Constant in two-dimensional p-compliance-network problem

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March 2014, 9(1): 161-168. doi: 10.3934/nhm.2014.9.161

Constant in two-dimensional $p$-compliance-network problem

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

Received April 2013 Revised October 2013 Published April 2014

Abstract
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We consider the problem of the minimization of the $p$-compliance functional where the control variables $\Sigma$ are taking among closed connected one-dimensional sets. We prove some estimate from below of the $p$-compliance functional in terms of the one-dimensional Hausdorff measure of $\Sigma$ and compute the value of a constant $\theta(p)$ appearing usually in $\Gamma$-limit functional of the rescaled $p$-compliance functional.

Keywords: shape optimization., $\Gamma$-convergence.

Mathematics Subject Classification: Primary: 49J45; Secondary: 49Q10, 74P0.

Citation: Al-hassem Nayam. Constant in two-dimensional $p$-compliance-network problem. Networks & Heterogeneous Media, 2014, 9 (1) : 161-168. doi: 10.3934/nhm.2014.9.161

References:

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References:

[1]	D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations,, Proc. Roy. Soc. Edinburgh - A, 128 (1998), 945. doi: 10.1017/S0308210500030006. Google Scholar
[2]	G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks,, Networks and Heterogeneous Media, 2 (2007), 761. doi: 10.3934/nhm.2007.2.761. Google Scholar
[3]	G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem,, ESAIM Control Optimization and Calculus of Variations, 12 (2006), 752. doi: 10.1051/cocv:2006020. Google Scholar
[4]	G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993). doi: 10.1007/978-1-4612-0327-8. Google Scholar
[5]	S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem,, J. Conv. Anal., 12 (2005), 145. Google Scholar
[6]	P. Tilli, Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length,, Networks and Heterogeneous Media, 7 (2012), 127. doi: 10.3934/nhm.2012.7.127. Google Scholar
[7]	W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,, Graduate texts in Mathematics, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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