Equivalent norms for the Sobolev space W ${}_{0}^{m,p}$ (Ω)

Abstract

A new proof is given for a theorem by V. G. Maz'ya. It gives a necessary and sufficient condition on the open set Ω in R^N for the functions in W ${}_{0}^{m,p}$ (Ω) to have the ordinary norm equivalent to the norm obtained when including only the highest order derivatives in the definition. The proof is based on a kind of polynomial capacities, Maz'ya capacities.