Equivalent norms for the Sobolev spaceW 0 m,p (Ω)

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 inR ^N for the functions inW 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.