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We prove density of smooth functions in subspaces of Sobolev- and higherorder $BV$-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 withsuitably 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 orderbounded variation ($BV^m$) if its $m$-th order partial derivatives in the senseof distributions are finite Radon measures. This takes up earlier results by C.Tietz and the author concerning functions with merely one order ofdifferentiability which emerged in the context of a variational problem relatedto image analysis. In the connection of our methods we also investigate aquestion concerning the boundary traces of $W^{1,p}(\Omega)\capL^q(\Omega)$-functions.

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