A density result for Sobolev functions and functions of higher order bounded variation with additional integrability constraints

Abstract

We prove density of smooth functions in subspaces of Sobolev- and higher order $BV$-spaces of kind $W^{m,p}(\Omega)\cap L^q(\Omega-D)$ and $BV^m(\Omega)\cap L^q(\Omega-D)$, respectively, where $\Omega\subset\mathbb{R}^n$ ($n\in\mathbb{N}$) is an open and bounded set with suitably smooth boundary, $m<n$ is a positive integer, $1\leq p<\infty$ s.t. $mp<n$, $D\Subset\Omega$ is a sufficiently regular open subset and $q> np/(n-mp)$. Here we say that a $W^{m-1,1}(\Omega)$-function is of $m$-th order bounded variation ($BV^m$) if its $m$-th order partial derivatives in the sense of distributions are finite Radon measures. This takes up earlier results by C. Tietz and the author concerning functions with merely one order of differentiability which emerged in the context of a variational problem related to image analysis. In the connection of our methods we also investigate a question concerning the boundary traces of $W^{1,p}(\Omega)\cap L^q(\Omega)$-functions.