Differentiable functions with surjective Clarke Jacobians
Fonctions différentiables ayant un Jacobien de Clarke surjectif.
Résumé
We construct, for any n, m ∈ N \ {0}, a differentiable locally Lipschitz function f : R n → R m which is C 1 on the complement of an H 1 -null set E ⊂ R n and has the property that the range of its limiting Jacobian on E contains the family of all nonempty compact connected sets of (m×n)-matrices. As a consequence, the Clarke Jacobian J c f is surjective, that is, its range contains every nonempty compact convex subset of (m×n)-matrices. This reveals a significant difference between differentiable functions and C 1 -functions, since for a C 1 -function the Clarke Jacobian is always a singleton. As a by-product, we also obtain examples of C 1 -smooth functions from R n to R m (for any n, m ∈ N\{0}) with surjective derivative, that is, Im(Df ) = R m×n .
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