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A counterexample concerning formal integration by parts. (English. Abridged French version) Zbl 0723.46028

Summary: Let \(\Omega \subset {\mathbb{R}}^ n\), \(u\in H^ 1_ 0(\Omega)\cap L^{\infty}(\Omega)\), \(b\in L^ p(\Omega;{\mathbb{R}}^ n)\) with div b\(=0\) and div bu\(\in H^{-1}(\Omega)\). For \(n\geq 3\), \(p<3/2\) an example is given for which \(<div bu,u>_{H^{-1},H^ 1_ 0}\neq 0\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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