arXiv subjects
Publications and source records attributed to Roman V. Dribas.
We discuss the relationship between the chain rule for the divergence operator and the renormalization property for weak solutions of the continuity equation. We construct an example of bounded divergence-free vector field on the plane, which demonstrates that in general the first property is not sufficient for the second one.
If $f\colon [0,1]^2 \to \mathbb{R}$ is of class $C^2$ then Sard's theorem implies that $f$ has the following relaxed Sard property: the image under $f$ of the Lebesgue measure restricted to the critical set of $f$ is a singular measure. We show that for $C^{1,α}$ functions with $α<1$ this property is strictly stronger than the weak Sard property introduced by Alberti, Bianchini and Crippa, while for any monotone continuous function these two properties are equivalent. We also show that even in the one-dimensional setting Hölder regularity is not sufficient for the relaxed Sard property.