Functions of class $C^1$ subject to a Legendre condition in an enhanced density set

Abstract

We investigate certain second-order differential properties of functions and forms of class $C^1$ at the points around which a suitable Legendrian condition is “very densely verified”. In particular we provide a generalization of the classical identity $d^2=0$ on differential forms and some results about second-order osculating properties of graphs. Particular emphasis is placed on the case when the condition is verified in a locally finite perimeter set. A conjecture about the $C^2$-rectifiability of the horizontal projection of a Legendrian rectifiable set is discussed.