Measures of non-compactness of classical embeddings of Sobolev spaces

Let Omega be an open subset of R-n and let p is an element of [1, n). We prove that the measure of non-compactness of the Sobolev embedding W-0(k,p)(Omega) --> L-P* (Omega) is equal to its norm. This means that the entropy numbers of this embedding are constant and equal to the norm. The same is true, when lambda(n)(Omega) is small enough, for the embedding of W-0(1,n) (Omega) into the Orlicz space with Young function exp (t(n/(n-1))) - 1. The position is different for the embedding of W-0(1,p) (J) in C-0,C-1-1/p (J), J = (0, 1), when p is an element of (1, infinity): in this case the measure of noncompactness is less than the norm. (C) 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.