## 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:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Keywords:

formal integration by parts