Approximation by Solutions of Elliptic Equations

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded, smooth domain, $\Gamma$ a closed, smooth, $(n-1)$-dimensional surface with boundary 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 order $2m$ with smooth coefficients. Let $(B_1,\dots ,B_m)$ be normal system of boundary operators on $\partial \Omega$, which fulfils the classical root condition. $L_V(\Gamma )$ denote the space of the restrictions on $\Gamma$ of the functions from

It is proved that $L_V(\Gamma )$ is dense in the space $W_p^{2m-1/p}(\Gamma )(p>1)$.