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

François Murat; Cristina Trombetti

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 3, page 581-595
  • ISSN: 0392-4041

Abstract

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We state and prove a chain rule formula for the composition T ⁢ u of a vector-valued function u ∈ W 1 , r ⁢ Ω ; R M by a globally Lipschitz-continuous, piecewise C 1 function T . We also prove that the map u → T ⁢ u is continuous from W 1 , r ⁢ Ω ; R M into W 1 , r ⁢ Ω for the strong topologies of these spaces.

How to cite

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Murat, François, and Trombetti, Cristina. "A chain rule formula for the composition of a vector-valued function by a piecewise smooth function." Bollettino dell'Unione Matematica Italiana 6-B.3 (2003): 581-595. <http://eudml.org/doc/195023>.

@article{Murat2003,
abstract = {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.},
author = {Murat, François, Trombetti, Cristina},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {581-595},
publisher = {Unione Matematica Italiana},
title = {A chain rule formula for the composition of a vector-valued function by a piecewise smooth function},
url = {http://eudml.org/doc/195023},
volume = {6-B},
year = {2003},
}

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 dell'Unione Matematica Italiana
DA - 2003/10//
PB - Unione Matematica Italiana
VL - 6-B
IS - 3
SP - 581
EP - 595
AB - 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.
LA - eng
UR - http://eudml.org/doc/195023
ER -

References

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  1. AMBROSIO, L.- DAL MASO, G., A general chain rule for distributional derivatives, Proc. Amer. Math. Soc., 108 (1990), 691-702. Zbl0685.49027MR969514
  2. BOCCARDO, L.- MURAT, F., Remarques sur l'homogénéisation de certains problèmes quasi-linéaires, Portugaliæ Math., 41 (1982), 535-562. Zbl0524.35042MR766874
  3. BOUCHUT, F., Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rat. Mech. Anal., 157 (2001), 75-90. Zbl0979.35032MR1822415
  4. KINDERLHERER, D.- STAMPACCHIA, G., An introduction to variational inequalities and their applications, Academic Press, New York (1980). Zbl0457.35001MR567696
  5. LANDES, R., On the existence of weak solutions of perturbated systems with critical growth, J. reine angew. Math., 393 (1989), 21-38. Zbl0664.35027MR972359
  6. MARCUS, M.- MIZEL, V. J., Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rat. Mech. Anal., 45 (1972), 294-320. Zbl0236.46033MR338765
  7. MARCUS, M.- MIZEL, V. J., Nemitsky operators on Sobolev spaces, Arch. Rat. Mech. Anal., 51 (1973), 347-370. Zbl0266.46029MR348480
  8. STAMPACCHIA, G., Equations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures, 16, Les Presses de l'Université de Montréal, Montréal (1966). Zbl0151.15501MR251373

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