×

An optimal partition problem for eigenvalues. (English) Zbl 1123.65060

For a bounded, smooth domain \(\Omega\) in \(\mathbb R^n\), the authors study the problem of finding \(m\) disjoint subsets \(\Omega_j\) such that \(\overline \Omega = \bigcup \overline \Omega_j\) and the sum \(\sum \lambda_1( \Omega_j )\) is minimized. Here, \(\lambda_1( \Omega_j )\) denotes the first eigenvalue of the Laplacian on \(\Omega_j\) with Dirichlet boundary data. It is shown that such a minimizing partition exists and that the interfaces are regular.

MSC:

65K10 Numerical optimization and variational techniques
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bucur, P.; Buttazzo, G.; Henrot, A., Existence results for some optimal partition problems, Adv. Math. Sci. Appl. Tokyo, 8, 2, 571-579 (1998) · Zbl 0915.49006
[2] Buttazzo, G.; Dal Maso, G., Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim., 23, 17-49 (1991) · Zbl 0762.49017 · doi:10.1007/BF01442391
[3] Buttazzo, G.; Dal Maso, G., An existence result for a class of shape problems, Arch. Ratio Mech. Anal., 122, 183-195 (1993) · Zbl 0811.49028 · doi:10.1007/BF00378167
[4] Bucur, D.; Zolesio, J. P., N-dimensional shape optimization under capacitary constraints, J. Diff. Eqs., 123, 504-522 (1995) · Zbl 0847.49029 · doi:10.1006/jdeq.1995.1171
[5] Chavel, I. (1984). Eigenvalues in Riemannian Geometry, Pure Appl. Math. Vol. 115, Academic Press, Inc., Orlando, F1. 362 pp. · Zbl 0551.53001
[6] Cafferrelli, L. A., and Lin, F. H. Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. preprint. · Zbl 1194.35138
[7] Chang, S. M.; Lin, C. S.; Lin, T. C.; Lin, W. W., Segregated nodal domains of two-dimensional multi-spices Bose-Einstein condensates, Phys. D., 196, 3-4, 341-361 (2004) · Zbl 1098.82602 · doi:10.1016/j.physd.2004.06.002
[8] Garofalo, N.; Lin, F. H., Monotonicity properties of variational integrals, Ap-weights, and unique continuation, Indiana Univ. Math. J., 35, 245-267 (1986) · Zbl 0678.35015 · doi:10.1512/iumj.1986.35.35015
[9] Lin, F. H., Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44, 287-306 (1991) · Zbl 0734.58045 · doi:10.1002/cpa.3160440303
[10] Lin, F. H., and Yang, X. P. (2002). Geometric Measure Theory, An Introduction, Adv. Math. Vol. I, Int’l. Press, Boston. · Zbl 1074.49011
[11] Morey, C. B., Multiple Integrals in The Calculus of Variations (1966), New York: Springer-Verlag, New York · Zbl 0142.38701
[12] Sverak, V., On optimal shape design, J. Math. Pures Appl., 72, 537-551 (1993) · Zbl 0849.49021
[13] Ziemer, W., Weakly Differentiable Functions (1989), Berlin: Springer-Verlag, Berlin · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.