Geodesic
A chain rule formula for the composition of a vector-valued function by a piecewise smooth function
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 581-595.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We state and prove a chain rule formula for the composition $T(u)$ of a vector-valued function $u\in W^{1, r}(\Omega;\mathbb{R}^{M})$ by a globally Lipschitz-continuous, piecewise $C^{1}$ function $T$. We also prove that the map $u \to T(u)$ is continuous from $W^{1, r}(\Omega;\mathbb{R}^{M})$ into $W^{1,r}(\Omega)$ 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 $u\in W^{1, r}(\Omega;\mathbb{R}^{M})$ con una funzione $T$ globalmente lipschitziana e $C^{1}$ a tratti. Si dimostra inoltre che l'applicazione $u \to T(u)$ è continua da $W^{1, r}(\Omega;\mathbb{R}^{M})$ in $W^{1,r}(\Omega)$ rispetto alle topologie forti di questi spazi. 
@article{BUMI_2003_8_6B_3_a5,
     author = {Murat, Fran\c{c}ois and Trombetti, Cristina},
     title = {A chain rule formula for the composition of a vector-valued function by a piecewise smooth function},
     journal = {Bollettino della Unione matematica italiana},
     pages = {581--595},
     publisher = {mathdoc},
     volume = {Ser. 8, 6B},
     number = {3},
     year = {2003},
     zbl = {1179.46037},
     mrnumber = {969514},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a5/}
}
TY  - JOUR
AU  - Murat, François
AU  - Trombetti, Cristina
TI  - A chain rule formula for the composition of a vector-valued function by a piecewise smooth function
JO  - Bollettino della Unione matematica italiana
PY  - 2003
SP  - 581
EP  - 595
VL  - 6B
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a5/
LA  - en
ID  - BUMI_2003_8_6B_3_a5
ER  - 
%0 Journal Article
%A Murat, François
%A Trombetti, Cristina
%T A chain rule formula for the composition of a vector-valued function by a piecewise smooth function
%J Bollettino della Unione matematica italiana
%D 2003
%P 581-595
%V 6B
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a5/
%G en
%F BUMI_2003_8_6B_3_a5
Murat, François; Trombetti, Cristina. A chain rule formula for the composition of a vector-valued function by a piecewise smooth function. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 581-595. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a5/

[AD] L. Ambrosio-G. Dal Maso, A general chain rule for distributional derivatives, Proc. Amer. Math. Soc., 108 (1990), 691-702. | MR | Zbl

[BM] L. Boccardo-F. Murat, Remarques sur l'homogénéisation de certains problèmes quasi-linéaires, Portugaliæ Math., 41 (1982), 535-562. | fulltext mini-dml | MR | Zbl

[B] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rat. Mech. Anal., 157 (2001), 75-90. | MR | Zbl

[KS] D. Kinderlherer-G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York (1980). | MR | Zbl

[L] R. Landes, On the existence of weak solutions of perturbated systems with critical growth, J. reine angew. Math., 393 (1989), 21-38. | MR | Zbl

[MM1] M. Marcus-V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rat. Mech. Anal., 45 (1972), 294-320. | MR | Zbl

[MM2] M. Marcus-V. J. Mizel, Nemitsky operators on Sobolev spaces, Arch. Rat. Mech. Anal., 51 (1973), 347-370. | MR | Zbl

[S] G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures, 16, Les Presses de l'Université de Montréal, Montréal (1966). | MR | Zbl