A chain rule formula for the composition of a vector-valued function by a piecewise smooth function

Murat, François; Trombetti, Cristina

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 581-595 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

We state and prove a chain rule formula for the composition $T (u)$ of a vector-valued function $u \in W^{1, r} (Ω; R^{M})$ by a globally Lipschitz-continuous, piecewise $C^{1}$ function $T$ . We also prove that the map $u \to T (u)$ is continuous from $W^{1, r} (Ω; R^{M})$ into $W^{1, r} (Ω)$ for the strong topologies of these spaces.

Si enuncia e si dimostra una formula di derivazione per funzioni $T (u)$ ottenute componendo una funzione a valori vettoriali $u \in W^{1, r} (Ω; R^{M})$ con una funzione $T$ globalmente lipschitziana e $C^{1}$ a tratti. Si dimostra inoltre che l'applicazione $u \to T (u)$ è continua da $W^{1, r} (Ω; R^{M})$ in $W^{1, r} (Ω)$ rispetto alle topologie forti di questi spazi.

Publié le : 2003-10-01

MR 2014820 | Zbl 1179.46037

@article{BUMI_2003_8_6B_3_581_0,
     author = {Fran\c cois Murat and Cristina Trombetti},
     title = {A chain rule formula for the composition of a vector-valued function by a piecewise smooth function},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {6-A},
     year = {2003},
     pages = {581-595},
     zbl = {1179.46037},
     mrnumber = {2014820},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2003_8_6B_3_581_0}
}

Murat, François; Trombetti, Cristina. A chain rule formula for the composition of a vector-valued function by a piecewise smooth function. Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003) pp. 581-595. http://gdmltest.u-ga.fr/item/BUMI_2003_8_6B_3_581_0/

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