Local minimizers of functionals with multiple volume constraints

Oudet, Édouard; Rieger, Marc Oliver

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 780-794.

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 $\Gamma$-limit argument and illustrate this with numerical computations. Finally we construct explicitly local and global minimizers for problems with two volume constraints.

Classification : 49J, 65K10
Mots clés : volume constrained problems, numerical simulations, level set method, local minima

[1] E. Acerbi, I. Fonseca and G. Mingione, Existence and regularity for mixtures of micromagnetic materials. Proc. Royal Soc. London Sect. A 462 (2006) 2225-2244. | MR | Zbl

[2] N. Aguilera, H.W. Alt and L.A. Caffarelli, An optimization problem with volume constraint. SIAM J. Control Optim. 24 (1986) 191-198. | MR | Zbl

[3] G. Allaire, F. Jouve and A.M. Toader, A level-set method for shape optimization. C. R. Acad. Sci. Paris 334 (2002) 1125-1130. | MR | Zbl

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

[5] A. Braides, $\Gamma$-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). | MR | Zbl

[6] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems 65. Birkhäuser Boston (2005). | MR | Zbl

[7] G. Dal Maso, An Introduction to $\Gamma$-Convergence. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl

[8] 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. | MR | Zbl

[9] A. Henrot and M. Pierre, Variation et optimisation de formes, une analyse géométrique 48. Springer-Verlag, Paris (2005). | MR | Zbl

[10] M. Morini and M.O. Rieger, On a volume constrained variational problem with lower order terms. Appl. Math. Optim. 48 (2003) 21-38. | MR | Zbl

[11] S. Mosconi and P. Tilli, Variational problems with several volume constraints on the level sets. Calc. Var. Part. Diff. Equ. 14 (2002) 233-247. | MR | Zbl

[12] S. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J. Comput. Phys 171 (2001) 272-288. | MR | Zbl

[13] S. Osher and J.A. Sethian, Front propagation with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | MR | Zbl

[14] E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: COCV 10 (2004) 315-335. | EuDML | Numdam | MR | Zbl

[15] M.O. Rieger, Abstract variational problems with volume constraints. ESAIM: COCV 10 (2004) 84-98. | EuDML | Numdam | MR | Zbl

[16] M.O. Rieger, Higher dimensional variational problems with volume constraints - existence results and $\Gamma$-convergence. Interfaces and Free Boundaries (to appear). | MR | Zbl

[17] J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics 10. Springer (1992). | MR | Zbl

[18] P. Vergilius Maro, Aeneidum I. (29-19 BC).