no. 1
Abstract variational problems with volume constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 84-98

Existence results for a class of one-dimensional abstract variational problems with volume constraints are established. The main assumptions on their energy are additivity, translation invariance and solvability of a transition problem. These general results yield existence results for nonconvex problems. A counterexample shows that a naive extension to higher dimensional situations in general fails.

DOI : 10.1051/cocv:2003034
Classification : 49
Keywords: level set constraints, nonconvex problems, minimization 
@article{COCV_2004__10_1_84_0,
     author = {Rieger, Marc Oliver},
     title = {Abstract variational problems with volume constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {84--98},
     year = {2004},
     publisher = {EDP Sciences},
     volume = {10},
     number = {1},
     doi = {10.1051/cocv:2003034},
     mrnumber = {2084256},
     zbl = {1068.49002},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2003034/}
}
TY  - JOUR
AU  - Rieger, Marc Oliver
TI  - Abstract variational problems with volume constraints
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2004
SP  - 84
EP  - 98
VL  - 10
IS  - 1
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2003034/
DO  - 10.1051/cocv:2003034
LA  - en
ID  - COCV_2004__10_1_84_0
ER  - 
%0 Journal Article
%A Rieger, Marc Oliver
%T Abstract variational problems with volume constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2004
%P 84-98
%V 10
%N 1
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2003034/
%R 10.1051/cocv:2003034
%G en
%F COCV_2004__10_1_84_0
Rieger, Marc Oliver. Abstract variational problems with volume constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 84-98. doi: 10.1051/cocv:2003034

[1] L. Ambrosio, I. Fonseca, P. Marcellini and L. Tartar, On a volume-constrained variational problem. Arch. Rational Mech. Anal. 149 (1999) 23-47. | Zbl | MR

[2] M.E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluids and immissible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6 (1996) 815-831. | Zbl | MR

[3] M. Morini and M.O. Rieger, On a volume constrained variational problem with lower order terms. Appl. Math. Optim (to appear). | Zbl | MR

[4] S. Mosconi and P. Tilli, Variational problems with several volume constraints on the level sets. Calc. Var. Partial Differential Equations 14 (2002) 233-247. | Zbl | MR

[5] P. Tilli, On a constrained variational problem with an arbitrary number of free boundaries. Interfaces Free Bound. 2 (2000) 201-212. | Zbl | MR

Cité par Sources :