On the existence of variations, possibly with pointwise gradient constraints

Simone Bertone; Arrigo Cellina

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 2, page 331-342
  • ISSN: 1292-8119

Abstract

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We propose a necessary and sufficient condition about the existence of variations, i.e., of non trivial solutions η ∈ W 0 1 , ∞ ( Ω ) to the differential inclusion ∇ η ( x ) ∈ - ∇ u ( x ) + 𝐃 .

How to cite

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Bertone, Simone, and Cellina, Arrigo. "On the existence of variations, possibly with pointwise gradient constraints." ESAIM: Control, Optimisation and Calculus of Variations 13.2 (2007): 331-342. <http://eudml.org/doc/249933>.

@article{Bertone2007,
abstract = { We propose a necessary and sufficient condition about the existence of variations, i.e., of non trivial solutions $\eta\in W^\{1,\infty\}_0(\Omega)$ to the differential inclusion $\nabla\eta(x)\in-\nabla u(x)+\{\bf D\}$. },
author = {Bertone, Simone, Cellina, Arrigo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Variations; differential inclusions; necessary conditions; Differential inclusion},
language = {eng},
month = {5},
number = {2},
pages = {331-342},
publisher = {EDP Sciences},
title = {On the existence of variations, possibly with pointwise gradient constraints},
url = {http://eudml.org/doc/249933},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Bertone, Simone
AU - Cellina, Arrigo
TI - On the existence of variations, possibly with pointwise gradient constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/5//
PB - EDP Sciences
VL - 13
IS - 2
SP - 331
EP - 342
AB - We propose a necessary and sufficient condition about the existence of variations, i.e., of non trivial solutions $\eta\in W^{1,\infty}_0(\Omega)$ to the differential inclusion $\nabla\eta(x)\in-\nabla u(x)+{\bf D}$.
LA - eng
KW - Variations; differential inclusions; necessary conditions; Differential inclusion
UR - http://eudml.org/doc/249933
ER -

References

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  1. V.I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics60, Springer-Verlag, New York, Heidelber, Berlin.  
  2. H. Brezis, Analyse fonctionnelle, théorie et applications. Masson, Paris (1983).  
  3. A. Cellina, On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993) 337–341.  
  4. A. Cellina, On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993) 343–347.  
  5. A. Cellina and S. Perrotta, On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient. SIAM J. Control Optim.36 (1998) 1987–1998.  
  6. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics19, American Mathematical Society, Providence, Rhode Island (1998).  
  7. L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc.137 (1999) 653.  
  8. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992).  
  9. M. Tsuji, On Lindelöf's theorem in the theory of differential equations. Jap. J. Math.16 (1939) 149–161.  

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