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 inR N for the functions inW m,p0 (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.
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References
Adams, D. R. andPolking, J. C., The equivalence of two definitions of capacity,Proc. Amer. Math. Soc. 37 (1973), 529–533.
Adams, R. A.,Sobolev Spaces, Academic Press, New York, 1975.
Maz'ya, V. G., On (p, l)-capacity, imbedding theorems, and the spectrum of a self-adjoint elliptic operator,Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 356–385 (Russian). English transl.:Math. USSR-Izv. 7 (1973), 357–387.
Maz'ya, V. G.,Sobolev Spaces, Springer-Verlag, Berlin-New York, 1985.
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Wannebo, A. Equivalent norms for the Sobolev spaceW m,p0 (Ω). Ark. Mat. 32, 245–254 (1994). https://doi.org/10.1007/BF02559531
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DOI: https://doi.org/10.1007/BF02559531