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A refinement of Rellich’s theorem. (English) Zbl 0666.46041
If $$u_ k$$ converges weakly to u in the Sobolev space $$H_ 0^{1,p}(U)$$, $$U\subset {\mathbb{R}}^ n$$, $$1\leq p\leq n$$, then the exceptional sets $$E_ k=\{x\in U:$$ $$| u_ k-u| \geq \epsilon \}$$ are analysed. Their relative capacities are shown to vanish in a weak sense, which leads to new compactness results for solutions of certain elliptic inequalities.
Reviewer: J.F.Toland

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)