Weighted Sobolev spaces and regularity for polyhedral domains

Overview & Synopsis

We prove a regularity result for the Poisson problem $-\Delta u = f$, $u \vert_{\pa \PP} = g$ on a polyhedral domain $\PP \subset \RR^3$ using the \BK\ spaces $\Kond{m}{a}(\PP)$. These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges \cite{Babu70, Kondratiev67}. In particular, we show that there is no loss of $\Kond{m}{a}$--regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a ``trace theorem'' for the restriction to the boundary of the functions in $\Kond{m}{a}(\PP)$.