June  2013, 8(2): 573-589. doi: 10.3934/nhm.2013.8.573

Asymptotics of an optimal compliance-network problem

1. 

International Centre for Theoretical Physics, Strada Costiera,11, I - 34151 Trieste, Italy

Received  October 2012 Revised  March 2013 Published  May 2013

We consider the problem of the optimal location of a Dirichlet region in a $d$-dimensional domain $\Omega$ subjected to a given force $f$ in order to minimize the $p$-compliance of the configuration. We look for the optimal region among the class of all closed connected sets of assigned length $l.$ Then we let the length $l$ tend to infinity and we look at the $\Gamma$-limit of a suitable rescaled functional, from which we get information of the asymptotic distribution of the optimal region. We also study the case where the Dirichlet region is a discrete set of finite cardinality.
Citation: Al-hassem Nayam. Asymptotics of an optimal compliance-network problem. Networks and Heterogeneous Media, 2013, 8 (2) : 573-589. doi: 10.3934/nhm.2013.8.573
References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York 2002.

[2]

G. Bouchité, C.Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris Sér. I, 335 (2002), 1-6.

[3]

D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equation and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005.

[4]

D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006.

[5]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Networks and Heterogeneous Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761.

[6]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optimization and Calculus of Variations, 12 (2006), 752-769. doi: 10.1051/cocv:2006020.

[7]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[8]

G. Dal Maso and F. Murat, Asymptotic behavior and corrector for the Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Sup. Pisa Cl. Sci Ser. (4), 24 (1997), 239-290.

[9]

V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.

[10]

S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158.

[11]

V. Šverak, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551.

show all references

References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York 2002.

[2]

G. Bouchité, C.Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris Sér. I, 335 (2002), 1-6.

[3]

D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equation and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005.

[4]

D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006.

[5]

G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Networks and Heterogeneous Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761.

[6]

G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optimization and Calculus of Variations, 12 (2006), 752-769. doi: 10.1051/cocv:2006020.

[7]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[8]

G. Dal Maso and F. Murat, Asymptotic behavior and corrector for the Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Sup. Pisa Cl. Sci Ser. (4), 24 (1997), 239-290.

[9]

V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.

[10]

S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158.

[11]

V. Šverak, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551.

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