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.
Let F ∈ C 1 (, R n) and f ∈ C 2 (), where is an open subset of R n with n even. We describe the structure of the set of points in at which the equality D f = F and a certain non-integrability condition on F hold. This result generalizes the second statement of Balogh (
SynopsisLet Ω be an open subset of Rn. It is well known that, given a suitable real-valued function f on Ω × Rk and a Rk -valued Borel measure µ on Ω, then one can define a real-valued measurefµ on Ω. The object of this note is to define the Ψ-strict convergence of the Rk-valued Borel measures µj to the Rk-valued Borel measure µ, where Ψ: Ω × Rk → [0, + ∞] is a continuous function which is positively homogeneous and convex in the Rk-variable, and to investigate the lower semicontinuity and continuity of the map µ → fμ with respect to the Ψ-strict convergence; here f is positively homogeneous in the Rk-variable and satisfies one suitable convexity condition (related to Ψ).
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