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Shape optimisation problems governed by nonlinear state equations

Published online by Cambridge University Press:  14 November 2011

Dorin Bucur  and
Paola Trebeschi
Dorin Bucur
Affiliation:
CNRS-Equipe de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France, e-mail: bucur@math.univ-fcomte.fr
Paola Trebeschi
Affiliation:
Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy, e-mail: trebesch@dm.unipi.it
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Abstract

The purpose of this paper is to give a compactness-continuity result for the solution of a nonlinear Dirichlet problem in terms of its domain variation. The topology in the family of domains is given by the Hausdorff metric and continuity is obtained under capacity conditions. A generalisation of Sverak's result in iV-dimensions is deduced as a particular case.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 5 , 1998 , pp. 945 - 963
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Bagby, T.. Quasi topologies and rational approximation. J. Funct. Anal. 19 (1992), 581–97.Google Scholar
3Boccardo, L. and Murat, F.. Almost everywhere convergence of the gradients of solutions of elliptic and parabolic equations. Nonlinear Anal. 10 (1972), 259–68.Google Scholar
4Bucur, D. and Zolésio, J. P.. N-dimensional shape optimization under capacitary constraints. J. Differential Equations 123 (1995), 504–22.CrossRefGoogle Scholar
5Bucur, D. and Zolésio, J. P.. Wiener criterion and shape continuity for the Dirichlet problem. Boll. Un. Mat. Ital. (7) 1–B (1997), 757–71.Google Scholar
6Buttazzo, G. and Dal Maso, G.. Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991), 17–49.CrossRefGoogle Scholar
7Buttazzo, G. and Dal Maso, G.. An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993), 183–95.CrossRefGoogle Scholar
8Chenais, D.. On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), 189–289.CrossRefGoogle Scholar
9Dal Maso, G.. Γ-convergence and μ-capacities. Ann. Scuola Norm. Sup. Pisa 14 (1987), 423–64.Google Scholar
10Dal Maso, G. and Defranceschi, A.. Limits of nonlinear Dirichlet problems in varying domains. Manuscripta Math. 61 (1988), 251–78.CrossRefGoogle Scholar
11Dal Maso, G. and Mosco, U.. Wiener's criterion and Γ-convergence. Appl. Math. Optim. 15 (1987), 15–63.CrossRefGoogle Scholar
12Dal Maso, G. and Murat, F.. Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators (Preprint SISSA 33/96, to appear).Google Scholar
13Heinonen, J., Kilpelainen, T. and Martio, O.. Nonlinear potential theory of degenerate elliptic equations (Oxford: Clarendon Press, 1993).Google Scholar
14Henrot, A.. Continuity with respect to the domain for the Laplacian: a survey. Control Cybernet. 23 (1994), 427–43.Google Scholar
15Pironneau, O.. Optimal shape design for elliptic systems (Berlin: Springer, 1984).CrossRefGoogle Scholar
16Rauch, J. and Taylor, M.. Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18 (1975), 27–59.CrossRefGoogle Scholar
17Sverak, V.. On optimal shape design. J. Math. Pures Appl. 72 (1993), 537–51.Google Scholar
18Ziemer, W.. Weakly differentiable functions (Berlin: Springer, 1989).CrossRefGoogle Scholar