Abstract variational problems with volume constraints

Marc Oliver Rieger

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 1, page 84-98
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Rieger, Marc Oliver. "Abstract variational problems with volume constraints." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 84-98. <http://eudml.org/doc/90722>.

@article{Rieger2010,
abstract = { 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. },
author = {Rieger, Marc Oliver},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Level set constraints; nonconvex problems; minimization.; level set constraints; minimization},
language = {eng},
month = {3},
number = {1},
pages = {84-98},
publisher = {EDP Sciences},
title = {Abstract variational problems with volume constraints},
url = {http://eudml.org/doc/90722},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Rieger, Marc Oliver
TI - Abstract variational problems with volume constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 1
SP - 84
EP - 98
AB - 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.
LA - eng
KW - Level set constraints; nonconvex problems; minimization.; level set constraints; minimization
UR - http://eudml.org/doc/90722
ER -

References

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  1. L. Ambrosio, I. Fonseca, P. Marcellini and L. Tartar, On a volume-constrained variational problem. Arch. Rational Mech. Anal.149 (1999) 23-47.  
  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.  
  3. M. Morini and M.O. Rieger, On a volume constrained variational problem with lower order terms. Appl. Math. Optim (to appear).  
  4. S. Mosconi and P. Tilli, Variational problems with several volume constraints on the level sets. Calc. Var. Partial Differential Equations14 (2002) 233-247.  
  5. P. Tilli, On a constrained variational problem with an arbitrary number of free boundaries. Interfaces Free Bound.2 (2000) 201-212.  

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