Non-rigidity of the absolutely continuous part of A-free measures

Abstract

We generalize a result by Alberti, showing that, if a first-order linear differential operator $A$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure μ satisfying $A\mu =0$. When $A$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of μ. Finally, we show that operators in the above class satisfy a Lusin-type property.