Pré-Publication, Document De Travail Année : 2025

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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Dates et versions

hal-04952836 , version 1 (17-02-2025)

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  • HAL Id : hal-04952836 , version 1

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Aris Daniilidis, Sebastian Tapia-Garcia. Differentiable functions with surjective Clarke Jacobians. 2025. ⟨hal-04952836⟩
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