On the existence of variations, possibly with pointwise gradient constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 331-342.

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.

DOI : 10.1051/cocv:2007017
Classification : 49K10, 35F30
Mots-clés : variations, differential inclusions, necessary conditions 
@article{COCV_2007__13_2_331_0,
     author = {Bertone, Simone and Cellina, Arrigo},
     title = {On the existence of variations, possibly with pointwise gradient constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {331--342},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {2},
     year = {2007},
     doi = {10.1051/cocv:2007017},
     mrnumber = {2306639},
     zbl = {1124.49012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2007017/}
}
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
PY  - 2007
SP  - 331
EP  - 342
VL  - 13
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2007017/
DO  - 10.1051/cocv:2007017
LA  - en
ID  - COCV_2007__13_2_331_0
ER  - 
%0 Journal Article
%A Bertone, Simone
%A Cellina, Arrigo
%T On the existence of variations, possibly with pointwise gradient constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 331-342
%V 13
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2007017/
%R 10.1051/cocv:2007017
%G en
%F COCV_2007__13_2_331_0
Bertone, Simone; Cellina, Arrigo. On the existence of variations, possibly with pointwise gradient constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 331-342. doi : 10.1051/cocv:2007017. http://www.numdam.org/articles/10.1051/cocv:2007017/

[1] V.I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, Heidelber, Berlin. | MR | Zbl

[2] H. Brezis, Analyse fonctionnelle, théorie et applications. Masson, Paris (1983). | MR | Zbl

[3] A. Cellina, On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993) 337-341. | Zbl

[4] A. Cellina, On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. | Zbl

[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. | Zbl

[6] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, Rhode Island (1998). | MR | Zbl

[7] L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999) 653. | MR | Zbl

[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). | MR | Zbl

[9] M. Tsuji, On Lindelöf's theorem in the theory of differential equations. Jap. J. Math. 16 (1939) 149-161. | JFM

Cité par Sources :