Uniform approximation by solutions of general boundary value problems for elliptic equations of arbitrary order I

Let $\Omega \subset \mathbb R^n$ be a bounded, smooth domain, $\Gamma$ a closed, smooth, $(n-1)$-dimensional surface in the interior of $\Omega$ and $V$ an open subset of the boundary $\partial \Omega$. In $\Omega$ we consider a properly elliptic differential operator $L$ of arbitrary order with smooth coefficients. Let $B_1, \dots, B_m$ be a normal system of boundary operators on $\partial \Omega$, which fulfils the classical roots condition. $L_V(\Gamma)$ denotes the space of the restrictions on $\Gamma$ of the functions from $$L_V(\Omega) = \{u: Lu = 0 \: \mathrm {in} \: \Omega, B_1 u|_{\partial \Omega} =\dots = B_m u|_{\partial \Omega} = 0 \: \mathrm {in} \: \partial \Omega \setminus V \}.$$ Among other things it is proved, that the space $L_V(\Gamma)$ is dense in the space $W^{m-1} (\Gamma)$ of the Whitney-Taylorfields of the order $m-1$, i.e. all derivatives up to the order $m-1$ can be uniformly approximated on $\Gamma$.