Weighted Sobolev spaces and regularity for polyhedral domains

摘要

We prove a regularity result for the Poisson problem $-\Delta u = f$, $u|\_{\pa \PP} = g$ on a polyhedral domain $\PP \subset \RR^3$ using the \BK\spaces $\Kond{m}{a}(\PP)$. These are weighted Sobolev spaces in which theweight 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 withsmooth coefficients. We also establish a "trace theorem" for the restriction tothe boundary of the functions in $\Kond{m}{a}(\PP)$.