Calculus of Variations and Geometric Measure Theory

G. Crippa - N. Gusev - S. Spirito - E. Wiedemann

Failure of the Chain Rule for the Divergence of Bounded Vector Fields

created by crippa on 09 Dec 2014


Submitted Paper

Inserted: 9 dec 2014
Last Updated: 9 dec 2014

Year: 2014


We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.