Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-06T13:58:07.402Z Has data issue: false hasContentIssue false
  • English
  • Français

Local minimizers of functionals with multiple volume constraints

Published online by Cambridge University Press:  07 February 2008

Édouard Oudet  and
Marc Oliver Rieger
Édouard Oudet
Affiliation:
Université de Savoie, France; Edouard.Oudet@univ-savoie.fr
Marc Oliver Rieger
Affiliation:
University of Zürich, ISB, Switzerland; rieger@isb.uzh.ch
Get access

Abstract

We study variational problems with volume constraints, i.e., with level sets of prescribed measure. We introduce a numerical method to approximate local minimizers and illustrate it with some two-dimensional examples. We demonstrate numerically nonexistence results which had been obtained analytically in previous work. Moreover, we show the existence of discontinuous dependence of global minimizers from the data by using a Γ-limit argument and illustrate this with numerical computations. Finally we construct explicitly local and global minimizers for problems with two volume constraints.

Keywords

Volume constrained problemsnumerical simulationslevel set methodlocal minima
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 780 - 794
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E., Fonseca, I. and Mingione, G., Existence and regularity for mixtures of micromagnetic materials. Proc. Royal Soc. London Sect. A 462 (2006) 2225–2244. CrossRef
Aguilera, N., Alt, H.W. and Caffarelli, L.A., An optimization problem with volume constraint. SIAM J. Control Optim. 24 (1986) 191–198. CrossRef
Allaire, G., Jouve, F. and Toader, A.M., A level-set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125–1130. CrossRef
Ambrosio, L., Fonseca, I., Marcellini, P. and Tartar, L., On a volume-constrained variational problem. Arch. Ration. Mech. Anal. 149 (1999) 23–47. CrossRef
A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002).
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems 65. Birkhäuser Boston (2005).
G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser Boston Inc., Boston, MA (1993).
Gurtin, M.E., Polignone, D. and Vinals, J., Two-phase binary fluids and immissible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6 (1996) 815–831. CrossRef
A. Henrot and M. Pierre, Variation et optimisation de formes, une analyse géométrique 48. Springer-Verlag, Paris (2005).
Morini, M. and Rieger, M.O., On a volume constrained variational problem with lower order terms. Appl. Math. Optim. 48 (2003) 21–38. CrossRef
Mosconi, S. and Tilli, P., Variational problems with several volume constraints on the level sets. Calc. Var. Part. Diff. Equ. 14 (2002) 233–247. CrossRef
Osher, S. and Santosa, F., Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys 171 (2001) 272–288. CrossRef
Osher, S. and Sethian, J.A., Front propagation with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12–49. CrossRef
Oudet, E., Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: COCV 10 (2004) 315–335. CrossRef
Rieger, M.O., Abstract variational problems with volume constraints. ESAIM: COCV 10 (2004) 84–98. CrossRef
M.O. Rieger, Higher dimensional variational problems with volume constraints – existence results and Γ-convergence. Interfaces and Free Boundaries (to appear).
J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics 10. Springer (1992).
P. Vergilius Maro, Aeneidum I. (29–19 BC).