Let $Ω_1, Ω_2\subset \mathbb R^n$ and $1\leq p <\infty$. We study the optimal conditions on a homeomorphism $f:Ω_1$ onto $Ω_2$ which guarantee that the composition $u\circ f$ belongs to the space $BV(Ω_1)$ for every $u\in W^{1,p}(Ω_2)$. We show that the sufficient and necessary condition is an existence of a function $K(y)\in L^{p'}(Ω_2)$ such that $|Df|(f^{-1}(A))\leq \int_A K(y)\,dy$ for all Borel sets $A$.
We study the optimal conditions on a homeomorphism $f:Ω\subset \R^n\to \R^n$ to guarantee that the composition $u\circ f$ belongs to the space of functions of bounded variation for every function $u$ of bounded variation. We show that a sufficient and necessary condition is the existence of a constant $K$ such that $|Df|(f^{-1}(A))\leq K\Ln(A)$ for all Borel sets $A$. We also characterize homeomorphisms which maps sets of finite perimeter to sets of finite perimeter. Towards these results we study when $f^{-1}$ maps sets of measure zero onto sets of measure zero (i.e. $f$ satisfies the Lusin $(N^{-1})$ condition).